Mechanics of Indentation for an Elastic Half-Space by Punches
SANDIP SAHA1, VIKASH KUMAR2, AWANI BHUSHAN3, APURBA NARAYAN DAS4,*
1Division of Mathematics, School of Advance Sciences,
Vellore Institute of Technology Chennai,
Tamilnadu-600127,
INDIA
2Department of Mathematics,
Sarala Birla University,
Ranchi,
INDIA
3School of Mechanical Engineering,
Vellore Institute of Technology,
Chennai,
INDIA
4*Department of Mathematics,
Alipurduar University, Alipurduar,
West Bengal-736121,
INDIA
Abstract: - The dynamic and static problems of finding stress components under four moving punches (
, located close to each other over an elastic half-plane (, are solved. Employing
the Fourier integral transform, the problem is reduced to a set of integral equations in both cases. Using the
Hilbert transform technique, the integral equations are solved to obtain the stress and displacement
components. Finally, exact expressions for the stress components under the punches and the normal
displacement component in the region outside the punches have been derived. Numerical results showing the
variations in stress intensity factors (SIF) at the punch ends, and the absolute value of torque applied over the
contact regions with different values of the parameters used in the problems have been presented in the form
of graphs.
Key-Words: - Punch, Fourier transform, Integral equations, Hilbert transform technique, Stress intensity
factor, Elastic half-space, Moving punches.
Received: April 14, 2023. Revised: October 27, 2023. Accepted: November 25, 2023. Published: December 31, 2023.
1 Introduction
Contact problems are common in engineering and
material sciences. Several punch problems in
elastodynamics have been discussed in detail in the
books by [1], [2]. [3], solved the moving punch
problems with the aid of the complex variable
method. [4], considered the in-plane problem of
indentation of an elastic layer over a rigid base by
moving punches. [5], considered the problems of
anti-plane indentation of an elastic layer by a pair of
moving punches. [6], solved the same problems
with two pairs of moving punches, [7], [8].
Structures formed as a solid foundation inserted
under the ground, are examples of large-scale
indentation. A problem of indentation of an elastic
half-plane by a wedge shaped punch, taking into
account the frictional and tangential-displacements
effects has been solved by [9]. [10], [11], [12], [13],
solved a few problems on symmetric, non-
symmetric, and frictionless indentations employing
the method of homogeneous function. [14], have
reviewed recent works on the inclusions of infinite,
and semi-infinite spaces under different forms of
loading. Different methods of solving the problems
of one or more inclusions have been presented in
their work. [15], solved an axisymmetric problem of
unilateral frictionless indentation of a semi-infinite
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elastic medium. [16], studied the problem of moving
punch over a layer under plane strain conditions.
The problem has been solved numerically after
converting it into a Cauchy-type singular integral
equation. [17], solved the case of indentation of an
orthotropic layer on an isotropic half-plane by a
steadily moving rigid cylindrical punch with the aid
of the Fourier integral transform technique, Galilean
transformation, and Gauss-Jacobi integral formula.
[18], have solved the problem of a moving punch
associated with the normal component of
displacement as an even degree polynomial. They
studied the effect of the degree of the polynomial
function on SIF at the punch end and on the torque
over the contact region. The residual stresses are
identified by instrumented elliptical indentation, and
inverse analysis by [19]. The simulation for JKR-
type adhesive contact of rough elliptical punches is
done by [20], using Boundary Element Method
(BEM), and the Fast Fourier Transform. The
problem of adhesion of a cylindrical punch with
varied elastic properties is studied by [21]. [22],
studied the Sevostianov-Kachanov approximation
for incremental compliance of non-elliptical
contacts. [23], has solved the Boussinesq and
Cattaneo problems for an ellipsoidal power-law
indenter analytically, [24]. The problem of
axisymmetric contact of two different power-law
graded elastic bodies has been solved by [25], after
reducing it to an integral equation with two different
kernels. Small contacts are also occurring in
different engineering fields. Such cases of
penetration are taken into account for studying the
distribution of stress under the indenter. These
studies have applications in designing geotechnical
and footing engineering and in indentation tests for
characterizing matters, [26], [27], [28], [29], [30],
[31], [32].
In this paper, the integral transform technique
has been utilized to solve both dynamic and static
problems for finding stress components under four
punches ( , located close
to each other on an elastic half-space (.The
Fourier integral transformation has been employed
to transform the problem into a set of five integral
equations. The use of the Hilbert transform
technique has been made for solving the integral
equations, and the stress component under the
punches and the normal displacement component in
the region outside the punches have been derived.
Finally, SIF at the punch ends and torque over the
contact regions are calculated; and the variations in
those with velocity of punch for different values of
the contact region of the inner pair of the punches
are presented graphically.
2 Formulation and Solution of
Problem I
We consider an isotropic (Figure 1), homogeneous,
and semi-infinite medium given by, which is
stress-free, and no displacement is prescribed on any
part of the boundary. Thus, the initial
conditions are zero. Four punches, located at
are assumed to be
moving at a constant speed,along positive
direction of the axis. The equations of motion
(neglecting body force) in terms of displacements
are
,
(1a,b)
wheredenote the displacement components
along theand axes respectively, are Lame’s
constants, and denotes the partial derivative with
respect to . We introduce the Galilean
transformation
and
withand as the moving coordinate system as
shown in Figure 1. Therefore, the deformation about
the y–axis will remain symmetric throughout the
motion.
Following the detailed analysis of the method,
[6], [7] and using the boundary conditions (due to
symmetry about )
for
for
for, (2a-c)
we obtain the following integral equations in
for (3a,b)
for (4a-c)
where is a constant and
.
To solve these equations, we assume
(5)
Next, using (5) in the expressions given by (4a-
c), we note that the expression of is
independent of the choice of the unknown functions
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. Using (5) in the equations given by
(3a,b) , we get
for (6)
Fig. 1: Geometry and coordinate system
On differentiation with respect to , this gives:
for
(7)
Using the method of solutions of the above
integral equations, [6], [7], we get:
,
. (8a,b)
Multiplying the equation (6) by
and
integrating taking limits to and
multiplying the same equation by
,
and integrating taking limits to we get
a system of linear equations involving.
Solving those, we get:
,
(9a,b)
where
(10a-d)
with
,
,
,
and
. (11a-d)
The normal component of stress in the plane of
the punches and just below those are given as:
. (12a,b)
with
and
and the normal displacement component
outside the contact regions can now shown to be
given by:
for(13a-c)
It is to be mentioned that the stress component
depends on the velocity of the moving punch.
However, in the plane of the punches, the normal
displacement component is independent of that.
Further, we note from equation (13) that the normal
displacement component decreases gradually as
tends to infinity.
The SIF at the ends of the punches is defined
by:
,
and using the expressions (12a,b) those are found
as:
,
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,
,
and
. (14a-d)
The torque applied over the contact regions are
given by:
and
and using (8a,b) and (12a,b) in the above
expressions, we obtain:
,
, (15a,b)
where
and
are elliptic integrals of first and third kind
respectively, and
.
2.1 Problem II
In this section, we consider a semi-infinite
homogeneous, isotropic material with punches
located at The
equations of equilibrium (neglecting body force), in
terms of displacements are:
(16a,b)
where , have already been defined earlier.
Using the same technique as adopted in the
problem 1, and using the boundary conditions (on
account of symmetry about )
for
for
for (17a-e)
we obtain the following integral equations in
for (18a,b)
for (19a-c)
where is a constant.
It is to be mentioned that the above integral
equations cannot be obtained using the
corresponding expressions of the dynamic problem
given by the equations (3a,b) by setting.
Now, employing the same method as adopted in the
problem I, one can easily obtain
, (20a,b)
where are same as given by (8a,b)
with the exception that are to be replaced by
,which satisfy (9a,b), when
is
replaced by .
The SIF at the ends of the punches are found as:
,
,
,
and
.
(21a-d)
Using the results:
and
the torque applied over the contact regions are found
as:
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,
(22a,b)
3 Numerical Discussions
In this section, numerical results for problem I for
the values of the parameters associated with the
problem have been presented graphically.
Computations of SIF and torque applied over the
contact regions have been done taking and
, i.e., taking the variations in
the position of inner edge of the first pair of punches
only. As the velocity of the punch is less than
Rayleigh wave velocity, we take the value of
From Figures 2, it is clear that the value of
the SIF at the inner edge of the first pair of punches
decreases, while the same at all other edges
increases with the increase in the values of
. In
other words, as the length of the contact region of
the inner pair of punches reduces keeping length of
the other punches fixed, the value of the SIF
decreases. From these graphs, we note that the
values of the SIF gradually decrease as
increases
and tend to zero as
tends to 0.9194, as expected.
Variations in absolute values of the torque applied
over the contact regions of both pairs of punches
have been presented in Figure 3. It is seen from the
figures that variations in absolute value of the torque
over both the contact regions are of similar
character, i.e., the value of absolute value of the
torque over both the contact regions decreases with
the increase in the values of
and tends to 0 as
tends to 0.9194. Nevertheless, the magnitude of the
absolute value of the torque over the outer pair of
punches is significantly higher than that over the
inner pair of punches.
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Fig. 2: Variations of S.I.F. with velocity of the punches
Fig. 3: Variations of torque with velocity of the punches
4 Conclusion
In this work, both the dynamic and static problems
of finding stress components under four punches,
located closely to each other over an elastic half
space, and moving steadily in a fixed direction are
solved. The integral transform technique has been
employed to study the behavior of SIF at the punch
ends, and the absolute value of the torque over the
contact regions with the variation in the parameters
involved in the problem. As the shape of an indenter
may vary, and the rigidity of the semi-infinite
medium over which the indenter acts is not always
uniform, it is reasonable to assume the normal
component of the displacement along the contact
regions as a function. However, to avoid complexity
in mathematical calculations, we have assumed that
the frictions less indenters are flat. The effect of the
variation in the velocity of the punches on the SIF at
the punch ends, and on the absolute value of the
torque over the contact regions have been studied.
How the reduction in length of the inner pair of
punches, keeping their outer edge fixed, affects the
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SIF at the punch ends and the torque over the
contact regions, have also been presented
graphically. Outcomes of this work are obtained
considering the indentation of a half-plane by four
rigid flat indenters, but this form of indentation
problem with an even number of punches (6,8,…)
can be solved. In those cases, the function (see
equation 5) is to be changed by adding more similar
integrals, and the method of computation will be
more complex.
Future scope of the study: After going through
this study, one can study the impact of the variations
in length of the contact regions of the outer pair of
punches, by considering the variations in the values
of
or
. The effect of alteration of the distance
between the pair of punches, by considering the
variations in the values of
or
, on the SIF and
absolute value of the torque can also be studied.
This work also suggests that a similar problem of
indentations can be extended with any even number
of punches, and can also be solved with different
shapes of indenter instead of the flat one.
Acknowledgment:
Authors are grateful to the referees for their valuable
comments on the earlier version of the work.
References:
[1] L. A. Galian, Contact problems in elasticity
theory, Rayleigh: North Carolina Stage
College, 1961.
[2] G. M. L. Gladwell, Contact problems in the
classical theory of elasticity, Sijth, and
Noordhooff, The Netherlands, 1980.
[3] C. Sve& L. M. Keer, Indentation of elastic
layer by moving punches. I. J. Solid Structure,
5, 795-816, 1969.
[4] R. J. Tait&T. B. Modde, Complex variable
method and closed form solutions to dynamic
crack and punch problem in the classical
theory of elasticity, I. J. Engineering Science,
19, 221-229, 1981.
[5] B. M. Singh & R. S. Dhaliwal, Closed form
solutions to dynamic problems by integral
transformation method, Z. Angew. Math.
Mech, 64, 31-34, 1984.
[6] A. N. Das, Indentation of an elastic layer by
four moving punches, Eng. Frac. Mech, 40(5),
815-823, 1993.
[7] A. N. Das & M. L. Ghosh, Four Coplaner
Grifth Cracks in an elastic medium, Eng. Frac.
Mech, 43(6), 941-955, 1992.
[8] J. C. Cooke, The solution of some integral
equations and their connections with dual
integral equations and trigonometric series,
Glas. Math. J, 11, 9-20, 1970.
[9] H.G.Georgiadis, L.M.Brock, A.P.Rigatos,
Dynamic indentation of an elastic half-plane
by a rigid wedge: frictional and tangential-
displacement effects. Int. J. Solids
Structures,32(23),3435-3450,1995.
[10] L. M. Brock, Symmetrical frictionless
indentation over a uniformly expanding
contact region-I. Basic analysis. Int. J. Engng
Sci. 14, 191-199, (1976).
[11] L. M. Brock, Symmetrical frictionless
indentation over a uniformly expanding
contact region-lI. Perfect adhesion. Int. J.
Engng Sci. 15, 147-155, (1977).
[12] L. M. Brock, Dynamic analysis of non-
symmetric problems for frictionless
indentation and plane crack extension. J.
Elasticity 8, 273-283. (1978).
[13] L. M. Brock, Exact transient results for pure
and grazing indentation with friction, J.
Elasticity 33, 119-143. (1993).
[14] K. Zhou, J. HsinHoh, X.Wang, L. M. Keer,
H.L.JohnPeng, S. Bin,Q. Jem Wang, A review
of recent works on inclusions. Mech.
Materials, 60, 144-158,2013.
[15] I Ivan Argatov, JFederico Sabina. Small-scale
indentation of an elastic coated half-space:
The effect of compliant substrate. Int.J.Engng.
Sci., 104, 87-96,2016.
[16] I.Comez, K.B.Yilmaz, M.A.Guler,
B,Yildirim. Frictionless contact problem
between a rigid moving punch and a
homogeneous layer resting on a Winkler
foundation.J.Struc. Engng. And
Appl.Mech.,2(2) 75-87,2019.
[17] I.Comez, M.A.Guler. On contact problem of a
moving rigid cylindrical punch sliding over an
orthotropic layer bonded to an isotropic half
plane. Math and Mech. of solids. 25(10)1924-
1942, 2020
[18] S. Saha, V. Kumar, A.N.Das. An elastic half-
space with a moving punch. WEAS Tran. on
Appl. and Theo. Mech., 16,245-249,2021.
[19] Vladimir Buljak, Giulio Maier, Identification
of residual stresses by instrumented elliptical
indentation and inverse analysis, Mechanics
Research Communications, Volume 41, 2012,
Pages 21-29,
[20] Qiang Li, JosefineWilhayn, Iakov A.
Lyashenko, Valentin L. Popov, Adhesive
contacts of rough elliptical punches,
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.29
Sandip Saha, Vikash Kumar,
Awani Bhushan, Apurba Narayan Das
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Volume 18, 2023
Mechanics Research Communications,
Volume 122, 2022, 103880, ISSN 0093-6413,
[21] Attila Kossa, René Hensel, Robert M.
McMeeking, Adhesion of a cylindrical punch
with elastic properties that vary
radially,Mechanics Research
Communications, Volume 130, 2023,104-123.
[22] Argatov I.I. On the Sevostianov–Kachanov
approximation for the incremental
compliances of non-elliptical contacts. Math.
Mech. Solids. 2022.
[23] Willert E. Analytic contact solutions of the
Boussinesq and Cattaneo problems for an
ellipsoidal power-law
indenter. Meccanica. 2023;58:109–117.
[24] Barber J.R. Contact Mechanics. Springer
International Publishing AG; Cham,
Switzerland: 2018. 585.
[25] Antipov Y.A., Mkhitaryan S.M.
Axisymmetric contact of two different power-
law graded elastic bodies and an integral
equation with two Weber–Schafheitlin
kernels. Q. J. Mech. Appl.
Math. 2022;75:393–420.
[26] Heß M., Popov V.L. Method of
Dimensionality Reduction in Contact
Mechanics and Friction: A User’s Handbook.
II. Power-Law Graded Materials. Facta Univ.
Ser. Mech. Eng. 2016;14:251–268.
[27] Popov V.L., Heß M., Willert E. Handbook of
Contact Mechanics—Exact Solutions of
Axisymmetric Contact Problems. Springer;
Berlin/Heidelberg, Germany: 2019. 347p.
[28] Li Q., Wilhayn J., Lyashenko I.A., Popov
V.L. Adhesive contacts of rough elliptical
punches. Mech. Res. Commun. 2022; 122:
103880.
[29] Guduru R P., Detachment of a rigid solid from
an elastic wavy surface: Theory. J. Mech.
Phys. Solids, 2007; 55: pp. 445-472.
[30] Guduru R P. Bull C. Detachment of a rigid
solid from an elastic wavy surface:
Experiments. J. Mech. Phys. Solids, 2007; 55:
pp. 473-488.
[31] Wu J J. Numerical simulation of the adhesive
contact between a slightly wavy surface and a
half-space. J. Adhes. Sci. Technol. 2012; 26:
pp. 331-351.
[32] Li Q., Pohrt R., Popov V L., Adhesive
strength of contacts of rough spheres. Front.
Mech. Eng. 2019; 5: p. 7.
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