Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 14, N

o
 3, 2016, pp. 241 - 249 

DOI: 10.22190/FUME1603241L 

Original scientific paper 

INDENTATION OF FLAT-ENDED AND TAPERED INDENTERS 

WITH POLYGONAL CROSS-SECTIONS 

UDC 539.3 

Qiang Li, Valentin L. Popov 

Department of System Dynamics and the Physics of Friction, TU Berlin, Germany 

Abstract. Using the Boundary Element Method, we numerically study the indentation of 

prismatic and tapered indenters with polygonal cross-sections. The contact stiffness of 

punches with flat bases in the form of a triangle and a square as well as a number of 

higher-order polygons is determined. In particular, the classical results of King (1987) for 

indenters with triangle and square base shapes are revised and more precise numerical 

results are provided. For tapered indenters, the equivalent transformed profile used in the 

Method of Dimensionality Reduction (MDR) is determined. It is shown that the 

MDR-transformed profile of polygon-based indenters with power function side is given by the 

power function with the same power; it differs from the 3D profile only by a constant 

coefficient. These coefficients are listed in the paper for various types of indenters, in 

particular for pyramidal and paraboloid ones. The determined MDR-transformed profiles 

can be used for study of other contact problems such as tangential contact, normal contact 

with elastomers, and, in an approximate way, to adhesive contacts. 

Key Words: Indentation, Contact Stiffness, Polygonal Indenter, Boundary Element 

Method, MDR Transformed Profile 

1. INTRODUCTION 

Indentation test is a very common way of probing mechanical properties of materials such 

as hardness, contact stiffness, elastic modulus and strain-stress relation [1-3]. There is a variety 

of indenter geometries used in macro- and microindentation; the most popular are spherical and 

pyramidal indenters (e.g. for the Vickers hardness test and Brinell hardness test) [4]. The 

contact stiffness of indenters with regular geometries is also important for the foundation 

design [5]. The analytical solution for contact between a rigid cylindrical flat punch and an 

elastic half space was given by Galin in 1953 (English translation see [6]). His results were later 

published by Sneddon and, in this way, made public to the western world [7]. Based on this 

                                                           
Received September 10, 2016 / Accepted November 04, 2016 

Corresponding author: Qiang Li  

Institute of Mechanics, Berlin Institute of Technology, Strasse des 17. Juni 135, 10623 Berlin, Germany  
E-mail: qiang.li@tu-berlin.de 



242 Q. LI, V.L. POPOV 

solution, Oliver and Pharr proposed an analysis method to determine the hardness and elastic 

modulus from the load-displacement curves of indentation test [8]. General relations among 

contact stiffness, contact area, and elastic modulus during indentation have been analytically 

derived only for axisymmetric indenters. For a non-axisymmetrical geometry, a correction 

coefficient is needed [9], which can be still found only numerically.  

In this paper we numerically investigate the indentation of rigid bodies with various 

geometries: the flat-ended punches in Section 2 and tapered indenters in Section 3. In both 

cases we consider different polygonal bases including triangle and square. Note that the 

assumption of a rigid indenter is no restriction as the normal frictionless contact of two 

elastic bodies with elastic moduli  E1 and E2 and Poisson numbers ν1 and ν2 can always be 

reduced to the contact of a rigid indenter and an elastic medium with an effective elastic 

modulus E
*
 determined as [10]  

 
2 2

1 2

*

1 2

1 11

E EE

  
  . (1) 

In the present paper, the indentation test is numerically simulated by the high resolution 

Boundary Element Method (BEM), which has recently been generalized to arbitrary 

contact problems including tangential contact and adhesive contact [11, 12].  

2. INDENTATION OF PRISMATIC INDENTERS WITH POLYGONAL BASE 

The normal contact stiffness between a rigid flat cylinder and an elastic half space is 

given by k=2aE
* 

[7], where a is the radius of the cylinder, and E
*
 is the effective elastic 

modulus, Eq. (1). In the case of a prismatic indenter with an arbitrary base form, the normal 

contact stiffness is given by [5]: 

 
*

2
A

k E


  ,  (2) 

where A is the contact area of the base. Obviously the value of β is equal to 1 for the 

flat-ended cylinder. It was proven that Eq. (2) is also valid for indenters which have a cross 

section other than a circle [5]: β=1.034 for triangle and β=1.012 for square. These results 

were numerically obtained by King in 1987. Due to the limitation of computer technology 

at that time, King used only 200 elements for simulating a triangle indenter, and the  

 

Fig. 1 Prismatic indenters with polygonal bases: m=3 (triangle),  

m=4 (square), m=5 (pentagon) and m=∞ (circle) 



 Indentation of Flat-Ended and Tapered Indenters with Polygonal Cross-Sections 243 

triangular area looked quite „rugged‟. Note that the stiffness of a flat punch and 

correspondingly factor β are related to the so-called harmonic capacity of the base form of 

the punch. This analogy was discussed by Argatov (2010) [13]. 

Below we repeat the calculations of King using the current high-resolution BEM and 

provide corrected values. 

Using the boundary element method we have numerically carried out the indentation 

test for different shapes of cross section of indenters: from triangle (m=3), square (m=4), 

pentagon (m=5) to circle (m=∞) as shown in Fig.1. In the simulation, the whole area was 

divided into 1024x1024 elements where at least 200000 elements were in the contact area. 
It is at least 1000 times more than in the King‟s simulations; therefore, a much more precise 

result could be obtained. The values of coefficient β for different m are presented in Fig.2 

and Table1. For the two most popular indenter shapes, the values are: 

 
1.061,  for triangle,

1.021,  for square,








  (3) 

which is larger than the values reported by King [5]. It can be seen that with the same area 

of cross section, the stiffness of triangular indenter is for 6% larger than that of a flat 

cylinder. 

 

Fig. 2 Factor β for different polygonal indenters. The two stars indicate the results obtained 

numerically by King in 1987 [5] 

Table 1 Values of constant β 

m 

polygon 

3 

(triangle) 

4 

(square) 

5 6 7 8 ∞ 

(cylinder) 

β 1.061 1.021 1.010 1.005 1.003 1.002 1.000 



244 Q. LI, V.L. POPOV 

3. INDENTATION OF TAPERED INDENTERS WITH POLYGONAL BASE  

AND POWER FUNCTION SIDE SURFACE 

Now we consider the tapered indenters which have a regular polygonal base, as shown 

in Fig. 3. We begin with the most common type – a pyramid, and then extend it to indenters 

whose side profile is an arbitrary power function.  

3.1. Pyramidal indenters  

For the contact between a rigid cone with profile f (r) = tanθ·r and an elastic half space 

with effective elastic module E
*
, the dependence of normal force on indentation depth was 

analytically found by Galin [6] (see also Sneddon [7]):  

 
*

22

tan
N

E
F d

 
 , (4) 

where d is indentation depth and θ is defined in Fig. 3(c). This solution can be easily 

reproduced using the method of dimensionality reduction (MDR). In the framework of the 

MDR [14], any contact problem of an axis-symmetrical profile f(r) with an elastic 

half-space can be mapped onto a contact of a modified (MDR-transformed) profile g(x):  

 
| |

0 2 2

( )
( ) | | d

x f r
g x x r

x r





 ,  (5) 

with properly defined elastic foundation. For a conical profile, f(r) = tanθ·r, the substitution 

in Eq. (5) and integration provides the MDR-transformed profile: 

 ( ) ( / 2) | | tang x x    . (6) 

A short calculation (see. e.g. [14]) leads to Eq. (4).  

 

Fig. 3 Pyramid indenters for n=1 (a)-(c) and parabolic indenters for n=2 (d)-(f)  

with polygonal base, m=3 (triangle), m=4 (square),and m=∞ (cycle) 



 Indentation of Flat-Ended and Tapered Indenters with Polygonal Cross-Sections 245 

In [15], it was shown that an equivalent MDR-transformed profile does exist not only 

for axis-symmetrical indenters but also for indenters of arbitrary shape. As shown in [15] 

and [16], for this sake, quantity l=k/(2E
*
) (where k=dFN /dd is the incremental normal 

stiffness) should be determined numerically as function of indentation depth d. Inverse 

function d(l) is then exactly the unknown MDR transformed profile g(x). Let us illustrate 

this simple procedure on the example of conical indenter. By differentiating Eq. (4) with 

respect of d we get stiffness k=4E
*
d/(πtanθ) and length l=2d/(πtanθ). Inverse relation 

d=l(π/2)tanθ coincides exactly with the MDR transformed profile (6). This procedure is 

applicable regardless of whether dependence FN(d) was obtained analytically, numerically 

or experimentally. In the following, we determine dependence FN(d) numerically and 

extract from it the MDR-transformed profiles for a number of tapered profiles with 

polygonal cross-sections (Fig. 3).  

We start with consideration of pyramidal indenters. As shown in Fig. 3(a)(b), the bases 

of the indenter are regular polygons. Angle θ is defined as the angle between the ground 

plane and the 3D indenter side surface as shown in Fig.3. 

In the simulation we calculated the contacts of pyramid indenters with different 

polygonal bases varying from m=3 to 20, and for each type the angle ranges from =π/64 to 
31π/64. All the simulation results show that the one-dimensional profile is still a linear 

function which can be formulated as: 

 
1D

( ) | |g x c x  , (7) 

with c1D : 

 
1D

tanc    , (8) 

where α is dependent only on polygon order m. For the sake of comparison we can define a 

fictive rotationally symmetric 3D profile with the same inclination angle: 

 
3D 3D

( ) tanf r c r r    .  (9) 

Then we can write α =c1D/c3D. The values of α for different shapes of polygons are shown 

in Fig. 4(a) and Table 2. For a larger m, the shape of the pyramid indenter is close to a cone, 

the value of α is almost equal to π/2. 

 

Fig. 4 Coefficient of α for pyramidal indenter n=1 (a) and parabolic indenter n=2 (b)  

with different polygonal bases 



246 Q. LI, V.L. POPOV 

Table 2 Values of coefficient α 

m 

polygon 

3 

triangle 

4 

square 

5 6 7 10 20 30 ∞ 

cycle 

α (n=1) 

pyramid 
1.133 1.356 1.422 1.485 1.510 1.542 1.564 1.568 π/2 

α (n=2) 

paraboloid 
1.052 1.493 1.690 1.791 1.848 1.928 1.981 1.993 2 

3.2. Indenters with arbitrary power function geometry 

Let us now consider the case when the side surface of the indenter is not flat but is given 

by a power function. An example of parabolic indenter (shape with power 2) is shown in 

Fig. 3(d)-(f). We first remember the corresponding solution for an axisymmetric indenter 

with an arbitrary power function shape f(r) =cn·r
n
. According to Eq. (5) its one-dimensional 

MDR-transformed profile is given by: 

 ( ) | |
n

n n
g x c x  , (10) 

where: 

 
( 2)

2 ( 2 1 2)
n

n n

n





 

 
, (11) 

and Γ(n) is gamma function. In particular, for the cone (n=1) κ1= π/2 and for a paraboloid 

(n=2) κ2=2, corresponding to α =c1D/c3D for m=∞ as shown in Fig. 4 and Table 2. 

As in the previous Section, we define an axis-symmetrical shape with the same power-law 

shape as shown in detail in Fig. 3. To underline that we have to do with a three-dimensional 

body which is in contact with a three-dimensional half-space, we denote the corresponding 

reference shape as  

 
3 3

( )
n

D D
f r c r  . (12) 

This shape coincides with the vertical section of the polygonal indenters (shown by dashed 

lines in Fig. 3). 

The numerical indentation tests were carried out for different indenters with power 

function n from 1 to 20 and the polygonal base parameter m from 3 to 30. The results show 

that the 1D profile for an arbitrary power function is still a power function with the same 

power. Coefficient α =c1D/c3D for the same type of indenter (fixed n and m) is constant 

(independent of coefficient c3D). An example of parabolic indenter (n=2) is shown in Fig. 4 

(b), where the values of α for triangle, square and further polygonal based profile are 

presented. In the limiting case the indenter is a spherical cylinder, and α=2 corresponding 

to κ2=2 is well-known from the MDR theory [14]. 

If we use the following parameter instead of α 

 1

3

D

n D

c

c



 , (13) 

then in the limiting case m=∞, value ξ for any power function n will be equal to 1, ξm=∞=1. 

Some values of ξ, in particular for pyramid and parabolic indenter with triangle and square 

base are shown in Fig. 5 and Table 3. 



 Indentation of Flat-Ended and Tapered Indenters with Polygonal Cross-Sections 247 

 

Fig. 5 Coefficient of ξ for indenters with power function profile 

Table 3 Values of coefficient   

n 

m 

3 

(triangle) 

4 

(square) 

5 10 20 30 

1 (pyramid) 0.723 0.866 0.923 0.986 1.000 1.000 

2 (paraboloid) 0.526 0.747 0.845 0.964 0.991 0.997 

3 0.384 0.648 0.777 0.947 0.987 0.994 

10 0.043 0.241 0.058 0.835 0.957 0.983 

20 0.002 0.058 0.190 0.695 0.918 0.964 

 

3.3. Consideration of indenters with the same base area  

In Section 2 it is found that the contact stiffnesses of triangular, rectangular indenters 

and flat cylinder with the same cross-section area are almost the same, and differ at most by 

6%. It thus appears to be sensible to try as “reference” indenters the axisymmetrical 

profiles with the same area of cross-section. This definition is slightly different from the 

definition in the previous Section. For both initial polygonal profile and the reference 

axisymmetrical profile we carry out the MDR transformation and determine the equivalent 

1D-MDR profiles. Let us explain the exact procedure on the example of a pyramid indenter 

(n=1). First, we determine the area of the indenter at different height and construct a cone 

with exactly the same cross-section areas. Then we carry out the three dimensional 

indentation test of the polygonal indenter by the BEM simulation and extract corresponding 

MDR profile g(x)m-poly and corresponding coefficient c1D,m-poly as described in Section 3. For 

the reference axisymmetrical profile, the corresponding MDR transformed profile and the 

corresponding coefficient c1D,m=∞ are determined by (5). Finally we compare this c1D,m-poly and 

the coefficient of the axisymmetric conical profile using the ratio  

 
1 , -poly

1 ,

D m

D m

c

c




 . (14) 



248 Q. LI, V.L. POPOV 

In an absolute similar way comparisons were also carried out for other power function 

geometries. The results are shown in Fig.6 and Table 4. It can be seen that the coefficient 

c1D of pyramid indenter is close to that of conical indenter: it differs by at most 7% in the 

case of triangular base (c1D =0.927). It is noted that coefficient c1D cannot directly reflect the 

contact stiffness. Take an example of triangular indenter with power n=20 whose geometry is 

close to the flat triangular indenter (Fig.1a), its ζ is very small ζ =0.295 (m=3, n=20), but the 

contact stiffness at the large indentation depth is the same to the flat indenter.  

 

Fig. 6 Comparison of coefficient c1D among different indenters with the same base area 

Table 4 Coefficient ζ for different power n and polygon m 

n 

m 

3 

(triangle) 

4 

(square) 

5 6 10 20 

1 (pyramid) 0.927 0.974 0.988 0.994 1.000 1.000 

2 (paraboloid) 0.870 0.951 0.977 0.988 0.997 1.000 

3 0.817 0.931 0.966 0.981 0.996 1.000 

10 0.536 0.806 0.820 0.893 0.990 1.000 

20 0.295 0.651 0.814 0.889 0.973 0.997 

4. CONCLUSION 

Indentation of flat-ended and tapered indenters with polygonal base was numerically 

simulated using the boundary element method. The contact stiffnesses of prismatic 

punches with the same cross section area are almost same as the cylindrical indenter, where 

the triangular punch differs at most by 6%.  For pyramidal indenter and others with power 

function side, the one dimensional MDR transformed profile was generated based on the 

three dimensional simulation of indentation. It is found that the 1D profile is still a power 

function with the same power and it differs only by a constant factor. The factor was 

numerically calculated for the indenters with different power function side and different 

polygonal base. The generated MDR profiles can be used for the further contact problems, 

such as tangential contact or contact with linear viscoelastic bodies.  



 Indentation of Flat-Ended and Tapered Indenters with Polygonal Cross-Sections 249 

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