Computer simulation of pressing a ceramic ball into elastic-plastic material

Abstract

This article describes distributions of stresses and strains created by pressing a ball with 80 HRC hardness into soft 35HRC steel, which occurs in the technological process of burnishing. We present a computer model and a solution developed with the Nastran FX program, based on the finite elements method. The validity of the model has been checked using a small force applied on the ball, within the range of elastic deformations, where Hertz formulas are applicable. Then computations have been made for a pressure force of 30 000 N, which corresponds to hardness testing conditions by the Brinell method. The dimensions of plastic indentations from simulations and experiments are compared.

Full text

Zeszyty Naukowe Akademii Morskiej w Szczecinie 42 (114) 9
Scientific Journals Zeszyty Naukowe
of the Maritime University of Szczecin Akademii Morskiej w Szczecinie
2015, 42 (114), 9–13
ISSN 1733-8670 (Printed)
ISSN 2392-0378 (Online)
Computer simulation of pressing a ceramic ball into
elastic-plastic material
Stefan Berczyński1, Daniel Grochała1, Zenon Grządziel2
1 West Pomeranian University of Technology, Faculty of Mechanical Engineering and Mechatronics
70-310 Szczecin, Aleja Piastów 19, e-mails: {stefan.berczynski; daniel.grochala}@zut.edu.pl
2 Maritime University of Szczecin, Institute of Basic Technical Sciences
70-500 Szczecin, ul. Wały Chrobrego 1–2, e-mail: z.grzadziel@am.szczecin.pl
Key words: computer simulation, ceramic ball, elastic-plastic material, burnishing, Hertzian stresses, de-
formations, and contact
Abstract
This article describes distributions of stresses and strains created by pressing a ball with 80 HRC hardness
into soft 35HRC steel, which occurs in the technological process of burnishing. We present a computer model
and a solution developed with the Nastran FX program, based on the finite elements method. The validity of
the model has been checked using a small force applied on the ball, within the range of elastic deformations,
where Hertz formulas are applicable. Then computations have been made for a pressure force of 30 000 N,
which corresponds to hardness testing conditions by the Brinell method. The dimensions of plastic
indentations from simulations and experiments are compared.
Introduction
Pressing a hard ceramic ball into a soft material
is the first phase of surface treatment by burnishing
(Grochała, Berczyński & Grządziel, 2014). A deep
knowledge of stress and strain distributions running
perpendicular and parallel to the treated surface is
essential in selecting technological parameters of
burnishing (Grochała, 2011; Lopez et al., 2005;
2007). In the next phase, the burnishing elements
are shifted.
The physical model
This article presents a computer simulation of
the process of pressing a ball 10 mm in diameter,
made of the ceramic material ZrO2 (zirconium
dioxide) that has properties similar to steel:
Young’s modulus, E, equal to 210 Ga, Poisson’s
ratio,

equal to 0.29. A 10 mm ball is forced into
the center of a specimen base, 20 mm in diameter
and 10 mm in height (Figure 1). The cylindrical
specimen is made of steel, X42CrMo4, with these
properties: Young’s modulus, E, equal to 210 GPa,
Poisson’s ratio,

 equal to 0.29. The ceramic ball
is resistant to plastic deformations (very hard,
80 HRC), while the specimen being burnished is
made of soft material subject to elastic and plastic
deformations (hardness 35–40 HRC).
Figure 1. A physical model of a ball pressed into a cylindri-
cal sample
The computer model
As the geometry of the objects and load are
symmetrical, we took a section of a joint with 3
rotation angle to model the process of the ball
pressing into the specimen material, as illustrated
by Figure 2.

Stefan Berczyński, Daniel Grochała, Zenon Grządziel
10 Scientific Journals of the Maritime University of Szczecin 42 (114)
Figure 2. A section of a ball and cylinder divided into finite
elements: 6,860 elements and 13,802 nodes
The ball section model is composed of 5,042
elements and 10,125 nodes. The cylindrical speci-
men model consists of 1,818 elements and 3,677
nodes. Around the point of contact of the ball and
cylinder base we used a densified division into
finite elements due to the occurrence of contact
stresses (a side length of a finite element in this
area is 0.05 mm). All 3D elements were created by
rotating 2D elements by a 3° angle around the joint
axis. This resulted in 6-node elements fixed to the
axis of rotation. Eight-node elements were created
in the remaining volume.
Stress calculations in the elastic range
at low pressure
To verify whether the adopted model is correct,
we made calculations for low pressures to be able
to determine whether the results of reduced stresses
obtained directly from the Hertz formulae are valid
for an elastic model. Instead of a calculator, we
used the program HertzWin2.6.0 (Budynas &
Nisbett, 2008; Deeg, 1992). The program makes
use of the numerical integration method, and auto-
matically presents the results in tables and dia-
grams. Figure 3 shows the calculations made by the
program from physical model data, for the pressure
force of 600 N.
The same problem was solved by means of
the program Nastran FX (Nastran, 2010). The
program is based on the finite elements method
(FEM). Figure 4 presents a map of reduced stresses
obtained with this program. Distributions of re-
duced stresses are clearly visible across the entire
volume of the ball and the cylindrical specimen.
Figure 3. The results of calculations of the program HertzWin: ball radius 5 mm, force 600 N, with Young’s modulus and
Poisson’s ratio the same for the ball and cylindrical specimen: E = 210 GPa,

= 0.29

Computer simulation of pressing a ceramic ball into elastic-plastic material
Zeszyty Naukowe Akademii Morskiej w Szczecinie 42 (114) 11
The greatest values of these stresses do not occur
on the contact surfaces, but at some distance from
the surface, which can be seen in Figure 5, the
diagram showing the stresses occurring along the
symmetry axis of the joint. From a depth of
0.18 mm, the curves are overlapping.
Figure 4. A map of reduced stresses obtained from Nastran
FX calculations
Figure 5. Distributions of reduced stress occurring along
the axis of symmetry of the cylindrical specimen obtained
by the programs HertzWin and Nastran FX
Calculations of residual stresses
and strains
Calculations were made for stresses and strains
created under a load of 30,000 N. This is the force
to be imposed on the indenter in hardness meas-
urements using a ball 10 mm in diameter. Because
plastic strains occur under such conditions, the non-
linearities of the specimen material were accounted
for. The diagram of stretching the material of the
cylindrical specimen (steel X42CrMo4) is given in
Figure 6.
This curve represents the stress

as a function
of relative elongation,

, which was taken into
account in the program Nastran FX. Figure 7 shows
a map of stresses occurring in the ball and cylindri-
cal specimen under a pressure of 30,000 N, as
calculated by Nastran FX. The elastic stresses in the
ball disappear once the load is removed, while the
stresses in the cylindrical specimen are the sum of
elastic and plastic stresses. Maximum reduced
stresses in the ball occur 2 mm from the point of
contact, and reach about 1550 MPa. Maximum
reduced stresses in the cylindrical specimen occur
on the contact surface and are equal to 1855 MPa.
Figure 7. Reduced stresses in the ball and cylindrical
specimen under a load of 30,000 N
Figure 8 presents a diagram of reduced stresses
running along the axis of symmetry of the model.
Once the ball is unloaded, residual stresses will
remain in the material of the cylindrical specimen.
A map of the reduced residual stresses that remain
in the specimen is given in Figure 9.
Figure 10 presents a diagram of reduced residual
stresses along the axis of the cylindrical specimen.
Maximum reduced stresses are 800 MPa, and occur
at a depth of about 1 mm from the specimen sur-
face.
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 1000 2000 3000
Depth [mm]
Reduced stresses [MPa]
NASTRAN
HERTZWIN
Figure 6. Diagram of stretching test of steel X42CrMo4

Stefan Berczyński, Daniel Grochała, Zenon Grządziel
12 Scientific Journals of the Maritime University of Szczecin 42 (114)
The computations of Nastran FX also result in
maps of plastic deformations that remain after the
ball is unloaded. Figure 11 shows plastic defor-
mation in a longitudinal section of the cylindrical
specimen.
Figure 8. Reduced stresses in the ball and cylindrical
specimen along the axis of symmetry under a 30,000 N load
Figure 9. A map of reduced residual stresses that remain in
the cylindrical specimen after the ball is unloaded
Figure 10. A diagram of reduced residual stresses along the
specimen axis remaining in the material after ball unload-
ing
Figure 11. Deformations of the cylindrical specimen after
ball unloading
The experiments
Indentations (Figure 12) made on the surface of
steel specimens in hardness tests by the Brinell
method (ceramic ball with a diameter of 10 mm,
pressure force 30,000 N) were measured in a multi-
sensor machine for surface topography measure-
ments AltiSurf A520 (Sn/No:0513-A520-05/144),
which has a confocal sensor CL2, measuring range
of 400 µm and resolution of 22 nm. The cloud of
points obtained was processed using AltiMap
Premium software (version: 6.2.7200). The results
of diameter and indentation depth h are compared
in Figure 13.
Figure 12. Deformations of the cylindrical specimen after
ball unloading, a) view of the scanned surface and the
method of indentation diameter measurement, b) indenta-
tion depth measurement (as per ISO 5436-1)
To verify the correctness of the results, only the
plastic strains were checked. The indentation diam-
eter and depth measurement results are plotted in
Figure 13. The plots show a comparison of plastic
deformations determined by Nastran FX (FEM),
with a shape created after the pressing of an ideal
10 mm ball. The indentations from experiments and
simulations display very good agreement. On this
basis we can state that the distribution of residual
stresses is also correct.
-12
-10
-8
-6
-4
-2
0
2
4
6
0 500 1000 1500 2000
Depth [mm]
[MPa]
Reduced stresses under 30 000 N load
ball
cylindrical specimen
-12
-10
-8
-6
-4
-2
0
0 200 400 600 800 1000
Depth [mm]
Reduced residual stresses [MPa]
a)
b)

Computer simulation of pressing a ceramic ball into elastic-plastic material
Zeszyty Naukowe Akademii Morskiej w Szczecinie 42 (114) 13
Figure 13. Comparison of the test results, shape of the
radius and depth of the indentation
Conclusions
Developed in the Nastran FX software, a model
of pressing a ceramic ball into elastic-plastic mate-
rial may be used in further research into the optimi-
zation of technological burnishing parameters, and
for simulation of the values and distributions of
surface stresses in objects subjected to burnishing.
A computer model of a ball-flat surface contact
requires a densified division by finite elements in
the area of the point of contact.
Maximum values of reduced stresses in the elas-
tic and plastic ranges occur under the material
surface.
Preliminary calculations in the elastic range can
be performed by a user-friendly program HertzWin
that produces the results instantly.
Modern programs based on the finite elements
method very accurately calculate plastic defor-
mations. Experimental verification of residual
stresses, although possible by X-ray methods, is
very inconvenient (Senczyk, 2005).
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0
0.1
0.2
0 1 2 3 4
Depth of indentation [mm]
Radius of indentation [mm]
NastranFX
Ideal ball
Experiment