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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).

References

1. BUDYNAS, R. & NISBETT, K. (2008) Shigley’s Mechanical

Engineering Design. 8th edition. McGraw-Hill.

2. DEEG, E.W. (1992) New algorithms for calculating hertzian

stresses, deformations, and contact zone parameters. AMP

Journal of Technology. 2. November. pp. 14–24.

3. GROCHAŁA, D. (2011) Nagniatanie narzędziami hyrosta-

tycznymi powierzchni przestrzennych złożonych na frezar-

kach CNC. Dissertation, West Pomeranian University of

Technology, Szczecin

4. GROCHAŁA, D., BERCZYŃSKI, S. & GRZĄDZIEL Z. (2014)

Stress in the surface layer of objects burnished after mill-

ing. International Journal of Advanced Manufacturing

Technology. 72. pp. 1655–1663.

5. LOPEZ de LACALLE, L.N., LAMIKIZ, A., MUNOA, J. &

SANCHEZ J.A. (2005) Quality improvement of ball-end

milled sculptured surfaces by ball burnishing. International

Journal of Machine Tools and Manufacture. 45. pp. 1659–

1668.

6. LOPEZ de LACALLE, L.N., LAMIKIZ, A., SANCHEZ, J.A. &

ARANA J.L. (2007) The effect of ball burnishing on heat-

treated steel and inconel 718 milled surfaces. International

Journal of Machine Tools and Manufacture. 32. pp. 958–

968.

7. Nastran FX (2010) [Online] Available from:

http://midasfea.se/mechanical.php [Accessed: 7 July 2015]

8. SENCZYK, D. (2005) Podstawy tensometrii rentgenowskiej.

Poznań: Wydawnictwo Politechniki Poznańskiej.

9. Vink System Design & Analysis (2015) [Online] Available

from: http://en.vinksda.nl/toolkit-mechanical-calculations/

hertz-contact-stress-calculations [Accessed: 7 July 2015]

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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