# abaqus - ball indentation- high value of stress

Dear all,

I have modeled a cyclic ball indentation problem in

Abaqus. The sphere has been modeled as axisymmetric rigid surface while

the plate is axisymmetric deformable. The material data fed in for the

plate has been obtained from low cycle fatigue experiments and the

combined hardening model using half-cycle has beem employed.

Apart from this, the loading is displacement-controlled and sphere

is the master surface while the plate is slave surface. The other

parameters used are: surface to surface contact, small sliding, "Hard"

contact and friction coefficient as 0.2.

The meshing is done as is done is various published papers.

The

**PROBLEM**: I am getting Stress values (in the vertical direction: S22) as

very high. I am getting values as high as 1600 MPa underneath the

indenter, when the UTS of the material is only 540 MPa.

What could be the reason for this?

It would be kind of you to reply to me as soon as possible.

Thank You,

--

Aneesh Bangia

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## Why do you start from the most complex problem directly???

Dear Aneesh

I find this problem also in my students.

They think that FE commercial softwares can solve everything on earth, and they start from the final complex problem.

I am surprised you didn't use also temperature-dependence of plasticity modulus etc.

Going to the point. Let me see what you get in simplified cases:

1) cyclic indentation but elastic, and compare to Hertz analytical solution

2) cyclic loading of your material, but in a simple beam -- and compare to analytical solution

3) as 2, but concentrated force in halfspace ---and compare to analytical solution

4) remove friction, add friction

5) increase step of load, decrease step

6) try some of the ABAQUS demo first

I can think of another 50 experiments. But try these 5 first.

Hope this helps

Michele Ciavarella, http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

blogs http://rettorevirtuoso.blogspot.com/

YouTube Channel http://www.youtube.com/user/RettoreVirtuoso

## High stresses due to constraint

Dear Aneesh,

The high stresses you are seeing may actually be correct due to the generation of a large hydrostatic stress under the indenter, which can lead to axial stresses that are approximately 3 times the yield strength of the material.

Chad

## Well Chad is right, you could check that easily

Chad is rigth, and indeed the Brinell hardness test is based on that equation.

Check http://en.wikipedia.org/wiki/Brinell_scale. The BHN can be converted into the ultimate tensile strength

(UTS), although the relationship is dependent on the material, and

therefore determined empirically. The relationship is based on Meyer's

index (n) from Meyer's law. If Meyer's index is less than 2.2 then the ratio of UTS to BHN is 0.36. If Meyer's index is greater then the ratio increases.[1]

Since the hydrostatic stress depends largely on Poisson's ratio, try to change that.

You should not be sad however, since Heinrich Hertz himself made

this mistake you are making, as he assumed the plastic flow would start

from the surface in his third paper on the subject when he firstly proposed to define hardness from the local pressure on surface.

Contact mechanics

Memorial of Heinrich Hertz on the campus of the Karlsruhe Institute of Technology

Main article: Contact mechanics

In 1881–1882, Hertz published two articles on what was to become known as the field of contact mechanics. Hertz is well known for his contributions to the field of electrodynamics (

see below);however, most papers that look into the fundamental nature of contact

cite his two papers as a source for some important ideas. Joseph Valentin Boussinesq

published some critically important observations on Hertz's work,

nevertheless establishing this work on contact mechanics to be of

immense importance. His work basically summarises how two axi-symmetric

objects placed in contact will behave under loading, he obtained

results based upon the classical theory of elasticity and continuum

mechanics. The most significant failure of his theory was the neglect

of any nature of adhesion between the two solids, which proves to be

important as the materials composing the solids start to assume high

elasticity. It was natural to neglect adhesion in that age as there

were no experimental methods of testing for it.

To develop his theory Hertz used his observation of elliptical Newton's rings

formed upon placing a glass sphere upon a lens as the basis of assuming

that the pressure exerted by the sphere follows an elliptical

distribution. He used the formation of Newton's rings again while

validating his theory with experiments in calculating the displacement

which the sphere has into the lens. K. L. Johnson, K. Kendall and A. D.

Roberts (JKR) used this theory as a basis while calculating the

theoretical displacement or

indentation depthin the presenceof adhesion in their landmark article "Surface energy and contact of

elastic solids" published in 1971 in the Proceedings of the Royal

Society (A324, 1558, 301-313). Hertz's theory is recovered from their

formulation if the adhesion of the materials is assumed to be zero.

Similar to this theory, however using different assumptions, B. V. Derjaguin,

V. M. Muller and Y. P. Toporov published another theory in 1975, which

came to be known as the DMT theory in the research community, which

also recovered Hertz's formulations under the assumption of zero

adhesion. This DMT theory proved to be rather premature and needed

several revisions before it came to be accepted as another material

contact theory in addition to the JKR theory. Both the DMT and the JKR

theories form the basis of contact mechanics upon which all transition

contact models are based and used in material parameter prediction in

Nanoindentation and Atomic Force Microscopy. So Hertz's research from

his days as a lecturer, preceding his great work on electromagnetism,

which he himself considered with his characteristic soberness to be

trivial, has come down to the age of nanotechnology.

You may also like to visit Wikipedia

Contact mechanics

From Wikipedia, the free encyclopedia

Jump to: navigation, search

Continuum mechanics[show]Laws

Conservation of mass

Conservation of momentum

Conservation of energy

Entropy inequality

[show]Solid mechanics

Solids · Stress · Deformation · Finite strain theory · Infinitesimal strain theory · Elasticity · Linear elasticity · Plasticity · Viscoelasticity · Hooke's law · Rheology

[show]Fluid mechanics

Fluids · Fluid statics

Fluid dynamics · Viscosity · Newtonian fluids

Non-Newtonian fluids

Surface tension

[show]Scientists

Newton · Stokes · Navier · Cauchy · Hooke · Bernoulli

This box: view • talk • edit

Stresses in a contact area loaded simultaneously with a normal and a

tangential force. Stresses were made visible using photoelasticity.

Contact mechanicsis the study of the deformation of solids that touch each other at one or more points[1][2]. The physical and mathematical formulation of the subject is built upon the mechanics of materials and continuum mechanics and focuses on computations involving elastic, viscoelastic, and plastic bodies in static or dynamic contact. Contact mechanics is foundational to the field of mechanical engineering; it provides necessary information for the safe and energy efficient design of technical systems.The original work in contact mechanics dates back to 1882 with the publication of the paper "On the contact of elastic solids" ("Ueber die Berührung fester elastischer Körper") by Heinrich Hertz. Hertz was attempting to understand how the optical properties of multiple, stacked lenses might change with the force

holding them together. Results in this field have since been extended

to all branches of engineering, but are most essential in the study of tribology and indentation hardness.

Hertzian contact stress refers to the localized stresses that develop

as two curved surfaces come in contact and deform slightly under the

imposed loads. This amount of deformation is dependent on the modulus of elasticity

of the material in contact. It gives the contact stress as a function

of the normal contact force, the radii of curvature of both bodies and

the modulus of elasticity of both bodies. In gears and bearings in

operation, these contact stresses are cyclic in nature and over time

lead to sub-surface fatigue cracks. Hertzian contact stress forms the

foundation for the equations for load bearing capabilities in bearings,

gears, and any other bodies where two surfaces are in contact.

Principles of contacts mechanics can be applied in areas such as locomotive wheel-rail contact, coupling devices, braking systems, tires, bearings, combustion engines, mechanical linkages, gasket seals, metalworking, metal forming, ultrasonic welding, electrical contacts, and many others. Current challenges faced in the field may include stress analysis of contact and coupling members and the influence of lubrication and material design on friction and wear. Applications of contact mechanics further extend into the micro- and nanotechnological realm.

The motion of a single body in space is described by the governing equations of continuum mechanics.

The approach used in contact mechanics is to restrict the the motion of

two or more bodies in space by additional constraints. These

unilateralconstraints ensure that bodies do not penetrate each other after coming

into contact. Once the general equations for a contact problem are set

up, different solution schemes can be used to simulate the behaviour of

bodies in contact and to compute displacement and stress fields. A

distinction is usually drawn between contact with and without friction.

Contents

[hide]

Michele Ciavarella, http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

blogs http://rettorevirtuoso.blogspot.com/

YouTube Channel http://www.youtube.com/user/RettoreVirtuoso

## force vs indenter displacement

That was a relief, really.

thanks a lot Mike and Chad!

i cannot however relate BHN to UTS because the material parameter has to be obtained emperically.

In order to validate my results from simulation, I compared my Force vs Displacement curves with that obtained experimentally with the ones obtained by simulation.

The force value obtained through simulation was found to be about 1.5 times that of the value obtained through indentation experiments.( 2080 N compared to 1400 N)

This is one benchmark I need to confirm what I'm doing is correct.

Also, it must be pointed out here that the experiments were performed sometime back by some other student, and I am just using his experimental results.

## I don't understand, it is all VERY empirical anyway,except test!

Annesh

Frankly, now again I don't undertstand! Why do you beleive your LCF data are absolutely good, to be beleived as GOD? Actually, on the contrary, the more DISTANT you are from the conditions you simulate in FEM tests, the more OBVIOUS is that you do not fit correctly !

To relate hardness to UTS, what do you mean you cannot do it?

Here, it is ALL empirical fitting. Once you decide what to best fit, the rest is not exact, but approximate.

Ideally, the easiest to fit is the test itself! In other words, you generate a plastic material that is generating that indentation test. Of course when you use back into LCF fatigue data, you are lost again.... ;)

So please reformulate. As I told you, to relate indentation laws with Meyer's "law" --- the term "law" as usual is misused by engineers as if it were a law of physics. In fact, since it is quite general, you can fit almost ANYTHING to it --- this is perhaps to say this is really not a law, or else that this is the common mistake with power laws, unfortunately these fitting equations are considered "laws"!

Perhaps your assignement is not well posed. I additionally do not understand if you want to do cyclic indentation fit testing with LCF data or not. It reminds me of my work in Rolling Contact Fatigue, where contact stresses have been attempted to be related to LCF plasticity "laws" for ages, without much success -- please read these two papers of mine:

A re-examination of rolling contact fatigue

experiments by Clayton and Su with suggestions for surface durability

calculations

Wear,Volume 256, Issues 3-4,February,2004

Pages 329-334L. Afferrante, M. Ciavarella, G. Demelio

A note on Merwin’s measurements of forward flow in

rolling contact

Wear,Volume 256, Issues 3-4,February 2004,Pages 321-328A. R. S. Ponter, L.

Afferrante, M. Ciavarella

And read again WIKIPEDIA please.

Meyer's law

From Wikipedia, the free encyclopedia

Jump to: navigation,

search

Meyer's lawis an empiricalrelation between the size of a hardness test indentation

and the load required to leave the indentation.[1]

Contents

[hide]

//

[edit] Equation

It take the form:

where

n usually lies between the values of 2, for fully strain hardened materials, and 2.5,

for fully annealed materials. It is roughly related to the

strain hardening coefficient in the equation for the true stress-true

strain curve by adding 2.[1]

Note, however, that below approximately d = 0.5 mm (0.020 in) the value

of n can surpass 3. Because of this Meyer's law is often restricted to

values of d greater than 0.5 mm up to the diameter of the indenter.[2]

The variables k and n are also dependent on the size of the indenter.

Despite this, it has been found that the values can be related using

the equation:[3]

Meyer's law is often used to relate hardness values based on the fact

that if the weight is halved and the diameter of the indenter is

quartered. For instance, the hardness value for a test load of 3000 kg

and a 10 mm indenter is the same for a test load of 750 kg and a 5 mm

diameter indenter. This relationship isn't perfect, but its percent error is relatively small.[4]

A modified form of this equation was put forth by Onitsch:[5]

[edit] See also

[edit] References

[edit] Notes

abHardness Testing, http://www.key-to-steel.com/IT/fr/Articles/Art140.htm, retrieved 2008-10-07 .^Tabor, pp. 12-14.

^Tabor, p. 8.

^Tabor, pp. 10-11.

^Blau,P. J.; Lawn, Brian R.; American Society for Testing and Materials

Committee E-4 on Metallography, International Metallographic Society

(1986),

Microindentation Techniques in, ASTM International, p. 93, ISBN 0803104413, http://books.google.com/books?id=dzX7hkibzzUC .Materials Science and Engineering

[edit] Bibliography

The Hardness of Metals, OxfordUniversity Press, ISBN 0198507763, http://books.google.com/books?id=b-9LdJ5FHXYC .

Michele Ciavarella, http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

blogs http://rettorevirtuoso.blogspot.com/

YouTube Channel http://www.youtube.com/user/RettoreVirtuoso

## the project goal

Hi Mike,

Thanks for the direction, again.

I found out the Meyer's index which was below 2.2; and my UTS was 560.5

MPa and BHN came out to be 1569.3 (in MPa). UTS/BHN = 0.357

(which is close to 0.36, as written in wiki)

So, does this kind of validate the high stresses observed underneath the indenter?

I would now like to shed some light on what my project is about and what all I have done already.

I have to correlate the plastic dissipation energies for LCF and

cyclic indentation. For that the experiemental part has already been

performed by another student, and I am doing the simulation part. I

have already performed simulations for LCF and obtained good results

(hysteresis curves, plastic dissipation energy curves, and so on)

But, I've been facing problems in the case of indentation. Once I'm

sure that the stress values that I've obtained are actually correct and

other things such as Force vs displacement curves are matching with

indentation experimental curves to an extent (with little error), I can

go ahead with correlating the plastic dissipation energy for the two

processes.

Meanwhile, I am reading the papers as suggested by you.

Thanks and Regards,

Aneesh

## Check also these references

Modeling of the cyclic ball indentation test for small

specimens using the finite …

T Yamamoto, H Kurishita, H Matsui - Journal of Nuclear

Materials, 1999 - Elsevier

...Fig. 6 shows a typicalload-displacement curve from the

cyclicindentationtestconducted for

the as-machined specimen together with the FEM simulation of the tests.

The experimental results

showed much larger elastic deformations than any simulation result.

...Cyclic indentation in aluminum

F Yang, L Peng, K Okazaki - Journal of Materials Science,

2007 - Springer

...Al F, d Fig. 1 Schematic of thecyclicindentationtestof Al 123...Page 3. Results anddiscussion

Indentation fatigue curves During the

cyclicindentationtest,a cyclic indentation load was applied

to the indenter, which caused the indenter to move into the surface of

the specimen.

...Investigation

of Material Fatigue Behavior Through Cyclic Ball Indentation Testing

RV Prakash, P Bhokardole, CS Shin -

Journal of ASTM International, 2008 - astm.org

...Thepurpose of this work is to examine the fatigue behavior of materials

through an in-situ test

technique, viz.,

cyclicindentationtestmethod....This record corresponds to a

cyclicindentationtestthat was carried out between −50 N and −500 N force at afrequency of 0.25 Hz.

...Numerical investigation of indentation fatigue on

polycrystalline copper

BX

Xu, ZF Yue, X Chen - J. Mater. Res, 2009 - mrs.org

...loop in the indentation load–depth curve that depends on the dopant

level and the depth of

indentation.13 This study may shed some light on extracting sensitive

material information such

as the sen- sitive defect–solute interactions by using the

cyclicindentationtest, which are...Articoli

correlati - ACNP Posseduto Biblioteche - Tutte

e 4 le versioni

Berkovich

indentation-induced deformation microstructures in GaN thin films

CH Chien, SR Jian, CT Wang, JY Juang,

JC … - Journal of Physics D: …, 2007 - iop.org

...previous load, which completed the first cycle. It then was reloaded to a

larger chosen

load and unloaded by 90% for the second cycle. Figure 1 illustrated a

typical

cyclicindentationtestrepeated for 5 cycles. It is noted thatin each

...Articoli

correlati - ACNP Posseduto Biblioteche

An experimental methodology for characterizing fracture of

hard thin films under …

A

Yonezu, B Xu, X Chen - Thin Solid Films, 2009 - Elsevier

...Unfortunately, such signal may be difficult to detect during

cyclicindentationtest....Fig.1b shows the conventional

cyclicindentationtestwith a constant force range ΔF.

Detailed test conditions are described in the following sections.

...Citato

da 1 - Articoli

correlati - Tutte

e 2 le versioni

[PDF] Application of the indentation method for cracking

resistance evaluation of …

kirj.ee [PDF]A Sivitski, A Gregor, M Saarna, P

Kulu, F … - Estonian Journal of …, 2009 - kirj.ee

...475 6 200 – AlTiN –60…–150 4 × 10–3…1.2 × 10–2 (60…125)/(52...130)

430…450 6 150…200 –

2.3.

Cyclicindentationtestprocedure A servohydraulic fatigue test system INSTRON 8800 and

Vickers diamond pyramid indenter were used in the indentation

experiments.

...An indentation system for determination of viscoplastic

stress-strain behavior of …

gkss.de [PDF]N Huber, E Tyulyukovskiy, HC

Schneider, R … - Journal of Nuclear …, 2008 - Elsevier

The

development of fusion materials for the first wall in future fusion

reactors requires methods

for the investigation of irradiation effects on the mechanical

properties of materials which are

only available in small volumes. Depth and force reading hardness

measurement (or

...Citato

da 6 - Articoli

correlati - ACNP Posseduto Biblioteche - Tutte

e 8 le versioni

Study of ratcheting by the indentation fatigue method with a

flat cylindrical indenter. …

BX Xu, ZF Yue - Journal of Materials Research, 2007 -

mrs.org

The finite element method (FEM) was used to study the

flat cylindrical indentation fatigue behavior

using a kinematic hardening model (AF model). This study was motivated

by the experimental

work of the preceding paper [BX Xu and ZF Yue, J. Mater. Res. 21, 1793

(2006)], in which

...Citato

da 5 - Articoli

correlati - ACNP Posseduto Biblioteche - Tutte

e 8 le versioni

A study on determining hardening curve for sheet metal

H Tian, D Kang - International Journal

of Machine Tools and …, 2003 - Elsevier

The hardening curve

for sheet metal can be determined from the load–displacement curve of

tensile specimen with rectangular cross-section. The previous

researches, however, have paid

little attention to its use in large deformation. Moreover, it varies

with materials,

...Citato

da 2 - Articoli

correlati - ACNP Posseduto Biblioteche - Tutte

e 6 le versioni

[PDF] FINITE ELEMENT ANALYSIS OF SLIDING CONTACT BETWEEN A …

dynalook.com [PDF]SS Akarca, WJ Altenhof, AT Alpas -

dynalook.com

S. Subutay Akarca, University of Windsor 401

Sunset Ave. Windsor, ON, CANADA N9B 3P4

Telephone: (519) 253 3000 ext.2605 Facsimile: (519) 973 7007

...Dr. William J. Altenhof, University

of Windsor 401 Sunset Ave. Windsor, ON, CANADA N9B 3P4 Telephone: (519)

253 3000

...You probably have them all, but just to make sure you have !!

:)

Editor, Italian Science Debate, www.sciencedebate.it

blogs http://rettorevirtuoso.blogspot.com/

YouTube Channel http://www.youtube.com/user/RettoreVirtuoso

## plastic dissipation energy

those links were very helpful Mike, especially the first two.

thanks!

Now that the problem of high stress has been resolved, comes the issue of Plastic Dissipation energy.

As i have said before, my goal is to correlate the plastic dissipation energy for Low Cycle Fatigue with Cyclic Indentation.

The doubt right now is as follows:

I am getting a

huge difference in magnitude of PDEfor Low Cyc Fatigue and CI. For one cycle, PDE for LCF is around 35 J while it is only 7 mJ for indentation. That is, about 5000 times greater.Does this seem correct to you?

## Well first make me apart of your progress

You are not very kind. You say "they were very useful"... in what? Here, I am willing to help, for curiosity, but in return, I expect my curiosity to be satisfied!

So please provide full answer, I am not a cheap "vendor machine" of technical suggestions, I was expecting a civilized technical discussion between peers :)

For the next question, how do you compute PDE in the two cases?

Editor, Italian Science Debate, www.sciencedebate.it

blogs http://rettorevirtuoso.blogspot.com/

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## apologies

sorry Mike for being brusque.

The first paper was very useful because it clearly showed the

contours of Principal Stress whose values are shooting up in the range

that I am getting. (Although the material used was different from mine, these values were much greater than its Yield Strength). This vindicates my results and I am satisfied. Also,

it tells me the interpretation of the positive values of these stresses

and that these can be compared with the yield stress. The Force vs

Displacement curve's pattern matches with mine but i could get much out

of it.

The second paper was useful because it helped me in

improving my theoretical knowledge about Plastic Dissipation energy,

which is my prime area of concern. It also gave me a good idea about

how a load-controlled load should be provided, which is the case I may

examine next. The paper also plots PDE vs Force with PDE in micro-Joule

while Force in mN. So, I thinks it means that the order PDE I am

getting (in mJ for force in N) could be correct.

Also, Mike I

must point out that one of the authors is my faculty guide and the

other is whose experimental work I am continuing.

For the

papers, I must frankly say that I have only had a cursory glance

through them so far. But I look forward to go through them in detail in

the coming days. The list was exhaustive and I thank you Mike for

making the effort for me.

I am now pretty confident about the Stress vs Strain curves and the Force-displacement curves that I am getting because I've a good explaination for them.

Now coming to your question about PDE, I obtained it through the 'History Output Request' in ABAQUS.

I must say that I am really happy to include you in my progress of the project.

Regards,

Aneesh

## But I must say you should read your supervisor's papers !

All is well then. However, next time read your supervisor's papers, before asking imechanica!! I know it is nicer to find someone willing to help like me, and I am learning also in the process, but it sounds funny this very modern complex way to do the obvious thing to do!

Most supervisors ask for project works and thesis, continuation of work of previous students...

On the other hand, sometimes it is BETTER not to read the previous work, so that you can come out with better ideas, or new ideas. So don't follow your supervisor too much! This is why, for example, they say Einstein had those brilliant ideas at age 16-26 only because he was NOT working in academia, writing proposals, following supervisor's ideas :) Who would ever ask for phd to devise a relativity theory????

Editor, Italian Science Debate, www.sciencedebate.it

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