element+analysis+of+instrumented+sharp+indentations+into+pressure sensitive+materials

8/7/2019 Element+Analysis+of+Instrumented+Sharp+Indentations+Into+Pressure Sensitive+Materials

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J. Mater. Sci. Technol., Vol.23 No.2, 2007 277

Element Analysis of Instrumented Sharp Indentations into Pressure-

sensitive Materials

Minh-Quy LE1) and Seock-Sam KIM2)

1) Department of Mechanics of Materials & Structures, Faculty of Mechanical Engineering, Hanoi Universityof Technology, Dai Co Viet Road, Hanoi, Vietnam

2) School of Mechanical Engineering, Kyungpook National University, Daegu 702-701, South Korea[Manuscript received January 13, 2006, in revised form April 14, 2006]

Finite element analysis was carried out to investigate the conical indentation response of elastic-plastic solidswithin the framework of the hydrostatic pressure dependence and the power law strain hardening. A largenumber of 40 different combinations of elasto-plastic properties with n ranging from 0 to 0.5 and y/Eranging from 0.0014 to 0.03 were used in the computations. The loading curvature C and the averagecontact pressure pave were considered within the concept of representative strains and the dimensional analysis.Dimensionless functions associated with these two parameters were formulated for each studied value of thepressure sensitivity. The results for pressure sensitive materials lie between those for Von Mises materials andthe elastic model.

KEY WORDS: Finite element analysis; Indentation; Mechanical Properties; Pressure-sensitivematerials

1. Introduction

Indentations have been widely used to deter-mine the hardness of materials[1]. Actually, instru-mented indentation techniques are developing exten-sively to characterize various materials including met-als, metallic alloys, ceramics, glasses, polymers, andcoated materials, etc[2]. The dimensional analysis andthe concept of representative strain are widely used ininstrumented sharp indentations to formulate manydimensionless functions, which relate indentation pa-rameters to the indenter geometry and the indented

material

s mechanical properties such as elastic mod-ulus, yield stress and strain hardening exponent[36].However, most of previous research works are basedon Von Mises criterion[7], in which the influence ofhydrostatic pressure on material plastic deformationis neglected.

In practice, hard metals, ceramics, bulk metallicglasses and polymers have been known to exhibit hy-drostatic pressure dependent plastic behavior. Thepressure-sensitive yielding occurs from the basic flowmechanism in polymers[814] from phase transforma-tion in ceramics[15]. Voids and other forms of defectscan also result in macroscopic pressure sensitivity[16].

The Mohr-Coloumb and Drucker-Prager yieldfunctions have been suggested to represent the yieldbehavior of these materials[1722]. Moreover, finiteelement analyses (FEA) of sharp indentation withinthe framework of the above yield criteria have beenundertaken for elastic-plastic materials with linearstrain hardening[17], and for elastic-perfectly plasticmaterials[18,19,23]. Giannakopoulos and Larsson[17]

presented a general response of the pressure-sensitivematerials under sharp indentation. We showed thatthe pressure sensitivity and the strain hardeningincrease the loading curvature C and the averagecontact pressure pave, and decrease the residual in-dentation depth-maximum indentation depth ratio

Ph.D., to whom correspondence should be addressed,E-mail: [email protected].

(see Fig.1). However, the strain-hardening of materi-als has not explicitly involved under a functional re-lation in their work above. Ganneau et al.[23] for-mulated a functional form relating the indentationhardness to the cohesion and the frictional-angle ofmaterials.

In the present work, the dimensional analysis andthe concept of representative strain are extended tostudy instrumented sharp indentations of pressure-sensitive materials. Attention is focused on the influ-ence of pressure sensitivity into the evolution of theloading curvature C and the average contact pressure

pave. Their relationship with the material

s mechan-ical properties is formulated for each given value ofthe pressure sensitivity.

2. Theoretical Backgrounds

2.1 Yield criterionIn the classical plasticity theory, it is generally as-

sumed that hydrostatic pressure has no effect on ma-terial plastic deformation, and plastic dilatation is ne-glected. The Von Mises yield criterion is widely usedto model this class of materials such as metals andmetallic alloys[7]. However, for many materials in-cluding hard metals, ceramics, glasses, and polymers,

pressure-sensitive yielding and plastic volumetric de-formation are exhibited. They are so-called pressure-sensitive materials and modeled according to Drucker-Prager yield criterion as follows[24]:

(ij) = e + am (1 a/3)y = 0 (1)and

m = kk/3 and e =

3J2 (2)

where m and e are the hydrostatic stress and Misesequivalent effective stress, respectively; a is a mate-rial constant that measures the pressure sensitivity ofyielding; (a/

3 is the ratio of the plastic volumetric

strain to the effective plastic shear strain, a


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278 J. Mater. Sci. Technol., Vol.23 No.2, 2007

Fig.1 Schematic representation of a conical indentation: (a) axisymmetric model of the indenter and specimen,(b) typical indentation load-depth curve

Fig.2 Power law elastic-plastic stress-strain behavior

a=0. More information on the pressure-sensitive yieldcriterion can be found in literature [11,24].

A direct measurement of the pressure-sensitivityindex a relies on shear experiment under pressure.An alternative method to determine a is to performcompressive tests under pressure p; let py denote the

compressive yield stress when superimposed by hy-drostatic pressure p[25]:

a =3

3 + (3)

where =pyy

p. In the case of transformation ce-

ramics, Chen[15] reported that a is 0.95 for Mg-PSZand 1.33 for Ce-TZP, and a is about 1.19 for ZrO2-containing ceramics[25]. Further, a is in the rangefrom 0.024 to 0.11 for steels[26,27], and from 0.17 to0.43 for polymers[28].

2.2 Power-law elastic-plastic behaviorElastic-plastic behavior of many engineering solidmaterials can be modeled by a power law description,

as shown schematically in Fig.2. A simple elastic-plastic, true stress-true strain behavior is assumed to

be = E , ( y) = K n, ( y) (4)

where E is the Youngs modulus, K a strength coeffi-cient, n the strain hardening exponent, y the initialcompressive uniaxial yield stress and y the corre-sponding yield strain, such that

y = Ey = Kny (5)

Here the yield stress y is defined at zero offset strain.The total strain, , consists of elastic strain e andplastic strain p:

= e + p (6)

The representative strain r, defined by Dao et al.[3]

corresponds to the strain accumulated beyond theyield point y.

= y + r (7)

where y is the strain reached at the yield stress, y.With Eqs.(5) and (7), when >y, Eq.(4) becomes

= y

1 + rE

y

n(8)

To complete the material constitutive description,

Poissons ratio is designated as , and the incremen-tal theory of plasticity with Von Mises criterion (whena=0) or Drucker-Prager criterion is assumed.

3. Finite Element Model

Since the indentation problem of a rigid cone intohalf-space is axisymmetric (Fig.1(a)), only one-half ofthe system is used in the modeling. Therefore, elastic-plastic indentation was simulated using the axisym-metric capacities of the MARC finite element code.Given that the projected contact area for a conicalindenter, a Berkovich indenter and a Vickers inden-ter are A=h2tan2; A=24.56h2; and A=24.50h2, re-spectively; the indenter was thus modeled as a rigidcone with a half-included angle of=70.3. This angle


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Fig.3 Finite element mesh of: (a) half-space, (b) zoom of the contact zone

Table 1 Mechanical properties of material usedin the computations

E/GPa y/MPa y/E210 300 0.001428

10 30 0.00350 200 0.00490 500 0.00556

130 1000 0.0076910 100 0.01

130 2000 0.01538150 3200 0.02133

10 270 0.027

10 300 0.03

Notes: correspond to unstable results for a=0.3and a=0.58. Poissons ratio, is fixed at 0.3.Strain hardening exponent, n, is varied from 0,0.1, 0.3 to 0.5. a=0 (Von Mises materials), a=0.3and a=0.58

Fig.4 Validation of the finite element model: indenta-tion load-depth curve obtained by FEA and ex-perimental data digitized in literature [3]

gives the same area-to-depth ratio as the Berkovichor Vickers indenter, which are commonly used ininstrumented-indentation experiments[2]. Since anytypical indentation experiments would involve blunt-ing of the indenter tip, the cone tip was smoothed bya sphere of radius much smaller than the indentationdepth. This also eliminates any possible convergenceerrors due to sharp corners.

The specimen was modeled as a large cylinder

represented by around 2100 large strain four-nodeaxisymmetric elements (Fig.3). The radius and theheight of the sample are equal or seventy times larger

than contact radius. These dimensions were found tobe large enough to approximate a semi-infinite half-space for indentations. This was evidenced by an in-sensitivity of calculated results to further increase inspecimen size.

Elements were finest in the central contact areaand became gradually coarser outwards. At the max-imum indentation depth, no less than 55 elementscame into contact. It enables an accurate determi-nation of the real impression size. Frictionless rollerboundary conditions were applied along the center-line and bottom. Outside surfaces were taken as freesurfaces. The interaction between the rigid indenterand specimen was modeled by contact elements with-out friction. The residual stresses were not taken intoaccount in the analysis. Displacement-controlled pro-cedure was used in this work.

Since indentation tests have been used for a greatvariety of materials, ranging metals, metallic al-

loys, ceramics, polymers, glasses, etc., it is necessaryto model indentation using general though simpli-fied descriptions for the mechanical properties of thematerial, including power-law strain hardening andpressure-sensitivity of yielding, which occurs even inmetals and metallic alloys[16,26,27]. Therefore, a largenumber of 40 different combinations of elastic-plasticproperties with n ranging from 0 to 0.5 and y/Eranging from 0.0014 to 0.03 were used in the compu-tations. This wide range of parameters covers mostlyengineering solid materials. Three different levels ofpressure-sensitivity were considered: a=0, a=0.3, anda=0.58. The mechanical properties of material usedin the computations are given in Table 1.

The finite element model was well tested for con-vergence and accuracy, and then validated by com-paring the indentation load-depth curves obtainedby FEA with experimental data for 7075-T651 alu-minum, which was investigated by Dao et al.[3]. Themechanical properties of 7075-T651 aluminum aretaken as[3]: E=70.1 GPa, =0.33, y=500 MPa, andn=0.12. The experimental indentation data were dig-itized. Figure 4 shows a good agreement between theresults obtained by FEA and those from experiments.Further, the results obtained by Dao et al.[3] for di-mensionless functions related to the loading curvatureC and the average contact pressure pave in the case

of Von Mises materials are reproduced in the presentwork as indicated in the following section. This alsoproves the validity of the finite element model.


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Fig.5 Dimensionless function II1 constructed for a=0.3 using three different values of the representative strainr: (a) r=0.01, (b) r=0.0164, (c) r=0.033. A representative strain r=0.0164 allows the construction ofII1 to be independent of strain hardening exponent n

4. Results and Discussion

4.1 Loading curvature CFigure 1(b) shows the typical indentation load-

depth response of an elastic-plastic material to sharpindentation. From the dimensional analysis and ge-

ometrical similarity of a conical/pyramid indenter,Cheng and Cheng[6] have demonstrated that the in-dentation load P during loading is proportional to thesquare of the indentation depth h:

P = Ch2 (9)

where C is the loading curvature, which is a measureof the resistance of the material to indentation. C canbe obtained by a least square fitting procedure fromthe loading part of an indentation P-h curve. It wasverified, for all computational simulations with mod-eled materials, that the P-h curve is well reproducedby Eq.(9).

For Von Mises materials (a=0), which was consid-ered by Dao et al.[3], the authors defined the dimen-sionless function II1 as below:

II1

Er

, n

=C

r(10)

where r is the stress corresponding to a representa-tive strain r, and E

is the reduced elastic modulusfor a rigid indenter:

E =E

1 2 (11)

In the case of a Berkovich indenter, a representa-tive strain of 3.3% allowed Dao et al.[3] to construct afunction II1 independent of the strain hardening ex-ponent. The function II1 was constructed with theresults of a parametric study of 76 cases with variouselastic-plastic parameters representing the behavior oftypical engineering metals (Eq.(A1)). Recently, Bu-caille et al.[4] constructed the function II1 by a sim-ilar numerical study on 24 cases. Further, the au-thors showed that three times fewer simulations aresufficient to reproduce the results presented by Daoet al.[3] and constructed the function II1 for indenterangles 60, 50, and 42.3.

The function II1 for a equal to 0.3 and 0.58 areconstructed in the present study (see the appendixfor the list of functions). Figure 5 clearly shows the

influence of the choice of the representative strain onthe evolution of the function II1, for a=0.3. For eachvalue of a, the representative strain has been definedas a strain level where II1 can be constructed indepen-dently of the strain hardening exponent n: r=0.033[3]

for a=0 (Von Mise materials), r=0.0164 for a=0.3,

and r=0.0076 for a=0.58.The ratio C/r is higher for larger a. As E

/rincreases and reaches high values, C/r becomes con-stant. For each value ofa, the curve ofC/r vs E

/ris equivalent for elastic-work hardening materials[29].It includes three successive parts: a linear part (elasticbehavior: C/E=(2/)tan), a curved part (elastic-plastic behavior) and a horizontal line (full plastic-ity) as indicated in Fig.6. As a increases, the sizesof the linear part and the curve part increase andthe size of the fully plastic part decreases. Whena=0.58, the fully plastic part seems to disappear andthe elastic-plastic regime is dominant. Figure 7 shows

the evolution of C/E

vs E

/r. With increasing a,C/E increases and reaches a limit in values as 1.39,1.56 and 1.75 for a=0 (Von Mises materials), a=0.3and a=0.58, respectively. It can be concluded thatif C/E is higher than 1.4, the pressure-sensitivity isprobably involved.

Figures 6 and 7 show that the results for pressure-sensitive materials are limited by those for Von Misesmaterials as lower bound and the elastic model as up-per bound. With increasing a, the results for pressure-sensitive materials can approach the elastic model(C/E=1.78). Therefore, the material has more elas-tical behavior with increasing a. Thus, the representa-tive strain associated with the dimensionless function

II1=C/r decreases as a increases.

4.2 Average contact pressureTabor[1] used the slip-line theory to derive the rela-

tion between hardness and yield stress for rigid plasticsolids as follows:

H pave = C0Y (12)where C0 is a constraint factor (C0=3). For elastic-perfectly-plastic, Tabor[1] showed that C0 is around3. Based on experimental observations for Von Misesmaterials, earlier studies by Tabor[1] and Johnson[29]

showed that the hardness can be also estimated bythe following formula:

H pave = C0r (13)


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Fig.6 Variation of the dimensionless function II1 vsE/r with three different levels of the pressuresensitivity

Fig.7 Variation of C/E vs E/r with three differentlevels of the pressure sensitivity

Fig.8 Variation of the dimensionless function II2 vsr/E

with three different levels of the pressuresensitivity

where r is calculated at a representative strainr=0.08.

Experiments showed that the factor C0 in Eq.(12)

is about 4.5 for silica and glasses[30]

, about 8.2 forZrO2-containing ceramics[25] and in order of 20-30 for

cementitious materials[3133].

Fig.9 Variation ofpave/E vs r/E

with three differentlevels of the pressure sensitivity

To study the average contact pressure, the di-mensionless function II2=pave/r suggested by Daoet al.[3] was used as follows:

II2

Er

, n

=paver

(14)

The function II2 for different levels of pressure-sensitivity are constructed within the concept of rep-resentative strain in the present study as shown inFig.8. Similar to the procedure taken to obtain therepresentative strain for II1, it is found that for a=0(Von Mises materials), the results reported by Daoet al.[3] is reproduced with a representative strainof 8.2%, and r=0.068 for a=0.3, and r=0.048 fora=0.58 (see the appendix for the list of functions). It

is found that the representative strain decreases as aincreases.Equation (14) was used to consider the evolution

of the dimensionless function pave/E vs r/E

forvarious values of n ranging from 0 to 0.5 (Fig.9).The pressuresensitivity a increases pave/r as well aspave/E

. With increasing a, the dimensionless func-tion II2=pave/r increases. pave/r varies from 2.66to 4.20 and from 3.79 to 6.65 for a=0.3 and a=0.58,respectively. The ratio pave/E

increases with increas-ing r/E. However, pave/r decreases with increas-ing r/E

especially for high value of a.Similar to the observation on the loading cur-

vature C, results on the average contact pressure

for pressure-sensitive materials are limited by thosefor Von Mises materials as lower bound and theelastic model as upper bound as shown in Fig.8(pave/E

=1/2tan0.178[29]). With increasing a, theresults for pressure-sensitive materials tend to theelastic model. However, the upper bound is very farfrom the results for pressure-sensitive materials in thiscase.

5. Conclusion

The loading curvature C and the average contactpressure pave are studied within the context of repre-

sentative strain and dimensional analysis. The resultson these two parameters for pressure-sensitive mate-rials lie between those for Von Mises materials and


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the elastic model. C/E and pave/E increase with

increasing pressure sensitive level. While the pressuresensitivity increases, C/E for pressure-sensitive ma-terials can approach the elastic model (C/E=1.78).The representative strains associated with the twostudied dimensionless functions decrease with increas-ing the pressure sensitivity. Dimensionless functionsassociated with the two studied parameters are estab-lished for each studied value of the pressure sensitiv-

ity.

AcknowledgementThis work was partially supported by the Research

Program 2005 of Hanoi University of Technology, Viet-nam.

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[4 ] J.L.Bucaille, S.Stauss, E.Felder and J.Michler: ActaMater., 2003, 51(6), 1663.

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[6 ] Y.T.Cheng and C.M.Cheng: Mater. Sci. Eng.: R,2004, 44(4-5), 91.

[7 ] R.Hill: The Mathematical Theory of Plasticity,Clarendon Press, Oxford, 1950.

[8 ] W.Whitney and R.D.Andrews: J. Polym. Sci. Pol.Sym., 1967, 16C, 2981.

[9 ] S.S.Stenstein and L.Ongchin: Am. Chem. Soc. Poly.,1969, 10, 1117.

[10] S.Rabinowtz, I.M.Ward and J.S.C.Party: J. Mater.Sci., 1970, 5, 29.

[11] D.C.Drucker: Metall. Trans., 1973, 4, 667.

[12] J.A.Sauer, K.D.Pae and S.K.Bhateja: J. Macromol.Sci. Phys., 1973, 88, 631.

[13] W.A.Spitzig and O.Richmond: Poly. Eng. Sci., 1979,19, 1129.

[14] L.M.Carapellucci and A.F.Yee: Poly. Eng. Sci., 1986,26, 920.

[15] L.W.Chen: J. Am. Ceram. Soc., 1991, 74, 2564.[16] H.Y.Jeong and J.Pan: Int. J. Solids Struct., 1995, 32,

3669.[17] A.E.Giannakopoulos and P.L.Larsson: Mech. Mater.,

1997, 25(1), 1.[18] R.Vaidyanathan, M.Dao, G.Ravichandran and

S.Suresh: Acta Mater., 2001, 49(18), 3781.[19] M.N.M.Patnaik, R.Narasimhan and U.Ramamurty:

Acta Mater., 2004, 52(11), 3335.[20] P.E.Donovan: Acta Metall., 1989, 37(2), 445.[21] P.B.Bowden and J.A.Jukes: J. Mater. Sci., 1972, 7,

52.[22] R.Quinson, J.Perez, M.Rink and A.Pavan: J. Mater.

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gineers, Springer-Verlag, New York, 1988.[25] I.W.Chen and P.E.Reyes-Morel: J. Am. Ceram. Soc.,

1986, 69, 181.[26] W.A.Spitzig, R.J.Sober and O.Richmond: Acta Met-

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Trans., 1976, 7A, 1703.[28] A.J.Kinloch and R.J.Young: Fracture Behaviour of

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Appendix

Dimensionless function II1 for a=0 (Von Mises Materials, r=0.033):

II1 =C1

0.033= 1.131

ln E

0.033

3+ 13.625

E0.033

2 30.594

ln E

0.033

+ 29.267 (A1)

for a=0.3 (r=0.0164):

II1 =C1

0.0164= 2.9584

ln E

0.0164

3+ 41.6068

E0.0164

2 136.3520

ln E

0.0164

+ 147.2762 (A2)

for a=0.58 (r=0.0076):

II1 =C1

0.0076= 11.1418

ln E

0.0076

3+ 185.6190

E0.0076

2 879.5642

ln E

0.0076

+ 1353.2351 (A3)

Dimensionless function II2 for a=0 (Von Mises Materials, r=0.082):

II2 =pave

0.082= 15.4944

0.082E

2 15.1699

0.082E

+ 2.7497 (A4)

for a=0.3 (r=0.068):

II2 =pave

0.068= 210.9303

0.068E

2 31.3284

0.068E

+ 4.1581 (A5)

for a=0.58 (r=0.048):

II2 = pave0.048

= 373.59260.048E

2 96.68500.0648

E

+ 6.6378 (A6)

element+analysis+of+instrumented+sharp+indentations+into+pressure sensitive+materials

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