Analysis of indentation of pressure sensitive plastic solids using the expanding cavity model

Mechanics of Materials 36 (2004) 633–645

www.elsevier.com/locate/mechmat

Analysis of indentation of pressure sensitive plastic solidsusing the expanding cavity model

R. Narasimhan *

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India

Received 17 June 2003

Abstract

The objective of this paper is to apply the expanding cavity model to study the conical or spherical indentation

response of hydrostatic pressure dependent plastic solids. To this end, the elastic–perfectly plastic stress and dis-

placement fields in a hollow sphere subjected to internal pressure are first obtained by assuming that the material obeys

the Drucker–Prager yield criterion. Attention is focussed on the influence of the pressure sensitivity index of the ma-

terial on the above fields as well as the development of plastic flow in the sphere. These results are then employed to

extend the expanding cavity model of indentation to pressure dependent plastic solids. It is found that the size of the

plastically deformed region surrounding the indented zone enhances significantly with the pressure sensitivity index of

the material. Also, the mean contact pressure normalized by the compressive yield stress increases strongly when the

material exhibits a pressure dependent plastic behaviour.

� 2003 Elsevier Ltd. All rights reserved.

Keywords: Pressure dependence; Plasticity; Indentation; Expanding cavity model

1. Introduction

Indentation tests, which are non-destructive

and simple to perform, have been traditionally

employed to measure the hardness (Tabor, 1951;

Johnson, 1970, 1985). In these tests, a spherical,

conical or pyramidal indenter is pressed onto the

flat surface of a specimen and the mean contact

pressure pm (also referred to as the Meyer hardness(Fischer-Cripps, 2000)) is determined from mea-

* Tel.: +91-80-2932959; fax: +91-80-3600648.

E-mail address: [email protected] (R. Narasi-

mhan).

0167-6636/$ - see front matter � 2003 Elsevier Ltd. All rights reserv

doi:10.1016/S0167-6636(03)00075-9

surement of the contact load P and the size of theresidual imprint. The mean contact pressure re-quired to cause bulk yielding in an indentation test

is much larger than that measured in a uniaxial

compression test (denoted here by rco) because ofthe high constraint (hydrostatic pressure) gener-

ated by the elastically strained material sur-

rounding the plastic zone. Thus, for ductile metals,

both experiments and theory predict that pm ¼Crco, where the constraint factor C is around 3(Tabor, 1951).

In tests involving spherical indenters, valuable

information about the elastic and plastic proper-

ties of the material can be obtained by plotting pm(or ‘‘indentation stress’’) against the ratio a=R,

ed.

mail to: [email protected]

634 R. Narasimhan / Mechanics of Materials 36 (2004) 633–645

where a and R are the radius of the contact zone

and the indenter, respectively (Fischer-Cripps,

1997). The quantity a=R is called as ‘‘indentation

strain’’. Theoretical analyses employing various

models have been proposed to understand the

mechanics of plastic indentation and to ultimatelyinterpret the indentation stress versus strain re-

sponse. These models are based on rigid plastic

slipline theory (Hill et al., 1947), an expanding

cavity encapsulating the indented zone (Marsh,

1964; Johnson, 1970) and elastic displacements of

the material remote from the indented zone (Shaw

and DeSalvo, 1970). The present paper deals with

the application of the expanding cavity model toanalyze the indentation behaviour of pressure

sensitive plastic solids.

Samuels and Mulhearn (1957) noted that for

blunt indenters, the mode of deformation at the

fully plastic state involves compression of the

material rather than cutting as is the case with

sharp indenters (Hill et al., 1947). Marsh (1964)

compared this mode of deformation to that whichoccurs during the radial expansion of a spherical

cavity subjected to internal pressure. In this con-

text, it must be mentioned that the elastic–plastic

solution for an internally pressurised hollow

sphere had been obtained by Hill (1950) using the

Von Mises yield criterion. Marsh (1964) proposed

a relationship between the indentation pressure

and the ratio E=rco of the Young�s modulus to yieldstress of the material. However, Johnson (1970)

argued that the indentation pressure should de-

pend also on the indenter angle and hence incor-

porated a modification in the expanding cavity

model. He suggested that the material in the in-

dented zone can be viewed as being encased in an

expanding hemispherical core and enforced

compatability between the volumetric expansionof the core and the volume of material displaced

by the indenter.

The expanding cavity model involves certain

key assumptions such as a hemispherical shape for

the plastically deformed region surrounding the

zone of indentation and radial displacement of

material particles. Other shortcomings of this

model have been pointed out in the literature andsome remedial measures have been suggested

(Chiang et al., 1982; Ghosal and Biswas, 1993).

One of these defects pertains to the non-vanishing

normal traction outside the contact zone on the

specimen surface, which is an outcome of applying

Hill�s (1950) solution for a spherical cavity directlyto the indentation problem. Notwithstanding these

inherent assumptions and limitations, the ex-panding cavity model is still being employed to

interpret indentation data in ductile solids (see, for

example, Fischer-Cripps, 1997; Kramer et al.,

1999). The reason for the enduring popularity of

this model is its simplicity and ability to predict

with reasonable accuracy important experimental

results such as the indentation stress versus strain

curve (Johnson, 1970; Fischer-Cripps, 1997) andevolution of plastic zone size with load (Kramer

et al., 1999). Further, it must be mentioned that

the stress distribution beneath the indenter ob-

tained from finite element investigations (see, for

example, Sinclair et al., 1985) is quite similar to

that predicted by the simple cavity model. Also,

computational studies (Fischer-Cripps, 1997; Pat-

naik and Narasimhan, 2003) have shown that theplastic zone associated with spherical indentation

approaches a hemispherical shape (as assumed in

the cavity model) as Ea=ðrcoRÞ increases.In recent years, there has been a growing in-

terest in the indentation response of ceramics

(Zeng et al., 1996; Giannakopoulos and Larsson,

1997), bulk metallic glasses (Vaidyanathan et al.,

2001) and polymers (Briscoe and Sebastian, 1996;Briscoe et al., 1998), which are known to exhibit

hydrostatic pressure dependent plastic behaviour.

The Mohr–Coloumb and Drucker–Prager yield

functions have been suggested to represent the

yield behaviour of these materials (Giannakopo-

ulos and Larsson, 1997; Vaidyanathan et al., 2001;

Donovan, 1989; Bowden and Jukes, 1972; Quin-

son et al., 1997). Indeed, finite element analyses ofBerkovich and Vickers indentation of elastic–

plastic solids obeying the above yield criteria have

been undertaken by Giannakopoulos and Larsson

(1997) and Vaidyanathan et al. (2001) with the

view of examining the indentation response of ce-

ramics and bulk metallic glasses. These studies

show that the indentation load at a given inden-

tation depth increases with the pressure sensitivityindex of the material which in turn implies an

enhancement in the mean contact pressure. In

R. Narasimhan / Mechanics of Materials 36 (2004) 633–645 635

order to understand these observations, as well as

other related experimental results, it is desirable to

extend the cavity model to analyze the indentation

of pressure sensitive plastic solids.

In an early work, Chadwick (1959) derived the

stress and displacement fields prevailing around aspherical cavity in an infinite solid by employing

the Mohr–Coloumb yield criterion. Puttick et al.

(1977) attempted to incorporate linear dependence

of hydrostatic pressure as well as strain rate within

the framework of the expanding cavity model in

order to interpret data on mean contact pressure

obtained from spherical indentation tests on

PMMA. However, there are some deficiencies intheir work. First, the hydrostatic pressure em-

ployed in their paper in deriving the stress field for

an internally pressurised spherical cavity is incor-

rect. Consequently, their solution does not reduce

to that given by Hill (1950) in the limit of pressure

independent plastic behaviour. Secondly, the ex-

pression for the radial displacement at the cavity

surface was not obtained and employed in the in-dentation analysis in the spirit of Johnson (1970).

Finally, and most importantly, since the objective

of their work was restricted to interpreting the

mean contact pressure measured in indentation

tests on PMMA, no systematic study of the role of

pressure sensitive yielding on the indentation be-

haviour was conducted. The above considerations

provide the impetus for undertaking the presentinvestigation.

In this paper, the stress and displacement fields

in a hollow sphere subjected to internal pressure

are first obtained within the framework of the

hydrostatic pressure dependent (Drucker–Prager)

yield criterion. These fields are similar to those

presented by Chadwick (1959). Further, it is

shown that the stresses and displacement reduce inthe limit as the pressure sensitivity index of the

material vanishes to Hill�s (1950) solution for theVon Mises case. These results are then employed

to extend the expanding cavity model to analyze

the (conical or spherical) indentation response of

pressure dependent plastic solids. The influence

of pressure sensitive yielding on the development

of plastic flow around the indented zone and themean contact pressure is studied. Also, the stress

distribution beneath the indenter is examined.

2. Internally pressurised hollow sphere

In this section, the stress and displacement fields

prevailing in a hollow sphere subjected to internal

pressure are presented following Chadwick (1959).A small strain elastic–perfectly plastic constitutive

model is employed along with the Drucker–Prager

yield condition. The equation governing the

growth of the plastic zone in the hollow sphere

with internal pressure is derived.

A hollow sphere with inner radius a and outerradius b, subjected to internal pressure p is con-sidered. The radius of the plastic zone at any givenstage of loading is denoted by rp. The Drucker–Prager yield function is given by (Chen and Han,

1988):

UðrijÞ ¼ re þ rm tan a � 1�

� 13tan a

�rco ¼ 0; ð1Þ

where

rm ¼ 13rkk and re ¼

ffiffiffiffiffiffiffi3J2

pð2Þ

are the hydrostatic stress and Mises equivalent

stress, respectively. Further, a is a pressure sensi-tivity index and rco is the yield stress in uniaxialcompression. Due to spherical symmetry, the

stress state in spherical coordinates ðq; h;/Þ is

given by ðrq; rh; rhÞ. The yield condition (1) cor-responding to this stress state becomes:

ðrh � rqÞ þ 13ðrq þ 2rhÞ tan a ¼ 1

�� 1

3tan a

�rco:

ð3Þ

2.1. Elastic region

The stresses and radial displacement in theelastic region, rp6 q6 b, are described by (Hill,

1950):

rq ¼ �Ab3

q3

�� 1

�; rh ¼ A

b3

2q3

�þ 1

�ð4Þ

and

uq ¼ AE

ð1�

� 2mÞq þ ð1þ mÞb32q2

�; ð5Þ

where E and m are the Young�s modulus andPoisson�s ratio, respectively. From the requirement

that the above stress components must satisfy the


yield condition (3) at q ¼ rp, the constant A is

determined as

A ¼1� 1

3tan a

� �rco

32

brp

� �3þ tan a

: ð6Þ

2.2. Elastic–plastic region

The stress, strain and displacement fields in the

plastic zone, a6q < rp, are presented in this sec-tion. On using the yield condition (3) along with

the equilibrium equation,

drq

dq¼ 2ðrh � rqÞ

q; ð7Þ

it can be shown (see Chadwick, 1959) that the

stresses are given by:

rq ¼ �Drcorpq

� �2q

þ Bq

rco ð8Þ

and

rh ¼ Dðq� 1Þrcorpq

� �2q

þ Bq

rco: ð9Þ

Here, B, q and D are given by:

B ¼ ð1� tan a=3Þð1þ 2 tan a=3Þ ; ð10Þ

q ¼ tan að1þ 2 tan a=3Þ ; ð11Þ

D ¼ Bqþ ð1� tan a=3Þ

32

brp

� �3þ tan a

� � brp

� �3"

� 1

#: ð12Þ

It must be noted that in deriving the above stress

fields, the continuity of rq at q ¼ rp has been en-forced. From the boundary condition rq ¼ �p atq ¼ a, the relationship between the cavity pressurep and the plastic zone size rp can be expressed as:

p ¼ Drcorpa

� �2q� B

qrco: ð13Þ

An additive decomposition of the infinitesimal

strain components into elastic and plastic parts is

employed along with the associated flow rule.

Since the normals to the yield surface which are

given by:

gq ¼ �1þ tan a3

; gh ¼ g/ ¼ 1

2þ tan a

3ð14Þ

are constant (i.e., independent of p or rp), theplastic straining is proportional. Hence, the total

strain components can be written as:

�q ¼ �eq þ �pq ¼ 1

Eðrq � 2mrhÞ þ wgq;

�h ¼ �/ ¼ 1

Eðð1� mÞrh � mrqÞ þ wgh:

ð15Þ

In order to determine w, the compatability

condition,

d�h

dqþ ð�h � �qÞ

q¼ 0 ð16Þ

is employed. On substituting (15) along with (8)and (9) into (16), and noting that w ¼ 0 at q ¼ rp,it can be shown that w is given by:

w ¼ LrcoE

rpq

� �2q"

� rpq

� �s#; ð17Þ

where

L ¼ 2ð2q� 3Þð1� mÞð3� 4qghÞ

Dq and s ¼ 3

2gh

: ð18Þ

Finally, on using (15)2 along with the strain–

displacement equation �h ¼ uq=q, the radial dis-placement uq is obtained as:

uq ¼ rcoE

Mrprpq

� �2q�1"

� Lrpgh

rpq

� �s�1

þ ð1� 2mÞBq

q

#; ð19Þ

where

M ¼ Lgh þ ð1� mÞðq� 1ÞDþ mD: ð20Þ

It must be noted that rq, rh and uq are continuous

at the plastic zone boundary (q ¼ rp).

2.3. Stress and displacement fields for small a

The stresses and radial displacement given by

Eqs. (8), (9) and (19) reduce to Hill�s (1950)

0 0.2 0.4 0.6 0.8 1 0

1

2

3

4

5

uρ(a)/a

p/ σ oc

Von Mises = 0.1o

= 10o

= 20o

ααα

Fig. 1. Variation of normalized internal pressure with nor-

malized radial displacement of inner surface of a hollow sphere

having b=a ¼ 10. The yield strain in uniaxial compression and

Poisson�s ratio are taken as 0.02 and 0.36, respectively.


solution for the Von Mises case in the limit as

a ! 0. Indeed, on noting the following limiting

relations:

lima!0

rpq

� �2q

¼ 1þ 2q lnrpq

� �;

lima!0

D ¼ 1

aþ 1

31

�� 2

rpb

� �3�;

lima!0

L ¼ �2ð1� mÞ;

it can be shown that (8), (9), (19) reduce to the

following expressions:

lima!0

rq ¼ �2rco lnrpq

� �þ 1

31

�� rp

b

� �3�;

ð21Þ

lima!0

rh ¼ rco

� 2 ln

rpq

� �þ 1

31

�þ 2

rpb

� �3�;

ð22Þ

lima!0

uq ¼ rcoE

ð1"

� mÞrprpq

� �2

� 2ð1� 2mÞq lnrpq

� ��

þ 1

31

�� rp

b

� �3��#: ð23Þ

Further, in the limit as a ! 0, the relationship

between the applied pressure and plastic zone size

(Eq. (13)) becomes:

p ¼ 2rco lnrpa

þ 1

31

�� rp

b

� �3�: ð24Þ

It must be noted that Eqs. (21)–(24) coincide with

those given by Hill (1950) for the Von Mises case.

2.4. Results

In this section, selected results obtained from

the above solution for a hollow sphere with

b=a ¼ 10 are presented. The yield strain in uniaxial

compression rco=E and Poisson�s ratio m are as-

sumed as 0.02 and 0.36, respectively. The pressuresensitivity index a is chosen in the range from 0� to20�. These values are typically representative ofpolymer materials like PMMA or PS (Bowden and

Jukes, 1972; Quinson et al., 1997) and Pd-based or

Zr-based bulk metallic glasses (Donovan, 1989;

Vaidyanathan et al., 2001). Attention is focussed

in the following on the influence of a on the stressdistribution and evolution of plastic yielding in the

sphere.

The variation of normalized pressure, p=rco, withnormalized radial displacement of the inner surface

of the sphere, uqðaÞ=a, is shown in Fig. 1 pertainingto different a values as well as for the Von Misescase. It can be seen from this figure that as aincreases, a higher level of pressure needs to be

applied in order to attain a certain radial dis-

placement at the inner surface of the sphere. Thus,

for example, corresponding to uqðaÞ ¼ 0:3a, thenormalized pressure p=rco increases from 2.8 to 3.9

as a changes from 0� to 20�. In Fig. 2, the nor-malized displacement of the inner surface is plotted

against the normalized plastic zone radius rp=a. Itshould be noted from this figure that correspond-

ing to a given increment in uqðaÞ the plastic zoneradius grows by a larger amount when the sphere

displays pressure sensitive yielding. For example,when uqðaÞ changes from 0:2a to 0:3a, the plasticzone radius rp enhances by 0:35a if the sphere ex-hibits pressure independent yielding, whereas it

grows by 0:55a for the case a ¼ 20�. This obser-vation has an important bearing on the evolu-

tion of the plastic zone size during indentation

1 2 3 40

0.2

0.4

0.6

0.8

1

rp / a

u ρ(a

) / a

Von Mises = 0.1o

= 10o

= 20o

ααα

Fig. 2. Variation of normalized radial displacement of inner

surface of a hollow sphere (having b=a ¼ 10) with normalized

plastic zone radius. The yield strain in uniaxial compression and

Poisson�s ratio are taken as 0.02 and 0.36, respectively.


of a pressure sensitive plastic solid as will be seen

in Section 4.

The distribution of normalized stresses rq=rcoand rh=rco through the hollow sphere are shown inFigs. 3 and 4, respectively, corresponding to the

stage when rp=a ¼ 4. It should be first noted from

these figures that both rq and rh are compressive

with peak magnitude at the hole surface. By con-

trast, rh is tensile at the hole surface (q ¼ a), whenthe entire sphere is elastic. Indeed, it is found that

rh=rco at the hole surface is around +0.3 at incip-ient yielding (rp ¼ a), and changes sign when

1 2 3 4 5 6 -5

-4

-3

-2

-1

0

σ ρ/ σ

oc

ρ / a

rp / a = 4

Von Mises = 0.1o

= 10o

= 20o

ααα

Fig. 3. Distribution of normalized stress rq=rco through a hol-low sphere (having b=a ¼ 10) at the stage when rp=a ¼ 4.

rp=a � 1:2. It can be seen from Fig. 3 that the

magnitude of rq at the hole surface increases with

a which corroborates with the observation made

earlier in connection with the applied pressure (see

Fig. 1). Interestingly, the reverse trend applies for

q > 2a (or, equivalently for q > 0:5rp). In partic-ular, it can be deduced from Eq. (8) that at the

elastic–plastic boundary the radial stress is given

by:

rqðrpÞ ¼ �rcoð1� tan a=3Þ32

brp

� �3þ tan a

� � brp

� �3"

� 1

#;

ð25Þwhich decreases in magnitude with increase in

pressure sensitivity index a. The above equationcan be used to obtain an alternate estimate of the

plastic zone size from the contact load measured in

an indentation test as will be seen in Section 3.

Finally, it should be noted from Fig. 4 that rh

becomes tensile as the elastic–plastic boundary is

approached with a peak value at q ¼ rp. It there-after decreases in magnitude (although it continues

to remain tensile) and displays little sensitivityto a.In all the results presented in Figs. 1–4 it can be

seen that the curve pertaining to a ¼ 0:1� is vir-tually indistinguishable from that for the Von

Mises case which was analyzed by Hill (1950). This

is not surprising in view of the fact that reduction

of the pertinent equations to the Von Mises case in

the limit as a ! 0 has been established in Section2.3. This renders confidence to the analytical re-

sults presented in this section.

3. Expanding cavity model for indentation analysis

The expanding cavity model developed by

Marsh (1964) and Johnson (1970) assumes that thesubsurface displacements produced by any blunt

indenter are approximately radial from the point

of first contact and the plastic strain contours are

hemispherical in shape. The contact surface is

taken to be encapsulated in a hemispherical core

of radius a (see Fig. 5). It is further assumed thata hydrostatic compressive stress of magnitude p

p

rp

a a

R

Core

da

z

r

Plastic Zone

Indenter

dr

a

Fig. 5. Schematic illustrating elastic–plastic indentation as idealized by the expanding cavity model. The contact zone is taken to be

encased in a hemispherical core of radius a, which in turn is surrounded by a hemispherical plastic zone of radius rp.

1 2 3 4 5 6

-2

-1

0

σ θ/σ o

c

ρ / a

Von Mises

α = 0.1o

= 10o

= 20o

rp / a = 4

α

α

Fig. 4. Distribution of normalized stress rh=rco through a hollow sphere (having b=a ¼ 10) at the stage when rp=a ¼ 4.


prevails inside the core (q6 a). The stresses anddisplacement outside the core (q > a) are taken tobe the same as that in an infinite elastic–plastic

body containing a spherical cavity under pressure

p. Thus, within the plastic zone (a < q6 rp), thestresses and displacement will be described by

Eqs. (8), (9) and (19) with b=rp ! 1 so that Dis modified as follows:

D ¼ Bqþ 2

3ð1� tan a=3Þ: ð26Þ

In the elastic region (q > rp), the stresses are givenby (see Eq. (4)):

rq ¼ � 23rcoð1� tan a=3Þ rp

q

� �3

and

rh ¼1

3rcoð1� tan a=3Þ rp

q

� �3

: ð27Þ

The relationship between the core pressure pand the plastic zone radius rp is governed by Eq.(13) with D defined according to Eq. (26). Finally,


the cavity model of indentation assumes that the

radial displacement of particles lying on the core

boundary (q ¼ a) during an increment of pene-

tration accomodates the volume of material dis-

placed by the indenter. Thus, it follows that for a

conical indenter with semi-angle of (p=2� b),

2pa2 duqðaÞ ¼ pa2 da tan b: ð28ÞOn using Eq. (19) in (28), the evolution equation

for the plastic zone size rp with contact radius a isobtained as:

rcoE

2qMrpa

� �2q�1� 3L2

rpa

� �s�1drpda

¼ tan b2

:

ð29Þ

It can be shown along the lines of Section 2.3 thatin the limit as a ! 0, the above equation reduces

to that derived by Johnson (1985), which is given

by:

rcoE

3ð1

� mÞ rpa

� �2� 2ð1� 2mÞ a

rp

drpda

¼ tan b2

:

ð30ÞAs suggested by Johnson (1985), for the case of a

spherical indenter, tan b in the above equationscan be replaced by a=R, where R is the indenter

radius. The above equations can be integrated

numerically from the contact radius corresponding

to initial yielding, ay , which is determined in Sec-tion 4, and the plastic zone size rp can be obtainedas a function of a. The core pressure p can then becomputed from Eq. (13).

An alternate estimate of the plastic zone radiusrp can be obtained from the normal load P mea-

sured in an indentation test by considering the

force equilibrium of the hemispherical region un-

dergoing plastic deformation (Kramer et al.,

1999). Thus,

P ¼ �pr2prqðrpÞ; ð31Þ

which is somewhat similar to that considered byFlamant (see, for example, Timoshenko and

Goodier, 1951), when establishing equilibrium for

the inwardly directed forces in the line (elastic)

contact problem. On substituting for rqðrpÞ fromEq. (25) (with b=rp ! 1) into the above equation

it follows that

rp ¼3P

2prcoð1� tan a=3Þ

1=2; ð32Þ

which coincides with the relation given by Krameret al. (1999) when a ¼ 0. Indeed, experimental and

computational studies (Bahr and Gerberich, 1996;

Kramer et al., 1999) have demonstrated that the

above relation (with a ¼ 0) provides a good ap-

proximation of the plastic zone size for pressure

independent plastic solids. Eq. (32) shows that for

a given indentation load P , the plastic zone radiuswill increase with the pressure sensitivity index a.It must be noted that the stress state immedi-

ately beneath the indenter will not be purely hy-

drostatic. As proposed by Johnson (1985), the

stresses below the indenter may be approximated

with reference to a cylindrical coordinate system

ðr; h; zÞ centered at the point of first contact (seeFig. 5) as follows:

rzz ’ �ðp þ rÞ; rrr ¼ rhh ’ �ðp � r=2Þ: ð33ÞOn substituting this assumed form for the stress

components into the yield condition (1), r is de-

termined as:

r ¼ 2rco3

1

þ tan a

prco

�� 1

3

�; ð34Þ

where the core pressure p is obtained as indicatedabove.

4. Results and discussion

In this section, some results obtained by ap-

plying the cavity model developed in Section 3 to

analyze the mechanics of indentation of pressure

sensitive plastic solids are presented. Attention isrestricted to spherical indentation only (with ind-

enter radius denoted by R), although similar ob-servations are expected to apply for conical

indenters as well. First, the stage at which yielding

commences in the region underneath the indenter

is determined. The evolution of the plastic zone

size and mean contact pressure during the inden-

tation process are examined. Finally, the stressdistribution beneath the indenter, as predicted by

the cavity model, is studied. The material proper-

ties such as rco=E, m and range of a are the same as

0 10 20 300

20

40

60

E a / (σoc R)

E r p

/ (σ

oc R)

Von Misesα = 0.1o

α = 10o

α = 20o

Fig. 6. Variation of normalized plastic zone size with normal-

ized contact radius corresponding to spherical indentation.


mentioned in Section 2.4. In presenting the results

below (in particular, the stress distributions), a

cylindrical coordinate system ðr; h; zÞ centered at

the point of first contact (see Fig. 5) will be em-

ployed. The discussion will focus on the role of

pressure sensitive yielding of the indented material.The stage at which yielding commences in the

indented material is determined by substituting the

elastic Hertz solution into the yield condition (1).

Thus, on employing the Hertz solution for spher-

ical indentation (Johnson, 1985), it is found that

initial yielding occurs at a certain depth z along theindenter axis. The normalized contact radius and

indentation load, Eay=ðrcoRÞ and Py=ðrcoR2Þ, at thisstage are summarized in Table 1 along with the

normalized depth jzj=ay . Results pertaining to

different levels of pressure sensitivity are presented.

It can be seen from this table that as a increasesfrom 0.1� to 20� there is a marginal increase in thecontact radius and load at initial yield. Further,

yielding commences at a slightly greater normal-

ized depth jzj=ay along the indenter axis when thematerial exhibits pressure sensitive yielding.

As mentioned earlier, the evolution of the

plastic zone size rp with contact radius a can beobtained by numerically integrating the differential

equation (29) or (30) (with tan b replaced by a=R)from the stage of initial yielding, a ¼ ay . A simple

Runge–Kutta algorithm is employed for this pur-

pose. The normalized plastic zone size Erp=ðrcoRÞ isplotted against the normalized contact radius

Ea=ðrcoRÞ in Fig. 6 corresponding to different val-ues of a as well as for the Von Mises case. It mustbe noted that the quantity Ea=ðrcoRÞ is a measureof the indentation strain normalized by the com-

pressive yield strain (Fischer-Cripps, 2000). It can

be seen from Fig. 6 that the size of the plastically

deformed region around the indented zone grows

Table 1

Values of normalized contact radius, indentation load at initial

yield and the location jzj=ay along the indenter axis at whichyielding commences corresponding to different a

a Eay=ðrcoRÞ Py=ðrcoR2Þ jzj=ay0.1� 2.31 0.0076 0.5

10� 2.46 0.0091 0.56

20� 2.63 0.011 0.62

faster with respect to the contact radius as the

pressure sensitivity of the material enhances. Thus,

for example, corresponding to Ea=ðrcoRÞ ¼ 30, the

plastic zone size increases by about 18.5% as achanges from 0� to 20�. On keeping in view Eq.(28), the above key result is attributed to the larger

growth in the plastic zone radius corresponding to

a given increment in the radial displacement of the

core boundary when a is higher (see Fig. 2 and

discussion in Section 2.4).

The variation of normalized core pressure,

p=rco, and mean contact pressure, pm=rco as given

by Eq. (33)1, with indentation strain are shownin Figs. 7 and 8, respectively. In Fig. 7, the core

pressure is plotted from the stage of initial yield-

ing, a ¼ ay , whereas in Fig. 8 the variation of themean contact pressure with Ea=ðrcoRÞ in the elasticregime (a < ay) is also included by using the Hertzsolution (Johnson, 1985). The latter is commonly

referred to as the indentation stress versus strain

curve (Fischer-Cripps, 2000). Experimental datareported by Fischer-Cripps (1997) for mild steel

and a mica containing glass–ceramic, as well as

those presented by Briscoe and Sebastian (1996)

for PMMA are also shown in Fig. 8. The Young�smoduli for mild steel and the glass–ceramic are 210

and 64 GPa, whereas, their yield strengths are 385

and 770 MPa, respectively (Fischer-Cripps, 1997).

The compressive yield strength for PMMA isaround 110 MPa at 20 �C (Bowden and Jukes,

0 10 20 300.5

1

1.5

2

2.5

E a / (σoc R)

p / σ

oc

Von Mises = 0.1o

= 10o

= 20o

ααα

Fig. 7. Variation of normalized core pressure with normalized

contact radius (or indentation strain) corresponding to spheri-

cal indentation.

0 10 20 300

1

2

3

4

E a / (σoc R)

p m /

σ oc

Von Mises = 0.1o

= 10o

= 20o

PMMA Mild steel Glass ceramic

Experimental data

ααα

Fig. 8. Variation of mean contact pressure (or indentation

stress) with normalized contact radius (or indentation strain).

Predictions based on the cavity model corresponding to differ-

ent a are shown along with experimental data for PMMA

(Briscoe and Sebastian, 1996), mild steel and a glass–ceramic

(Fischer-Cripps, 1997).


1972; Quinson et al., 1997) and its Young�s mod-ulus is about 3.8 GPa (Briscoe and Sebastian,

1996).

It can be seen from Fig. 7 that while the core

pressure is fairly independent of a immediately

following commencement of yielding beneath the

indenter, it increases significantly with a at later

stages of indentation. This is traced to the larger

plastic zone size as well as internal pressure asso-

ciated with the expansion of a spherical cavity

when a is higher (see Figs. 1 and 2). The above

trend is also reflected in Fig. 8 in the variation ofmean contact pressure with indentation strain. On

comparing Figs. 7 and 8, it can be noticed that the

elevation of the mean contact pressure with a is

more pronounced than that of the core pressure.

This added enhancement in pm occurs due to the

increase in the magnitude of the superposed com-

pressive axial stress r in the region beneath the

indenter with a (see Eq. (34)). For example, atEa=ðrcoRÞ ¼ 30, pm=rco increases from 2.55 to 3.5 as

a changes from 0 to 20�.It can be observed from Fig. 8 that the experi-

mental data for mild steel follow the curve corre-

sponding to the Von Mises case quite closely. By

contrast, the normalized indentation stress–strain

data for both PMMA and the glass–ceramic are

higher than the prediction based on the Von Misesyield criterion and lie between the curves pertain-

ing to a ¼ 10� and 20�. As already mentioned,

ceramics and polymers are known to exhibit

pressure sensitive yield behaviour (Heard and

Cline, 1980; Quinson et al., 1997). In fact, the ex-

perimental results of Bowden and Jukes (1972) and

Quinson et al. (1997) suggest a value of a of

around 20� for PMMA. Thus, it is clear from theabove discussion that the indentation stress versus

strain curve will be elevated due to pressure

sensitivity of the material. Further experimental

support for this conclusion is provided by micro-

indentation test results on various bulk metallic

glasses reported by Sargent and Donovan (1982)

which show values of hardness exceeding three

times the yield strength. Also, this corroborateswith the higher indentation load computed in finite

element studies (see, for example, Giannakopoulos

and Larsson, 1997) for pressure dependent plastic

solids. Finally, it must be observed from Figs. 6–8

that the solution obtained from the present anal-

ysis reduces to that given by Johnson (1985) for

the Von Mises case in the limit as a ! 0.

The variations of the normalized cylindricalstress components rzz=rco and rrr=rco with nor-

malized distance z=a along the indenter axis are

0 1 2 3 4

-4

-3

-2

-1

0

z/a

_σzz / σoc

Von Mises

α = 0.1o

= 10o

= 20oα

α

Fig. 9. Variation of the normalized cylindrical stress compo-

nent rzz=rco with normalized distance z=a along the indenter axiscorresponding to Ea=ðrcoRÞ ¼ 30.

0 1 2 -3

-2

-1

0

z/a

- σrr / σoc

Von Mises

α = 0.1o

= 10o

= 20oα

α

Fig. 10. Variation of the normalized cylindrical stress compo-

nent rrr=rco with normalized distance z=a along the indenter axiscorresponding to Ea=ðrcoRÞ ¼ 30.


displayed in Figs. 9 and 10, respectively, corre-

sponding to Ea=ðrcoRÞ ¼ 30. At this stage, the

plastic zone size rp is between 1.84 and 2:18a, de-pending on the value of a (see Fig. 6). It can beseen from Figs. 9 and 10 that in the region im-

mediately below the indenter, both rzz and rrr are

compressive and increase in magnitude with a.This trend is particularly pronounced for the axial

stress (see Fig. 9). The stresses decay strongly in

magnitude with distance for jzj > a. Pressure sen-sitivity enhances the magnitude of rzz at a given

z=a in this region as well. It is interesting to notefrom Fig. 10 that rrr becomes mildly tensile as the

plastic zone boundary is approached and then

decays with further increase in distance. In con-

trast to the above observations, the peak value of

this tensile radial stress decreases by about 16% as

a changes from 0� to 20�. A similar trend is dis-

played in the radial variation of the circumferen-

tial stress rhh along the surface of the indentedmaterial (z ¼ 0). The occurrence of this tensile

stress suggests that radial cracks may initiate on

the specimen surface at the plastic zone boundary

and propagate outwards into the elastically

strained region. Such fracture patterns have indeed

been observed in indentation experiments on some

polymeric materials such as PMMA which are

prone to crazing followed by brittle cracking when

subjected to tensile stress (Bowden and Jukes,1972; Puttick et al., 1977).

5. Conclusions

The main conclusions of this work are listed

below.

1. A higher level of internal pressure needs to ap-

plied in order to expand the spherical cavity as

the pressure sensitivity index a of the materialincreases. Also, the growth of the plastic zone

associated with a given expansion of the cavity

is larger when the sphere exhibits a pressure de-

pendent plastic behaviour.

2. As a consequence of the above, a significantenhancement in the size of the plastically


deformed region surrounding the indented zone

occurs with increase in a at a given contact ra-dius. An alternate estimate of the plastic zone

radius, rp, obtained by force equilibrium of

the plastically deformed region, also predicts asimilar behaviour.

3. The normalized mean contact pressure, pm=rco,experienced by the indented material increases

significantly with a. This is caused by the com-bined effect of a on both the core (hydrostatic)pressure as well as the superposed compressive

axial stress r acting on the indented zone. Re-

cent experimental data for a glass–ceramic andPMMA (Fischer-Cripps, 1997; Briscoe and Se-

bastian, 1996) corroborate with the above ob-

servation.

4. All stress components in the region beneath the

indenter are compressive and increase in magni-

tude with a. The circumferential stress on thespecimen surface (or the radial stress along the

indenter axis) attains a tensile peak at the elas-tic–plastic boundary. Although there is a drop

in the magnitude of this peak with increase in

a, it suggests that radial cracks may propagateoutward from the plastic zone boundary on

the specimen surface. This has been observed

in indentation tests on certain polymeric materi-

als like PMMA.

In closing, it must be mentioned that the de-

pendence of mean contact pressure and plastic

zone size on both the yield strength as well as the

pressure sensitivity index of the material suggest

that these key material properties can be obtained

from experimental data. Indeed, in recent experi-

mental investigations, atomic force microscopy

has been employed to determine the plastic zonesize around nanocontacts. These measurements

when combined with theoretical predictions based

on the cavity model, have provided reliable esti-

mates of the yield strength for various alloys

(Kramer et al., 1999). The results obtained in this

paper suggest that additional properties such as

the pressure sensitivity index for new materials like

bulk metallic glasses can be determined from depthsensing indentation tests like those reported re-

cently by Vaidyanathan et al. (2001).

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Documents

Analysis of indentation of pressure sensitive plastic solids using the expanding cavity model