determination of mechanical and rheological properties of ... · 37 40 48 53 42 45 50 47 32 50...

35 Dr. / Prof. Dharmendra C. Kothari, Prof. / Dr. Paul Luckham, Prof. / Dr. Christopher .J. Lawrence

International Journal of Computer & Mathematical Sciences

IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

Determination of Mechanical and Rheological Properties of

Cheese by Indentation

Dr. / Prof. Dharmendra C. Kothari,

Department Chemical Engineering &

Technology, Shri Shivaji Education

Society Amravati’s, College of

Engineering & Technology,

Babhulgaon (Jh.), NH 6, Dist.

AKOLA, M.S. INDIA.

Prof. / Dr. Paul Luckham, Professor of Particle Technology,

Department of Chemical Engineering

and Chemical Technology, Imperial

College, London,

SW7 2BY, UK,

Prof. / Dr. Christopher .J. Lawrence,

Department of Chemical Engineering

and Chemical Technology,

Imperial College, London,

SW7 2BY, UK,

ABSTRACT

The mechanical properties of five commercial cheeses

(Mozzarella, Brie, Shropshire Blue, Red Leicester and

Pecorino Romano) are reported based upon studies

using indentation experimentation involving loading-

unloading, and stress relaxation. The conventional

indentation hardness values ranged from “hard” to

“soft” in the order Pecorino Romano, Red Leicester,

Shropshire Blue, Brie and Mozzarella. The hardness is

universally proportional to the moisture content. The

elastic behaviour is characterised by the elastic modulus,

whereas the elasto-plastic behaviour is presented as a

power law with material index n and consistency k. The

elastic modulus of cheeses were found to be proportional

with the hardness values. The general stress relaxation

response approximately followed a power law viscous

relationship; nt kh , where h is the hardness and t is

lapsed time, and k and n are the intercept and power law

indices over a range of five decades in strain rate. The

values of flow index n are seen to be decreasing with the

hardness, whereas the consistency factor k increases

with the hardness values. However, it was observed that

the stress relaxation analysis appears to be sensitive to

the physical structure of the material.

Keywords: Cheese, Elasto-ViscoPlastic, Indentation,

Cone, Sphere.

1. INTRODUCTION

The process of cheese making is an ancient craft

that dates back thousands of years. By even today’s

standards of industrial technology, the process is

still a complicated one, which combines both “Art”

and “Science”, (Ranken, 1989). Cheese is a very

complex, physical, bio-chemical and

microbiological system. It can be considered as a

composite, or a filled gel, in which fat globules and

empty pockets or holes act as the filler entrapped in

a three-dimensional casein network. In general, the

mechanical properties of filled gels are influenced

by the structure of the network and the shape, size,

volume fraction, and deformability, of the filler

particles, as well as the extent and nature of the

filler interactions with the surrounding network,

(Malin and Tunick, 1995).

Most of the cheese cutting systems are empirically

designed and operated. Many cheeses systems use

technology developed in other and often require

pre-processing to give it required properties. This

added pre-processing is costly and time consuming,

and if not carried out correctly can result in serious

reductions in quality, yield and throughput of cut

product. Recently, effective studies of cheese

cutting with wire is carried out by (Kamyab, et. al.,

1998). To optimized the effective processing of the

cheeses it is essential to study its rheological

properties in which hardness is most important.

Hardness which may be defined in many ways is a

significant characteristic of a cheese, which is

important to both consumers and manufacturers.

Hardness clearly corresponds to a mechanical or

rheological response whose attributes will reflect

plastic flow, elasticity, creep and further processes.

It has long been realized that composition and

manufacturing variables, storage temperature and

time, influence the perceived rheological properties

of a cheese such as its firmness, springiness,

stickiness, crumbliness, greasiness, cohesiveness,

and viscosity. The rheological and fracture

properties of many types of cheese change

significantly during maturing mostly because of

protein hydrolysis (Masi, 1989). Therefore, it is

important for the food industry to have a good

understanding and knowledge of the rheological

properties of cheese and to interactive the subjective

assessment criteria with the successfully defined

mechanical characteristics.

MATERIALS

There are more than 2000 varieties of cheese

available around the world; however they are often



IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

classified into twelve basic categories. Of these

twelve, three main categories have been selected for

the present rheological studies. Five different

commercial cheeses, Pecorino Romano, Red

Leicester, Somerset Brie, Mozzarella (full fat) and

Shropshire Blue were selected on the basis of their

common texture classifications; “very-hard”,

“hard”, and “soft” as well as their moisture and fat

contents, Table [1]. One blue-veined cheese was

selected for the purpose of investigating the effects

of bacteria upon the rheological properties of

cheese.

The Pecorino Romano is an extremely hard cheese.

It is usually grated before being consumed and it

has a strong aroma. Red Leicester is a hard-pressed

cheese manufactured in a similar way to Cheddar up

to the stage of the handling of the curd. The

maximum scald temperature is 370 C and after whey

drainage, the curd is cut into small blocks and

turned at intervals for approximately 1 hour until

the acidity has reached 0.6%. The curd is then

milled, salted, filled into moulds and pressed. The

ripening period is two to three months. Red

Leicester has an open, flaky texture and a mild

mellow flavour. Its characteristic red colour derives

from the use of annatto. Somerset Brie is a soft,

creamy cheese, with a mild, aromatic flavour and a

close, smooth texture. It is manufactured in such a

way that the Brie is covered with a thin

distinguishable white mould-ripened surface layer.

Mozzarella is a soft, un-ripened plastic curd (pasta

filata) cheese. Mozzarella is well known for its

particular use as a cheese for pizza toppings where

its thermo-plasticity gives a characteristic

“stringiness” on heating. The protein network

within the cheese contributes to its thermoplastic

properties. Mozzarella has a soft, waxy body and a

mildly acid flavour. Shropshire Blue has a porous

texture, which allows air to penetrate into the

cheese and enhance the growth of the Penicillium

roqueforti bacteria.

Name Cheese

Type

Moisture

(%)

Fat

(%)

Pecorino Romano

Red Leicester

Somerset Brie

Mozzarella

Shropshire Blue

Very Hard

Hard

Soft

Soft

Blue-

veined

37

40

48

53

42

45

50

47

32

50

Table [1]: - Classification of the five different

cheeses (Approximate Composition, from Potter

and Hotchkiss, 1996).

Indentation responses of Elasto-Viscoplastic

contact

The objective of this paper is to assess the reliability

and reproducibility of indentation measurements

which may be used to characterise the flow

properties of various cheeses. In order to

adequately characterise the material properties, two

distinctly different protocols were followed and are

described. A schematic representation of the two

approaches and their corresponding methodologies

is illustrated in Figure [1]. This figure illustrates the

various indentation histories that may be imposed

on the cheese specimen, such as creep, relaxation,

and constant load rate or constant displacement rate

(Hill, 1992). In the first method, the material is

indented to a prescribed depth at a constant

penetrating velocity and then the direction of

motion is reversed, without any significant dwell

time; this method is well known as “loading and

unloading” (Johnson, 1970). In the second

procedure, once the indenter has reached the set, it

is left in position to sense the “stress relaxation” of

the specimen. Both of these procedures are the

principle flow forms of displacement-controlled

indentation (Briscoe and Sebastian, 1996).

Fig. [1]:- Diagram of experimental protocols, Load

L or Displacement h against time t.

The indentation method adopted provides a contact

compliance characteristic. The indenter moves into

the surface and the reactive load is sensed. For a

wholly elastic cheese, the loading and unloading

curves are identical and there is no stress relaxation.

Conversely, when the deformation is rigid-perfectly

plastic in nature, there is no elastic recovery in the

unloading curve; there is also no stress relaxation.

The nature of the unloading portion is naturally

governed by elastic properties of the material

although various time dependent components may

be evident (Adams and Briscoe, 1993). The elastic

relaxation component associated with regions

outside the contact zone predominates in the initial

portion of the unloading curve, which is

approximately linear. The elastic contribution is



IJCMS

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April 2016

manifested towards the final part of the loading

curve, when the surrounding surfaces has released

to the planer to be is usually assumed to be due to

the recovery of the actual deformation zone itself.

As a consequence, neither the depth of penetration

at the maximum load, ht, nor the residual depth of

penetration, hr, shown in Figure [2], provide

directly a means to compute the radius of the

plastically deformed contact region from the

compliance curves.

Figure [2]: - Schematic diagram of the contact

compliance curve: reaction force vs imposed

displacement.

The hardness (plastic component), H, is commonly

defined as

2a

L

A

LH

(1)

where L is the reaction force, A is the contact area,

projected onto the original surface, and a is the

corresponding radius for axially symmetric

indentations. The actual contact radius is

distinguished from the apparent radius of contact,

which corresponds to the value computed from the

total depth of penetration, ht. The means of

computing the hardness for various geometries is

discussed later.

The reduced elastic modulus, E*, is conventionally

obtained from the elastic contact stiffness of the

initial part of the unloading curve, Figure [2];

aESh

L *2

(2)

where )1( 2

*

EE , E is the Young’s modulus,

is Poisson’s ratio and S is the contact stiffness

upon unloading at the maximum penetration depth,

ht.

Evaluation of hp

A means of obtaining the plastic component of the

indentation depth hp, derived to compute the

hardness, H, is to extrapolate the initial linear

unloading portion of the load-displacement curve to

the zero-load axis as shown in Figure [2]. This

method was derived based upon the assumption that

the elastic deformation of the indented material

around the indent may be modelled as that

corresponding to the elastic indentation by an

equivalent cylindrical punch, having an identical

contact radius to that of the imposed indentation.

The computation is invariably subjective. A more

effective procedure for obtaining the value of hp

from the unloading curve is the adoption of an

appropriate curve-fitting method for the

experimental compliance data in order to compute

the intercept, hp.

The Box-Cox transformation method has the

advantages for its purposed and h is described in

detail by Briscoe and Sebastian (1996). The curve

fitting method adopted is of the form;

nhh kL )( 0 (3)

where h0 is the particular zero offset, k is a

coefficient and n is the characteristic load index.

For the loading curve, the equation can be

expressed as

1)( 011

nhhkL (3a)

where subscript 1 indicates loading.

For the unloading curve, the equation can be

expressed as;

2)(22

n

rhhkL (3b)

where subscript 2 indicates unloading.

The computed values of h0 and hr provide the

required numerical values of the zero points for the

loading and unloading curves of the indentation.

The magnitude of the parameter k depends upon the

material properties and also upon the actual indenter

geometry. The value of the index n depends upon

the mode of deformation and also upon the

geometry of the indenter. The value of n2 can be

evaluated by a simple linear regression of log (L) as

a function of log (h-hr). Differentiation of equation

(3b), substitution for Lt, and the simultaneous

solution of the resulting equation relationship with

equation (3b) give;

2nhh

hh

pt

rt

(4)

Contact radius

The computed contact radius is used to evaluate the

hardness of the material, as represented in Figure

[3], in which the cone indentation is represented in

Figure [3a] and the sphere in Figure [3b]. From a



IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

consideration of the geometry of the indenter used,

the contact radius is given approximately:

Fig. [3a]:-The CONE

indenter geometry.

Fig. [3b]: - The

SPHERE indenter

geometry.

Figure [3]: - Schematic diagram of the indenter

geometries. No surface relaxation on material

displacement is indicated.

for the cone, as: a = h tan (5)

and for the sphere as: a = 22 hRh (6)

where is the semi-included angle of the cone, R is

the radius of the sphere and h is the depth of

indentation, which can be represented as h = V (t-

t0), V is the penetration velocity, t0 is the time zero

offset and (t-t0) is the apparent time of the contact

between the indenter and the specimen. The contact

radius given by (5) and (6) does not account for the

“pile-up” or “sinking-in” of the deformed material

around the impression. To evaluate stress relaxation

the constant value of a corresponds to the last point

of loading, where the loading curve transfers to the

relaxation curve.

Plasticity index

A plasticity index, , is a parameter which

characterises the relative plastic and elastic

behaviour of the material under the action of

prescribed external strains. In the case of

indentation, the plasticity index can be usefully

expressed as the ratio of the elastic component of

the work done to the plastic component of the work

done, (Bower et. al, 1993). Using the parametric

relationship for the loading curves, the plasticity

index can be approximated in terms of the hardness

and the reduced elastic modulus.

H

E*

tan (7)

where is the angle of inclination of the indenter to

the sample surface and E* is again the reduced

elastic modulus. The effective contact strain is taken

as 0.2 tan (Briscoe and Sebastian, 1996).

For the cones the angle is not a function of depth

for large penetrations, the indentations are said to be

“self similar”. In the case of the sphere the

corresponding angle decreases with the depth of the

penetration; at a given depth, h, the strain in the h/R,

where R is the spheres values. Thus the effective

strains are constant for cone ~ 0.2 tan and are

typically ~ 0.35 for the sphere.

Stress Relaxation

An elasto-viscoplastic model for indentation is

shown in Figure [4], which comprise of the

conventional elastic and plastic descriptors. The

first term on the right-hand side of the equation

(Figure 4) represents the elastic component of the

strain, which responds instantaneously to the

applied stress, whilst the second term expresses the

viscous component, which depends on the duration

of the stress.

Figure [4]:- Schematic representation of a non-

linear Maxwell model for elasto-viscoplastic

materials.

If the stress is removed after some time, the spring,

no longer constrained, returns to its original length

immediately, but that part of the displacement due

to the dashpot creep remains. Extensive work has

been performed to characterise both the elastic and

viscous properties of soft-solid materials, using a

simple wedge-indentation stress relaxation method,

by Lawrence et. al, (1998).

In a general case, it is expected that the imposed

strain and strain rate will induce both elastic and

plastic contributions to the deformation, in different

proportions throughout the material. As long as a

material is uniform, it can be expected that the

nominal stress, , is related to some nominal strain

rate .

by a simple relationship.

)(

FCP . (8)

where CP is the plastic constraint factor ( a term to

include geometric factors and interface friction) and

F is a non-linear material function.

Similarly, for an essentially elastic response to

indentation, the nominal stress would be related to

some nominal strain by a simple relationship.



IJCMS

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Volume 5, Issue 4

April 2016

*ECE (9)

where CE is the elastic constraint factor which also,

like CP, embodies geometrical and frictional effects,

E* is an effective elastic modulus.

It is supposed that, during the indentation phase, the

elastic component of the strain is small, so that an

approximate relationship for the mean indentation

pressure is obtained as:

a

V FC

A

LP (10)

where V is the indentation velocity.

However, during the relaxation phase, the elastic

stress is equally important, and a relationship for the

indentation pressure is given as:

*

.

EC A

P FC

A

L

E

P (11)

where L is the load registered, and

.

L is the rate

of relaxation of the load. The equation (11) provides

quantitative values of the power law index q and the

intercept k, (Holdsworth, 1993).

q Lk

.

(12)

Equation (9) will be used in the stress relaxation

analysis in order to characterise elastic properties.

By the form of equations (10) and (11), it is noted

that plotting the mean indentation pressure against

nominal strain rate for the indentation phase, all the

different loading curves for a given material should,

in principle, collapse into a single master curve.

Similarly, a plot of the indentation pressure against

the rate of decrease of the load in the relaxation

curves phase for all different velocity curves should

also collapse upon a single master curve.

Furthermore the two master curves should have the

same shape. Subsequently, by choosing an

appropriate value of CEE*, the two sets of data may

be combined to give an overall single master curve.

The flow consistency and the material index are

then obtained from the equation of the master curve.

EXPERIMENTS

Sample Preparation

All the cheeses were removed from their

packages just before the experiment. In order to

prevent significant drying and consequent

rheological changes, once a cheese was un-

wrapped, the experiments were carried out within a

period of two hours. All the cheeses were cut into

blocks of 150 mm length, by 70 mm width by 70

mm height. The ambient temperature was in the

range 20-240C.

Figure [5]:- Schematic diagram of the experimental

set-up.

Description of the Apparatus Used

The principal device used was a standard universal-

testing machine (Instron 1122, Instron Ltd. High

Wycombe, UK), equipped with a 50N transducer

(bottom load cell) of accuracy 0.01N. A schematic

diagram of the experimental arrangement is shown

in Figure [5].

The instrument incorporates an electronic data

collection system with load cells detecting the load

applied to the specimen. Two geometries of

indenter were used: a stainless steel cone of

included angle 90o and a sphere of diameter 12.7

mm (1/2”). The specimens were supported on the

bottom platform. The machine was configured to

operate at constant indentation speeds of 2, 5, 10, 50

and 100 mm/min up to a prescribed depth of 5 mm,

i.e. the experiments were displacement controlled.

The reaction load was recorded at suitable intervals

automatically using analogue sensors. Force and

corresponding time data were recorded

simultaneously by a host computer, via a terminal

panel T31 (analogue to digital converter) connected

to the Instron interface.

Description of the Experimental Procedure

In a representative experiment the load cell output

was zeroed with the sample on the platform and the

indenter was brought to the sample surface by

moving the crosshead. Then, the required

indentation speed was set, and the indenter was

moved into the material at the chosen constant

speed. The displacement may be deduced from the

speed and lapsed time data. However, a separate



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April 2016

LVDT (linear variable displacement transducer)

was used to measure the imposed displacement.

The data acquisition rate was adjusted according to

the values of the imposed nominal strain rate.

RESULTS AND DISCUSSION

The indentation experiments were carried out on the

five chosen cheeses with two protocols i.e. loading-

unloading and loading-stress relaxation;

Loading and Unloading

The initial data were obtained in the form of time

series of load, and the corresponding displacement.

Figure [6] shows typical examples of loading and

unloading curves, where the total load (N) for the

constant depth is plotted against the indentation

displacement (mm) for a Red Leicester using

conical (Figure [6a]) and spherical (Figure [6b])

indenters at five different penetration velocities.

(6a): - For 900 cone

Indenter.

(6b) :- For ½” sphere

Indenter.

Figure [6] :- Loading-unloading curves of load

against displacement load for several indenter

speeds for Red Leicester Cheese.

The curves indicate that there is a little elastic

recovery of the material during unloading, which

tends to increase as the indentation rate is increased.

However, there is a significant energy loss and

permanent deformation during the indentation

cycle. The general trends for the other cheeses were

similar. All the compliance curves obtained

indicate, as expected, that the maximum indentation

force is much higher when using a spherical

indenter; because the contact area is larger. The

curves showed that Mozzarella was the most elastic

material of the set, with the least amount of plastic

deformation. The Pecorino Romano was the

hardest, with the highest indentation forces.

The parameters of the fitting curves

The Box-Cox transformation method was adopted

for the analysis of the unloading curves. A simple

linear regression of log (L) against log (h-hr) was

used to evaluate the parameter n2 (subscript 2

represents the unloading phase). Table [2] shows

the n2 values obtained from the fitted curves for all

the cheeses under both geometries of indenter. The

values of n2, for both geometric forms of indenter,

are generally less than 2. The results reflect the

relative insensitivity of the unloading curve to the

indenter geometry but values of the parameters are

dependent on the indenting speed. There may be

two reasons for the variation in the values of n2. The

determination of the residual depth of penetration hr

is subject to experimental uncertainties, and the set

depth of penetration was not reached exactly; both

influence the curve fitting. The latter was more

evident at higher indention speeds. These two

factors would have a significant effect on the values

of n2. The theoretically expected n2 values for the

case of purely elastic deformation are 1.5 and 2 for

a spherical and conical indenter respectively.

Hence from the values of n2, it can be concluded

that the deformation of all the cheeses was not

perfectly elastic, but a combination of elastic and

plastic.

Table [2]: - Parameter (n2) of fitted curves for

unloading.

Velocity Mozzarella. Brie Shrop. Blue Red Leicester Pecorino

mm/min

Cone Sphere Cone Sphe

re Cone

Sphere

Con

e

Sphere

Cone Sphe

re

2 0.44 0.57 0.36 0.65 0.30 0.61 0.86 0.54 0.69 0.76

5 0.47 0.65 0.81 0.79 0.56 0.44 0.63 0.44 0.75 0.61

10 0.42 0.36 0.69 0.87 0.54 0.37 0.75 0.55 0.70 0.49

50 0.71 0.65 0.86 0.53 0.55 0.39 0.84 0.55 0.70 0.96

100 1.54 1.53 1.27 1.68 1.82 0.60 1.82 1.47 2.96 1.66

The values of hp were evaluated using equation (4),

and are listed Table [3]. The results indicate an

obvious decreasing trend of hp with the increase of

the indentation rate. In other words, there is more

elastic recovery of the materials at the higher

indenting speeds. Such a trend is verified by the

nature of the loading and unloading curves.

Table [3]: - Values of hp, (mm). Velocity Mozzarella. Brie Shrop. Blue Red Leicester Pecorino

mm/min Cone

Sphere

Cone

Sphere

Cone

Sphere

Con

e

Sphere

Cone

Sphere

2 3.40 3.46 3.80 3.72 4.18 3.99 4.19 3.40 4.09 3.74

5 3.29 3.73 3.40 3.20 4.16 3.77 3.97 3.19 4.12 3.48

10 2.92 2.72 3.33 3.18 4.00 3.82 3.97 3.10 4.06 3.44

50 2.97 2.59 2.78 3.51 3.81 3.74 3.89 2.93 3.71 3.25

100 2.91 2.38 2.88 3.10 3.78 2.90 3.66 3.11 3.57 3.15

The difference in hp values between the indenters is

mainly due to their geometric differences. The

computed values of hp also confirm that Mozzarella

is the most elastic of the cheeses, as it has the

smallest amount of permanent plastic deformation.



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April 2016

Table [4]: - Computed values of mechanical

properties of cheeses using conical and spherical

indenters average of the five velocities. Cheese M.P. Mozzarell

a. Brie

Shrop.

Blue

Red

Leicester Pecorino

For 900 Conical Indenter

Elastic

Modulus,

MPa

0.334 0.149 0.842 1.227 1.313

Stiffness

MN/m

0.004 0.019 0.106 0.166 0.167

Plasticity

Index

0.5 0.7 0.9 0.8 0.8

Hardness,

MPa

0.009 0.026 0.134 0.222 0.227

For ½” Spherical Indenter

Elastic

Modulus MPa

0.058 0.297 1.135 1.327 1.462

Stiffness,

MN/m

0.089 0.484 0.182 0.141 0.242

Plasticity

Index

0.7 0.7 0.5 0.6 0.7

Hardness

MPa

0.0124 0.0574 0.0789 0.201 0.263

Hardness

The computed mean hardness values of all the

cheeses, obtained using both indenters, are

summarised in Table [4]. The variation in hardness

with speed is relatively small, as seen in Figure [6].

The result also reflects that the hardness is inversely

proportional to the moisture content of cheese,

which agrees with the general consensus; the higher

the moisture content, the softer the cheese. The

discrepancy in the hardness values measured by the

cone and sphere is mainly due to deflections with

imposed effective contact strain; see earlier.

Elastic Modulus

The elastic modulus is calculated according to

equation (2) and associated procedures. The

stiffness is computed from the slope of the

unloading curve, using the equation

SP

h h

t

t p

.

By definition, a material is said to be stiff when a

large force is required to produce a small strain;

Table [4]. The Poisson’s ratio was taken to be 0.45

for all the cheeses for the purpose of calculating the

Elastic modulus.

The value of the elastic modulus for different

indentation velocities for all the cheeses indicates

that there is a slight increase in the value with

indentation speed for the spherical indenter. This

variation indicates either that the strain imposed on

the cheese is dependent on the indentation rate, or a

degree of inaccuracy of the computed values at

higher speed. A more significant variation in the

computed values for the conical indenter is a

reflection of the difficulty of accurately specifying

the parameters required to define the slope at

unloading, as well as the corresponding values of

the contact area.

Loading and Relaxation

The relaxation response was observed by holding

the indenter within the sample for a period of 400

seconds after the set depth of 5 mm was reached. In

contrast to a typical loading and unloading curve,

Figure [7] shows a gradual elastic-plastic relaxation.

There are three aspects about the general shape of

the stress relaxation curve, which are worthy of

mention. First, as can be seen in the Figure [7], the

load raises during the indentation phase until the

prescribed depth is reached.

[7a]:- For a 900 cone

indenter.

[7b]:- For a ½” sphere

indenter.

Figure [7]:- Loading and relaxation curves for Red

Leicester, Using two indenters at 5 different speeds.

The curve is slightly concave for the cone

indentation and convex for the sphere indentation.

This is mainly due to the way in which the contact

radius changes with time. As was noted earlier, the

indentation load is proportional to the contact area

of an indenter. Secondly, during the relaxation

phase, when the loading motion ceases, the

indentation load falls rapidly at first, followed by a

long gradually decaying tail. This relaxation phase

indicates that the reaction to the load is partly

elastic and partly viscoplastic. The ratio of the

elastic and viscoplastic components depends on the

time-scale of the measurement.

The data of load and displacement were analysed

using equation (10) for the loading period and (11)

for the relaxation data. According to equation (10),

plots of the mean indentation pressure (nominal

stress) against nominal strain rate for different

speeds should collapse to a single master curve,

where strain rate is defined as the argument of the

function F. Similarly, by an inspection of equation

(11) the mean indentation pressure in the relaxation

phase is a function of the rate of decrease of the

load. First, the time dependent response was



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characterized by examining the relaxation behavior.

A typical result for the relaxation phase is plotted in

Figure [8] as the mean stress (MPa) against the

unloading rate (N/s) on log-log axes.

[8a]: - For 900 cone

indenter.

[8b]:- For ½” sphere

indenter.

Figure [8]:- Mean indentation pressure vs. load

relaxation rate for Red Leister, using two indenters

at 5 different speeds.

The various curves are seen to lie closely to each

other. It can therefore be concluded that all the

curves collapse onto a single master curve and the

trend of this master curve obeys a power law

relationship. The relationship is fitted

approximately, over three decades of strain rate,

using 19.0q for cone and 17.0q for the

sphere. Since the form of the function F must be the

same for the loading phase as well as for the

relaxation phase, the slope of the trend line in

Figure [8] was used to estimate an approximate

value of t0, which accounts for an arbitrary zero

offset in the time variable in the loading phase.

This term was incorporated into the time data for

equation (10). The t0 values were chosen separately

for each indentation speed to give the best fit of all

the curves to a single master curve. Figure [9] is a

plot of indentation pressure (MPa) against the

nominal strain rate (1/s) for Red Leicester on log-

log axes.

[9a]:- For 900 cone

indenter.

[9b]:- For ½” sphere

indenter.

Figure [9] :- Mean indentation pressure vs. strain

rate for Red Leicester, using two indenters at five

different speeds.

As is evident in Figure [9], the data collapse better

for the cone than for the sphere, and for larger

indentation depths (lower stress and strain rate). A

power trend was fitted to these data to represent the

single master curve. The value of the consistency k

was obtained from the equation of the trend line, as

the plastic flow intercept; k = 0.33 MPa.sn for the

cone and k = 0.05 MPa. sn.

Table [5]: - Results of stress relaxation experiment. Cheese

Selections

Sphere Cone Spher

e

Cone Sphere Cone

q K (MPa.sn) CEE (MPa)

Mozzaller

a

0.289 0.278 0.05 0.22 0.007 0.096

Somerset

Brie

0.186 0.247 0.10 0.40 0.075 0.051

Shropshire

Blue

0.163 0.211 0.03 0.13 0.155 0.089

Red

Leicester

0.166 0.192 0.05 0.33 0.080 0.036

Pecorino

Romano

0.182 0.184 0.09 0.43 0.170 0.159

The result of the stress relaxation analysis for Red

Leicester indicates that the power law relationship

is a sufficient description over the entire

experimental strain rate range of five decades; as is

shown in Figure [10]. It also confirms that the

assumption that n is reasonably constant over the

investigated range is valid. On the whole, the value

of n decreases with the hardness of the cheese;

while the value of k, on the other hand, increases

with the hardness. Thus both the n and k are related

to the hardness of the material.

[10a]:- For 900 cone

indenter.

[10b]:- For ½” Sphere

indenter.

Figure [10]:- Combined plot of data from

indentation and relaxation for Red Leicester, using

two indenters at five different speeds.

The forms of equation (10) and (11) both show the

shape of the viscosity function F. Choosing an

appropriate value of CEE, which is incorporated in

the relaxation equation, the various loading and

relaxation data may be combined as is shown in

Figure [10]. The values for red leicester and the

product of the elastic modulus and elastic constraint

factor was evaluated to be 0.106 MPa for sphere

and 0.044 for cone indenter. Values of the material

index and CEE for all the cheeses are tabulated in

Table [5].



IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

5. CONCLUSIONS

The various protocols for indentation studies, such

as loading and unloading and stress relaxation, were

investigated to characterise the elastic and

viscoplastic properties of several cheeses. The order

of cheese from the softest to the hardest was found

to be; Mozzarella, Brie, Shropshire Blue, Red

Leicester and Pecorino Romano. We have carried

out similar studies on Feta, Edam, Stilton, Chedder,

and Cheshire, and obtained similar results. The

computed values of hardness indicate that the

hardness of a cheese is dependent upon its moisture

content. The higher the moisture content is, the

softer is the cheese.

The Box-Cox transformation method provides the

hardness and the elastic properties from the

compliance data, as it eliminates some of the

ambiguity in selecting the linear portion of the

unloading curves for the usual graphical

extrapolation. It was noted that the values of

contact radius and the actual value of the set depth

reached played a significant part in determining the

computed results.

The stress relaxation analysis provides a means of

computing elastic properties of these elasto-

viscoplastic materials since it takes the inherent

large strain and strain rate of the material into

account. The elastic behaviour of a material is

characterised by the elastic modulus, whereas the

elasto-plastic behaviour is presented as a power law

with material index q and consistency k which are

the power law relationship parameters. All of the

cheeses were found to follow the power law

relationship over a range of five decades in strain

rate. The values of q decrease in an order with the

hardness of cheese, on the other hand the values of

k increase with the hardness. However, it was

observed that the stress relaxation analysis appears

to be sensitive to the physical structure of a

material. For instance, Shropshire Blue, which has a

porous structure, does not respond well to the

analysis because the cheese showed signs of

cracking; and “pile-up” effects were also noticed

during the indentation phase. As a result, a

significant discontinuity of the strain rate was seen

when combining the indentation and relaxation data

together.

NOMENCLATURE

A

a

nominal contact area

contact radius

CP

CE

D

E

E*

plastic consistency

elastic consistency

indenter diameters

Modulus of Elasticity

reduced elastic modulus

F

hE

h

h0

hp

hr

ht

H

function

elastic indentation depth or displacement

indentation depth, displacement

zero offset

plastic penetration depth

residual depth of penetration

penetration at maximum load

hardness

k

n

L

flow consistency or the flow intercept

flow index

Indentation load

R

S

t

t0

V

radius of spherical indenter

contact stiffness

time

zero offset

indentation velocity

poisson’s ratio

cone semi angle

stress

y yield stress

plasticity index

.

strain rate

strain

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IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

products”, a literature review, Food and Bioproduct

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