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3rd International Symposium on Food Rheology and Structure

267

RHEOLOGICAL CHARACTERISATION OF CHEESE

SM Goh1, MN Charalambides

1, S Chakrabarti

2, JG Williams

1

1Department of Mechanical Engineering, Imperial College London, SW7 2BX, U.K.

2General Mills Technology, Minneapolis, MN 55414

ABSTRACT

A scheme has been developed to characterise the strain

and the time dependent behaviour of non-linear

viscoelastic materials such as cheese in the form of a

non-linear constitutive model. The model consists of two

independent functions - a hyperelastic function to

characterise the strain dependent behaviour, and a Prony

series to characterise the time dependent behaviour. The

calibration of the model is made using data obtained from

monotonic uniaxial compression and stress relaxation

tests. In order to verify the material parameters obtained

from this scheme, experimental tests and finite element

simulations of the three point bend and wire cutting tests

of two cheeses were performed. The results from the

finite element simulations showed good agreement with

the experimental test data under various test conditions.

1. INTRODUCTION

Many foods such as cheese and dough exhibit large

strain, viscoelastic behaviour. For these foods, boththe strain and the time dependent mechanical

behaviour must be characterised. In order that the

constitutive models have good predictive

capabilities, they have to be calibrated using

consistent material data. However, for foods such

as cheese, consistent material data is not obtainable

because of the significant material variation between

different blocks and batches (Prentice et al. 1993).

Thus, the material data is not accumulative and has

to be collected for each batch. Successful methods

for characterising these foods would have to besimple, quick and be economical in terms of time

and materials. In this study, a method for

characterising the non-linear viscoelastic properties

of cheese was investigated. These properties were

then used in analysing three point bend and wire

cutting tests.

2. EXPERIMENTS

Mild Cheddar and Gruyere samples were bought

from a local supermarket and stored at 4C until

testing. A separate block of each cheese was usedfor the three point bend and the wire cutting tests.

In addition, a block of process cheese was supplied

by General Mills and was used in the wire cutting

tests.

The specimens were cut into rectangular shapes

using a wire cutter and into cylinders using a borer.

They were then wrapped in cling film and allowed to

equilibrate at room temperature (21C) for at least

two hours. All tests were performed at 21C using

the Instron 5543 testing machine.

Rectangular specimens of height 15mm, width30mm and length 60mm were prepared for the three

point bend test. The striker and the supports

consisted of steel rods of 10mm diameter with the

supports positioned at 50mm apart. For Gruyere,

the tests were conducted at 5, 50 and 500mm/min.

For mild Cheddar, the tests were conducted at two

crosshead speeds, 5 and 50mm/min. A further test

was also performed at 5mm/min using mild Cheddar

specimens which were notched to a depth of 7.5mm

at the plane of symmetry.

The wire cutting tests for Gruyere and mild Cheddarwere performed using wire diameters, d , of 0.25,

0.5 and 0.89mm as well as dowel pins of diameter

1.6 and 2mm. For the process cheese, wire

diameters of 0.25, 0.345, 0.5 and 0.89mm were

used. The dowel pins were sufficiently rigid, so the

crosshead displacement was an accurate measure

of the displacement of the pins. For the smaller wire

diameters, the crosshead displacement had to be

corrected for the deflection of the wire relative to the

crosshead to obtain the actual wire displacement

(Goh 2002).

The specimens for the wire cutting tests were

rectangular blocks of length 25mm, height 20mm

and width 15mm for the smaller wire diameters.

Blocks of length 30mm, height 30mm, and

thicknesses 20mm and 30mm were used for the

1.6mm and 2mm diameters respectively. Three

constant cutting speeds of 5, 50 and 500mm/min

were used.

The material calibration tests involved monotonic

uniaxial compression tests performed at true strain

rates of 0.25, 2.5 and 25/min, and relaxation tests


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268

performed at a true strain rate of 2.5/min up to a

strain of 0.04. The specimens were cylinders with

height 20mm and diameter 20mm. Prior to the start

of test, the platens were lubricated with Superlube

(Loctite Corp.) to eliminate the friction at the sample-

platen interface (Charalambides et al. 2001). Since

separate blocks of each cheese were used to studythe three point bend and the wire cutting tests, the

material data, in the form of true stress, , and true

strain, , were collected for each block separately.

The true stress was calculated based on the

assumption that the material was incompressible.

3. NUMERICAL SIMULATIONS

The numerical simulations were performed in the

commercial finite element code ABAQUS. In all

models, four noded, plane strain elements were

used to model the cheese. Because of symmetry,

only half of the specimen was required. The striker

and the support were modelled as rigid surfaces.

The striker was prescribed to move at speeds which

are identical to those in the experiments. The

contact surfaces were assumed to be frictionless

since preliminary results showed that the friction had

a negligible effect on the bending force.

For the wire cutting test, the focus of the finite

element analysis was on the indentation of the wire

into the specimen. This phase precedes the steady-

state cutting phase where the wire makes a cut

through the specimen. Only half of the specimen

was included due to symmetry. The contact

between the wire and the specimen was assumed to

be frictionless since the friction was found to have a

negligible effect on the indentation force.

4. THEORY

In ABAQUS (ABAQUS 1998), the viscoelastic model

consists of two independent components which

represent the strain and the time dependentbehaviour. During a step-strain relaxation test, the

relationship between the stress and the time and

strain can be expressed as,

( ) ( )tgf = (1)

where t is the time, and f and tg are the strain

and the time dependent functions respectively.

The time dependent behaviour in ABAQUS is

defined by the Prony series, which is expressed as,

( )

=

+=

N

i ii

tggtg

1

exp

(2)

where i are time constants and ig are dimension-

less numbers, and,

1

1

=+=

N

i

igg (3)

For non-linear, large deformations, the strain

dependent behaviour is defined by a hyperelastic

strain energy potential.

The Prony series and the hyperelastic potential canbe calibrated from ideal relaxation test data and the

stress-strain relationship corresponding to

instantaneous or long term deformation. However, it

is often not possible to perform experiments under

these ideal conditions, as is the case in this work.

Under non-ideal test conditions, the test data can be

described instead by the convolution integral,

( ) ( )( )

ss

fstgt

t

dd

d

0

=

(4)

The solutions of the convolution integral can befitted to experimental data to obtain the material

constants in f and tg . Although this direct

method is feasible for some hyperelastic functions,

such as the Mooney-Rivlin and polynomial strain

energy functions (e.g. Miller 1999), the convolution

integral can become intractable for other forms of

hyperelastic functions. An alternative procedure to

overcome this problem is to first calibrate the strain

dependent behaviour with a polynomial expression

given by,

( ) DCBAf +++= 234 (5)where A , B , C and D are constants. In

combination with the Prony series, the solutions to

the convolution integral can be obtained and

calibrated through a scheme as proposed in Goh et

al. (2002). After equation (5) has been calibrated, it

represents the stress-strain relationship under

instantaneous (i.e. t=0, tg =1 and f= in

equation (1)), uniaxial compression state, to which

other hyperelastic functions can be approximated.

The fitting of the hyperelastic functions is made byinputting the stress-strain data calculated using

equation (5) into ABAQUS. During the pre-

processing stage, ABAQUS automatically

approximates the input data with the chosen

hyperelastic function. The Van der Waals

hyperelastic potential was used in this work because

it led to a good approximation of the data.

Furthermore, it is also known to provide a more

accurate prediction of the general deformation

modes if the calibration of the material constants is

based only on one test (ABAQUS 1998).


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269

5. RESULTS AND DISCUSSION

Typical stress-strain curves of the cheeses used in

the wire cutting tests are shown in Figure 1. The

stress-strain data for the cheeses used in the three

point bend tests also share similar characteristics.

There was in general more scatter in the data for the

process cheese. This was due to the sagging of the

cheese under its own weight during the storage

period which led to a rather inhomogeneous mat-

erial.

0

10

2030

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6

strain

stress(kPa)

=25/min.

=2.5/min.

=0.25/min.

(a)

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6

strain

stress(kPa)

=25/min.

=2.5/min.

=0.25/min.

(b)

0

10

20

30

40

50

0 0.3 0.6 0.9 1.2 1.5 1.8strain

stress(kPa)

=25/min.

=2.5/min.

=0.25/min.

(c)

Figure 1 Stress-strain curves for (a) Gruyere (b) mild

Cheddar (c) process cheese

The results of the calibration of the polynomial and

the Prony series are shown in Table 1. Also

included in Table 1 are the values of the Van der

Waals hyperelastic constants. For the calibration,

the values of i were arbitrarily chosen as 0.1, 1,

10, 100 and 1000 seconds for i equal to one to five

respectively.

A (kPa) B (kPa) C (kPa) D (kPa) (kPa) m a

*mild Cheddar -1100 2040 -1330 560 172 3.11 1.51

*Gruyere -3100 4030 -2040 730 230 2.59 2.02

**mild Cheddar -3250 4050 -1950 610 190 2.54 2.19

**Gruyere -4440 5210 -2380 765 236 2.64 1.98

Processed 6.6 -8.5 -7 61.6 22.8 412 0.103

(a)g1 g2 g3 g4 g5 g

*mild Cheddar 0.312 0.289 0.109 0.101 0.109 0.080

*Gruyere 0.117 0.404 0.128 0.133 0.108 0.110

**mild Cheddar 0.304 0.303 0.114 0.106 0.089 0.084

**Gruyere 0.221 0.333 0.117 0.123 0.097 0.109

Processed 0 0.525 0.233 0.201 0.040 0.000

(b)

Table 1 Material parameters for (a) Strain dependent

function (b) Time dependent function *Three point bend

**Wire cutting/Indentation

The comparison between the experimental force-

displacement curves in the three point bend tests

and finite element predictions is shown in Figure 2.

A good agreement is observed in general.

0

2

4

6

8

10

12

0 2 4 6 8 10displacement (mm)

bendingforce(

N)

experimental

finite element

prediction

500mm/min

50mm/min

5mm/min

(a)

0

1

2

3

4

5

6

0 2 4 6 8 10displacement (mm)

bendingforce(N)

experimental

finite elementprediction

5mm/min

500mm/min

5mm/min

notchedspecimen

(b)

Figure 2 Bending force-displacement data (a) Gruyere (b)

mild Cheddar


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270

The numerical predictions of the indentation forces

are also in good agreement with the experimental

data for all wire diameters. The results for three

cases are shown in Figure 3. There was a small

scatter in the data for Gruyere and mild Cheddar so

the average values are shown. As mentioned

earlier, there was considerable scatter in the datafor process cheese and so the raw data are shown.

These results show that the proposed scheme for

obtaining the material constants is successful in

characterising the strain and the time dependent

material behaviour.

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5wire displacement (mm)

force

(N)

500mm/min

50mm/min

5mm/min

finite element

prediction

(a)

0

0.4

0.8

1.2

1.6

2

0 0.2 0.4 0.6 0.8 1wire displacement (mm)

force(N)

500mm/min

50mm/min

5mm/min


(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1 1.5 2wire displacement (mm)

force(N)

500mm/min

50mm/min

5mm/min


(c)

Figure 3 Indentation force-displacement data (a) Gruyere,

d=2mm (b) mild Cheddar, d=0.5mm (c) process cheese,

d=0.25mm

The wire cutting models were further investigated for

their ability to predict the steady-state cutting forces.

For this task, a simple fracture criterion based on a

critical strain was used. This fracture criterion was

adopted, because the fracture strains remained

relatively unchanged for the different strain rates for

the cheeses. Thus, the critical strain, crit , wasassumed to be equal to the fracture strain as

measured in the uniaxial compression test. For mild

Cheddar and Gruyere, the global fracture of the

specimens were observed to occur around the peak

in the curves. Thus, the fracture strains are

approximately 0.5 and 0.45 respectively. For the

process cheese, the specimens underwent a high

degree of compression and when the specimen

fractured, no drop in stress was recorded. From

visual observations, the specimens appeared to

fracture at strains of 1.4-1.6.In the finite element indentation models, the

maximum tensile strain, max,xx , occurs at the line of

symmetry in the direction normal to the movement

of the wire. With increasing indentation, the value of

max,xx increases monotonically. Thus, the changes

in max,xx along the line of symmetry were

monitored such that when crit was reached,

fracture was assumed to occur. The numerically

predicted indentation forces per unit width are

compared with the experimental steady-state cutting

data in Figure 4. Good agreement between thepredicted values and the experimental data is

observed. Thus, the critical strain criterion appears

to be valid for the prediction of the cutting force.

The validity of the indentation models to predict the

steady-state cutting force does require further

research. In the indentation models, it was found

that surface friction had a negligible effect on the

indentation load. However, theoretical considerat-

ions of the steady-state cutting stage (Kamyab et al.

1995) have predicted a large influence of the friction

on the cutting force. Furthermore, it has been

assumed that the fracture strain in tension was

equal to the fracture strain in compression. Since

the deformation of the material ahead of the wire is

highly constrained, it was also assumed that the

fracture strain was independent of the hydrostatic

stress. It will be necessary that other independent

tests such as the plane strain compression and the

tension tests be performed to investigate the

material behaviour in deformation states other than

uniaxial compression. The modelling of the steady-

state cutting stage will also be necessary tounderstand more fully the stress and deformation

states as well as the effect of friction.


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271

0

100

200

300

400

500

0 0.5 1 1.5 2wire diameter (mm)

cuttingfo

rce/width(J/m2)

5mm/min

50mm/min

500mm/min


(a)

0

100

200

300

400

500

0 0.5 1 1.5 2wire diameter (mm)

cuttingforce/w

idth(J/m2)

5mm/min

50mm/min

500mm/min


(b)

0

20

40

60

80

100120

140

160

0 0.2 0.4 0.6 0.8 1wire diameter (mm)

cuttingforce/width(J

/m2)

5mm/min

50mm/min

500mm/min

(c)

Figure 4 Prediction of steady-state cutting force for (a)

Gruyere (b) mild Cheddar (c) process cheese solid linerepresents finite element prediction using crit =1.6;

broken line represents finite element prediction using

crit =1.4

6. CONCLUSIONS

Finite element simulations have been performed to

model the mechanical behaviour of cheese. The

material models were calibrated through an indirect

approach, where the strain dependent behaviour

was first characterised by a polynomial, which was

then fitted with the Van der Waals hyperelastic

function in ABAQUS. The time dependent behaviour

was modelled using Prony series. The numerical

force-displacement curves were in good agreement

with the experimental data for three point bend and

indentation tests, suggesting that accurate

characterisation of the strain and the time

dependent behaviour of the cheeses was achieved.

The indentation models were also successful in

predicting the steady-state wire cutting force throughthe use of a critical fracture strain criterion.

REFERENCES

ABAQUSs user manual ver 5.8. Hibbitt, Karlssonand Sorensen (UK), Cheshire (1998)

Charalambides MN, Goh SM, Lim SL, JG Williams:The analysis of the frictional effect on stress-straindata from uniaxial compression of cheese, J.Mater. Sci. 36, 2313-2321 (2001)

Goh SM: An engineering approach to food texture

studies. Ph.D. thesis, Imperial College London(2002)

Goh SM, Charalambides MN, Williams JG: Largestrain time dependent behaviour of cheese, J.Rheol., submitted (2002)

Kamyab I, Chakrabarti S, Williams JG: Cuttingcheese with wire, J. Mater. Sci. 33, 2763-2770(1998)

Miller K: Constitutive model of brain tissue suitablefor finite element analysis of surgical procedures,J. Biomech. 32, 531-537 (1999)

Prentice JH, Langley KR, Marshall RJ: CheeseRheology, In Cheese: Chemistry, Physics andMicrobiology, Volume 1, 2

nded. PF Fox, (Ed.)

Chapman and Hall, London, 303-341 (1993)

ACKNOWLEDGEMENTS

The authors would like to thank the BBSRC for

financial support and General Mills for providing the

process cheese.

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