Enemamme PARAMETERS RELATED
TO THE HARDNESSOFCARROIS '

Dissertation for the Degree of Ph. D;
MECHEGAN STATE UNWERSITY
BASSAM AHMED SNOBAR
1973 ’

Mme.-...v..~.._
I ,‘n
VJ Ir» ~ . a.»
' La A's '. a

.2

hitchgfit ’. 1;”:
Ulll‘J'CIf

     

This is to certify that the

thesis entitled

ENGINEERING PARAMETERS RELATED TO
THE HARDNESS 0F CARROTS

presented by
BASSAM AHMED SNOBAR

has been accepted towards fulfillment
of the requirements for

_Eh.D—degree in AgLLaulLura 1 Engineering

or rofess

Date 11/6/1973

 

0-7639

   

     

LIBRARY BIND R

amomn
"AG & SUNS'
300K BIND!" 2mg.

'—
n
.-

__ ________._-n m—v-a- W‘“*‘v ‘-'-.-v~'

 

 

 

4"}? . . , . .t v t . ._ .. . .:.I!' \ .Autt . , t, .. 9... ..-.;....~ ‘1 .II“. . Fun . .I it, .. .I Luv.
.. . .
in . . t . SILK...

. ii . . .. n I... Lilli . ....v
4‘- . a 5%.“. '0 . ,4 II» hmly- ' a u . {iii .91. 5H .\7 ‘ . . put I-.

a A. v. in

... n tr 1.. Et‘ uterulla. It'll»

[slit-v. 1.3th 1 1-..: .- _ .
al- klfl...flwauu~fl.l¥k. ‘39.»... .. ....1..... .. . (say. . . .. 1 a a.” .llmlIlL _.v.|nt.l.nta..-..|-

 

ABSTRACT

ENGINEERING PARAMETERS RELATED
TO THE HARDNESS OF CARROI‘S
by
Bassam Ahmed Snobar

Studies were conducted to indentify and evaluate objectively a
textural parameter related to the long-term storage of carrots. Mech-
anical and rheological properties were used in this evaluation.

An equation was derived, using Hertz's Contact Theory, to calculate
the modulus of elasticity of a cylindrical sample canpressed in the
radial direction. This equation was used to calculate the tangent
modulus of a one-inch diameter carrot sample after it had been subjected
to a 0.1- inch displacement in the radial direction. Relaxation tests were
also conducted to study the relaxation behavior of cylindrical samples
as related to the long term storage of carrots.

The tangent modulus varied significantly as the moisture content
in an outer ring of the carrot decreased. The tangent modulus decreased
fran 869 psi to 27 psi as the moisture content decreased fran 86.6
percent to 72.5 percent (wet basis). Similar variations were observed
for the coefficients C1 in the relaxation equations

3 ..
F(t) = 2 c1 e01t

used to fit the relaxation data (generalized Maxwell Model). No sig-
nificant pattern was observed for the al.

The Texture Profile Analysis procedure was applied to carrot
samples using an axial compression load. No significant changes were
detected even though the physical appearance of the carrots changed
significantly. This fact was accounted fer by the fact that the center
core of a carrot is stronger than the outer ring of material and thus

biased the testing procedure.

Approved
aj r Pr 5

I
W 8 A m

Department Chairman

ENGINEERING PARAMETERS RELATED

TO THE HARDNESS OF CARROTS

by

Bassam Ahmed Snobar

A DISSERTATION
submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Agricultural Engineering

1973

"-5 ACKNOWIEDGEMENTS

The author wishes to express his sincere gratitude to Dr. L. J .
Segerlind (Agricultural Engineering), whose guidance, and encourage-
ment were invaluable.

Special sincere gratitude to Professor C. M. Hansen (Agricultural
Engineering) for his encouragement and his providing the assistantship.

The assistance from Dr. J. S. Frame (Mathematics) in deriving the
mathematical equation and serving as guidance committee member is also
acknowledged.

Equally, appreciations are also extended to Dr. D. R. Heldman
(Agricultural Engineering), who also served as guidance comnittee mem-
ber for developing the doctoral program and to Dr. G. E. Mase GMetallury,
ZMechanics and Material Science) for his assistance in the theoretical
development of the thesis.

The author wishes to express his gratitude to Dr. B. A. Stout,
Chainman of the Agricultural Engineering Department, fer his approving
the assistantship.

The author dedicates this work to his father, Ahmed M. Snobar, his
mother, Mrs. Yussra Nabhaan, and to his family. Their foresight, en-
couragement and personal sacrifice inspired him to undertake a career

in agricultural engineering.

ii

II.

III.

VI.

VII.

TABLE OF CONTENTS

INTRODUCTION AND OBJECTIVES
REVIEW OF LITERATURE

2.1 Methods of texture evaluation

2.2 Rheological Models

2.3 Hertz Contact Problem
PRELIMINARY STUDIES
THEORY

4.1 Derivation of an equation to determine modulus
of elasticity of a cylindrical specimen compressed

4.2 Fbppl's equation

4.3 Relaxation test
EXPERIMENTAL INVESTIGATION

5.1 Equipment

5.2 Experimental procedure
RESULTS AND DISCUSSION

6.1 Accuracy of the derived equation

6.2 Poisson's ratio

6.3 Deformation test

6.4 Relaxation tests
CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

7.2 Recommendations
APPENDIX
REFERENCES

iii

16
20
23
28

30
41
42
45
45
49
55
SS
56
58
59
68

69
71
88

Table
Table
Table
Table

Table

Table

Table

Table

Table

LIST OF TABLES
Standard hardness scale (Szczesniak et a1. 1963). 8
Procedure for evaluating texture (Brandt et a1. 1963). 10
Numerical values of coefficients for various commodities. 10

Modulus of elasticity for the center core and the outer
ring of six fresh carrot samples. 24

Moisture content of fresh carrots and of carrots stored
under different conditions and time periods. 24

Hardness parameter measurements using the TPA method

of axial deformation. The hardness is measured from
force—deformation curves produced upon twice caupressing
cylindrical samples of carrots previously stored in per-
forated plastic bags at 52°F and 95% Relative Humidity

for three weeks then removed fran the bags prior to testing
and stored in a roan at 70°F and 6095 Relative Humidity for
24 hours. 27

Summary of calculated values of E for different potato
specimen sizes and deformation using:

 

2 . 2
E1=0.955m,E2=1.91Pz , E3=§ii

L2 D
where A is cross sectional area. 57

Poisson's ratio, v, measurenents for fresh carrots
and for carrots stored in a roan at 70°F and 60%
relative humidity for 24 hours. 58

Values of moisture content, modulus of elasticity

as calculated from the derived equation, peak force

as the selected experimental curves indicated, and

the relaxation coefficients and exponential as

calculated from the assumed generalized Maxwell Model 60

iv

Figure

Figure

Figure

Figure

Figure

Figure

Figure
Figure
Figure
Figure
Figure

Figure

Figure
Figure
Figure

9.

10.
11.
12.

13.
14.
15.

LIST OF FIGURES

Correlation between the panel and the texturoneter
on the hardness scale (Szczesniak et a1. 1963).

Direct trace (heavy line) of force-distance curve
obtained for a G. F. Texture Profile on a cylinder
of pear tissue in the Instron machine. The test
consists of two complete compression-decompression
cycles. (Bourne, 1968a)

Compression device used for compressing carrots.
Part A is attached to the stationary crosshead

of the Instron. Part B is attached to the moving
cross-head. Space C is where the carrots are
placed to be compressed (Howard and Heinz, 1970) .
Typical compression force-distance curve obtained
for carrots. -(a) Load, with crosshead moving.

- (b) Distance proportional to the change in diameter
of the carrot compressed under a 2.5 pomds load
(Howard and Heinz, 1970) .

Kelvin and Maxwell models showing creep and stress
relaxation characterisitics (Sharma, 1964).

Typical force-deformation curve for TPA parameters
test.

Cylindrical sample before compression.
Cylindrical sample after compression.
Semi-ellipsoid pressure distribution

Pressure distribution over the surface of contact.
Area of contact.

One-feurth the area of contact divided into two
triangles.

Generalized Maxwell model.
Graphical representation of equation [18] .

Sampler, (A) is corer and (B) is Base.

12

15

15

18

26
32
32
33
33
37

37
44
44
46

Figure

Figure

Figure

Figure

Figure
Figure
Figure
Figure

Figure

Figure
Figure

Figure

Figure

Figure

Figure

Figure

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.
26.

27.

28.

29.

30.

31.

Trimmer, (C) is holding jug (D) is two—bladed
knife.

Die to be used in measuring the Poisson's Ratio,
(B) is base, (E) is sample holding die, and (F)
is spacer.

Corer with 7/8" outside diameter to be used to
separate 1/8" thick outer ring from the

1" diameter sample.

Aminco unit with chamber to store carrots at
given temperature and relative humidity.

Instron.Universa1IMachine.
Loading for the unrestrained test.
Loading for the restrain test.

Force-deformation curves for a cylindrical sample
of carrots with one-inch in diameter and one-inch

long.

Sample loading, in the radial direction or
parallel to the longitudinal axis.

Typical curve for deformation and relaxation tests.

Effect of moisture content on the applied radial ferce
in deformation test .

Mbdulus of elasticity of carrots as calculated by
the derived equation.

Effect of moisture content on the first coefficient
in relaxation test.

Effect of moisture content on the second coefficient
in relaxation test.

Effect omeoisture content on the third coefficient
in relaxation test.

Experimental and theoretical results for relaxation
test at different moisture content. Table (A-9)

vi

46

47

47

48

48

50

50

51

52
53

62

63

64

65

66

67

‘
6”,
L.:’

on

I. INTRODUCTION.AND OBJECTIVES

The carrot (Daucas carota) is a popular vegetable and is increasing
in importance, owing to the fact that its value in the diet is better
understood today than in past years. It is rich in carotine, a precursor
of vitamin A, and contains appreciable quantities of thiamine and ribo-
flavin. The carrot also has a high sugar content.

Concern has risen among growers and processors relative to the
changes in the texture of carrots particularly during storage. In order
to identify these changes, the texture parameters of carrots must be
defined and objectively measurable. Sensory, or subjective, analysis
has been used to measure the texture of carrots, but this requires a
trained taste panel which is not readily available to many investigators.

Lord Kelvin (1891) said, "I often say that when you can measure what
you are speaking about and express it in numbers you know something about
it, but when you cannot measure it, when you cannot express it in numbers,
your knowledge is of a meager and unsatisfactory kind, it may be the
beginning of knowledge , but you have scarcely, in your thoughts , advanced
to the stage of science, whatever the matter may be." There is a very
limited mnnber of reports in the literature related to the objective
evaluation of the texture of carrots. An objective procedure is needed
in order to bring consistency to the investigation of the effects of

storage time on the texture characteristics of carrots.

2

The specific objectives of this investigation are:

1. To define the important textural parameters related to the
long term storage of carrots.

2. To investigate methods of objectively measuring the textural
parameter hardness.

3. To investigate the relationship between mechanical properties

and moisture content .

 

II. REVIEW OF LITERATURE

2.1 Methods of texture evaluation

Texture is one of the three main attributes of foods that cause
pleasure in eating; the other two being flavor, and appearance. There
are many definitions for texture. Reidy (1970) reported a dictionary
definition of texture as "an identifying quality; the disposition or
manner of union of the particles of a body or substance". The
Institute of Food Technologists offered another definition for tex-
ture of food (Kramer, 1959) as: "The mingled experience deriving from
the sensation of the skin in the mouth after ingestion of a food or
beverage. It relates to density, viscosity; surface tension and other
physical properties of the material being sampled". A possible defini-
tion of texture in carrots may be stated as: "Texture of fresh carrots
is the feel of hardness or crispness of the tissue in the mouth".

The present methods for evaluation of textural characteristics are
classified into:

1. Subjective or sensory evaluation.

2. Objective or instrumental measurements.

Subjective estimations of the textural quality of foods have been per-
formed since mankind began eating food and they continue to this day.
This method of evaluation depends on human senses. Due to the fact that
hunan senses are subject to the influence of various factors which lead
to error, scientists began the search for objective or instrumental

methods of texture measurements.

4

Scott Blair (1958) classified objective methods of texture
measurement under the headings: fundamentz-Il, empirical and imitative.
Szczesniak (1966) further defined Scott Blair's classification system
as follows: fundamental methods measure the rheological properties,
such as elastic modulus and viscosity, and relate the nature of the pro-
duct to two basic rheological prototypes; a dashpot for a Newtonian
liquid and a metal spring for a Hookean solid. The springs represent
elastic moduli. The dashpots represent viscosities. Empirical tests
measure characteristics related to textural quality using penetration
force test, resistance to compression force test, and shearing force
test. Imitative tests are performed under conditions simulating those
to which the material is subjected in practice.

Bourne (1966a) classified the methods for objectively measuring the
textural properties of food under the headings: force-measuring,
distance-measuring, time-measuring, energy-measuring, ratio-measuring,
multiple measuring, and multiple-variable instruments.

The first simple instruments used to objectively assess food tex-
ture came with the advent of scientific research into food quality. An
examination of the literature of food texture measurement shows that
these simple mechanical devices generally compressed, sheared or punc-
tured the food in some way.

Experimental methods for measuring food texture date back at least
to 1905, when Hankoczy in Hungary designed an apparatus for measuring
the strength of gluten and in 1907 Lehmann described two instruments for
testing the tenderness of meat. Morris (1917) constructed a simple de-
vice for measuring the resistance of fruits to penetration. In 1925,
Magness and Taylor developed the Magness-Taylor fruit pressure tester.

This instrument which is still widely used, consists of a plunger with

5

either a S/l6-inch or 7/16-inch diameter tip attached to a calibrated
spring. The round tip is pressed into the fruit to a depth of 5/16-inch,
and the penetrating force is read on the scale. The skin is removed
from some fruits before the measurement is made. Kramer et a1. (1951)
and Decker et a1. (1957) developed the shear press which is one of the
popular instruments for measuring textural qualities of both fresh and
processed fruits and vegetables.

Kattan (1957) described an instrument for measuring firmness of
tomatoes based upon compression of the fruit by a concentric chain
which encircled the fruit.

Drake (1962) described an apparatus for automatic recording of mech-
anical resonance curves for test specimens of foodstuffs with the approx-
imate size of 6x12x50 mm. The simple evaluation procedure described
gave infbrmation on the modulus of elasticity (divided by the density)
and the degree of dampening.

Schomer et a1. (1963) developed an instrument called the "mechanical
thumb" which operates on a principle similar to the Magness-Taylor
tester. However, their test is nondestructive in that the fruit can.be
evaluated with the skin intact, and the depth of indentation (0.05 inch)
causes no significant damage to the carmodity. Parker et a1. (1966)
developed a simple, portable, inexpensive micrometer type device for
evaluating cherry firmness.

Bourne (1965) evaluated the perfbrmance of pressure testers by
making punch tests on apples with pressure tips mounted in a universal
testing machine. His study showed that the yield point is reached when
the pressure tip begins to penetrate the fruit tissue.

With the present realization of the importance of texture in con-

sumer acceptance, an.increasing amount of attention is being paid to

6

correlating experimental measurements with sensory methods of texture
evaluation. Friedman et a1. (1963) studies the correlation between
instrumental values using texturometer and subjective evaluation by a
trained texture profile panel. This study was applied to measurement
of the mechanical textural parameters: hardness, cohesiveness, viscosity,
elasticity, adhesiveness, brittleness, chewiness, and gumminess. This
study gave good correlation between objectively determined values and f“‘“‘
subjective evaluation. ~

Szczesniak et a1. (1963) developed a standard rating scale for mech-

anical parameters of texture and correlation between the Objective and

 

the sensory methods of texture evaluation. Standard rating scales of
hardness, brittleness, chewiness, gumminess, viscosity, and adhesiveness
were established for quantitative evaluation of food texture. Hard-

ness is judged organoleptically as the ferce required to penetrate a
substance with molar teeth. The evaluation was restricted to solids

and some semisolids because human perception of hardness is limited to
samples that can be confined between the teeth. In their study, Szczesniak
et a1. avoided fresh fruits and vegetables whose texture varies greatly
with variety, degree of maturity, and other factors, and items that
required cooking, baking, etc.

Table 1 shows the nine points which were selected to represent the
scale of hardness. Correlation was very good between taste panel and
objective evaluation on the hardness scale (Fig. l).

NUmerous methods of objectively measuring texture of agricultural
products have been developed, adapted, or studied by many scientists.

Szczesniak et a1. (1963) developed 3 Texture Profile Analysis (TPA)
technique by which textural parameters were derived from ferce vs. dis-

tance curves plotted on the General Food Texturometer. The (TPA)

parameters are: hardness, brittleness, chewiness, gumminess, viscosity,
cohesiveness, elasticity, and adhesiveness.

Brandt et al. (1963) developed a texture profile method that uses
the A.D. Little flavor profile method as a model. They defined tex-
tural profile as the organoleptic analysis of the texture complex of
a food in terms of its mechanical, geometrical, fat, and moisture
characteristics, the degree of each present, and the order in which they 'r‘“
appear from first bite through complete mastication. The procedure they _I
followed to evaluate texture was mechanical, and geometrical evaluation.

The mechanical parameters were evaluated with standard rating scales

 

developed by (Szczesniak et a1. , 1963). The geometrical characteristics
of texture were related to the size, shape, and arrangement of particles
within a food. Table 2 shows the procedure used in evaluating the
different textural characteristics with respect to their appearance.

Destructive and nondestructive techniques were developed to measure
the texture of foods. Mohsenin et a1. (1965) have suggested a "non-
destructive" technique for evaluating firmness of apples based upon the
appearance of a "yield point" within the fruit.

Bourne (1966b) designed a study to separate and measure the com-
pression components (proportional to area) and the shear components
(proportional to perimeter) of a simple puncture test. His puncture
test measured the force required to push different types of punches into
a food product. The test is characterized by: a) using a force-measuring
instrument; b) penetration of the punch into the food; and c) a penetra-
tion distance usually held constant. Bourne used two sets of punches
(one with a constant area and a variable perimeter, and the other with a
constant perimeter and a variable area) to measure the compression and

shear components in representative foods. The puncture force was

 

 

 

 

 

 

Tab_1_e_ll_Standard_hardness_sca_.16 szczesniak et a1 . 1963) .
Panel Brand or Sample
rating Product type Manufacturer size Temp.
1 Cream cheese Philadelphia Kraft Foods V." 45-SS°F
2 Egg white hardtooked ..... V." tip room
5 min
3 Frankfurters large, uncooked, Mogen David V;" SO—65'F
skinless Kosher Meat
Products Corp.
4 Cheese yellow, American, Kraft Foods V_»" SO—65°F
pasteurized
process
5 Olives exquisite giant Cresca Co. 1 olive 50—6$°F
size, stuffed
6 Peanuts cocktail type in Planters Peanuts 1 nut room
vacuum tin
7 Carrots uncooked, fresh ...... %" room
8 Peanut brittle candy part Kraft Foods room
9 Rock candy ..... Dryden 8; Palmer ...... room
300 ‘—
HARDNESS SCALE
m
L'.
z
3'
K 200 -
u
.—
u
2
o
a:
3
.—
X
m
.—
IOO —
l l l l l l l
O o 2 4 6
SENSORY RATING
Fig. 1. Correlation between the panel and the texturometer

on the hardness scale (Szczesniak et a1. 1963).

 

f""

9
expressed by the equation F = KS P + KC A + C, where Ks’ Kc’ and C
are constants, P is the perimeter of the punch, and A is the area of
the punch. KS represents the shear coefficient and KC the compression
coefficient of the food being tested. Bourne tested the validity of the
equation postulated above and found that the experimental data obtained
fitted the equation. Table 3 gives the numerical values of the coeffi-
cients KC, K s’ and C as measured by Bourne for various food commodities. bi
Bourne stated that K c and Ks can be a measure of the texture quality .-
of foods. Although Bourne did not specify the direction of applying

the compression force on the specimen, the work of Howard and Heinz (1970)

 

DIE nw‘ =31n‘h'h.’ "' . - '
i
l
.

vb

seems to indicate that the force was perpendicular to the longitudinal
axis.

Bourne was one of the first people to use the Instron Testing
Machine for measuring the properties of food products. In one of his
studies (Bourne 1967a) , he used this machine to study the deformation
rate of food under constant force. This test was used to determine the
softness of the food as means of measuring food quality.

Bourne and Mondy (1967b) measured the deformation of: 1) standard
cylinders of potato tissue and 2) whole potatoes under a metal punch,
using a constant force, as an indication of the fir-mess of whole
potatoes. They found that measuring deformation under a punch is pre-
ferred over measuring the deformation of a cylinder because a) it can
be performed more quickly and easily; b) it is not destructive; c) the
correlation with sensory evaluation is slightly improved. Both methods
of measuring deformation was found to be a useful objective index of
potato firmness.

Bourne (1967c) described a model system which closely represents

the deformation of a food as it is squeezed in the hand. The model

10

Table 2. Prgcedure_for evaluating texture (Brandt et a1. 1963)

 

 

EQELUErEEin-d on firsT‘ 131$)"

 

 

\

 

Geometrical

any, depending upon product structure

 

Marlicatory _
A__(_perceived during chewing)

 

 

 

_-_._-_-__/_
Mechanical
/ . . .
hardness Viscosuy brittleness
/
Mechanical
/ l \

gumminess chewiness adhesiveness

\

Geometrical

any, depending upon product structure

 

Residual

(changes made during mastication)

 

rate of breakdown type of breakdown

moisture absorption

1

mouthcoating

 

Table 3. Numerical values of coefficients for various commodities

(Bourne 1966b)

 

 

 

Compression Shear coeffi-
coefficient K. cient Ks Constant C
Commodity (Kg/cm”) (Kg/cm) (Kg)
Expanded polystyrene 4.86 0.34 —0.23
High-density polystyrene 13.14 2.20 —2.75
Polyurethane 3.57 0.29 -0.47
Apples (raw, Limbertwig variety) 7.52 0.16 0.03
Apples (raw, Fr. von Berl variety) 6.43 0.07 0.40
Banana (ripe, yellow) 0.43 0.06 —0.06
Creme filled wafers 1.06 0.14 0.64
Carrot (uncooked core tissue) 28.0 -0.03 2.18
Wiener (cold) 1.69 0.004 0.15
Potato (Irish, uncooked) 10.79 0.52 0.60
Rutabaga (uncooked) 29.58 0.86 —0.15
Sweet potato (uncooked) 19.8 0.90 0.35
1% agar gel 0.15 0.005 --0.01
2% agar gel 0.63 0.029 -0.02
3% agar gel 1.21 0.16 —0.33

 

11

consists of a set of true springs of differing heights and with differing
Hooke's constants arranged in parallel. No dashpots were needed in
this simple model. The model was restricted to represent the physical
response of the food to a single compression. He described a graphical
method for measuring the number, size, and Hooke's constants of the
springs in the model. The spring model showed that with some foods at
least, small compression forces measured differences in softness better
than large forces.

Bourne (1968a) described a method to determine TPA parameters
from force-distance curves produced upon twice compressing a specimen
to a fixed deformation on the Instron machine. The curve produced
for deriving the TPA parameters for pear parenchyma tissue from Instron
force vs. distance is shown in Fig. 2. Brittleness, hardness and
elasticity parameters are shown on the curve. Other TPA parameters

can be calculated as follows:

. A
Cohesrveness = 2
AT

Gtmminess = Hardness x Cohesiveness

Chewiness Gumminess x Elasticity
where
A1 is the area under the first curve and
A2 is the area under the second curve.

Ourecky and Bourne (1968) measured the texture of strawberry with
an Instron machine. In this test, skin toughness and flesh firmness
were determined by obtaining a puncture-force curve on the Instron.
Many of the curves consisted of two or three distinct peaks. The first

peak was defined as the mecture—force required to penetrate the skin.

The second or middle peak was interpreted as the resistance of the flesh

12

FORCE Kg

 

A

 

POW." ' ' W
‘7'" 5") 7i: 0 ‘M W 7.5
DISTANCE mm —

 

Fig. 2. Direct trace (heavy line) of force-distance
curve obtained for a G. F. Texture Profile
on a cylinder of pear tissue in the Instron
machine. The test consists of two complete
compressioi-decompression cycles .

(Bourne, 1968a)

13
or cortex and vascula cylinder to the penetrating probe. The third
peak was defined as the maximum force required to penetrate the fruit.
Fruits with a uniform flesh and core area gave no second peak.

Bourne and Meyer (1968b) reported studies on an extrusion type
of texture-measuring test cell as an attempt to measure the texture of
fresh peas. The extrusion cell was mounted on the Instron. The studies
included the effect of plunger speed, effect of annulus width, and effect
of sample size on the extrusion force. The force required fer this ex-
trusion was measured.using green peas as a test material. From.their
studies they found that this type of cell showed promise as a routine
testing instrument in commercial use because of its simplicity in
construction and operation, and.comparative low cost.

Bourne (1969) determined the possible relationship between deform-
ability, which is related to Young's modulus of elasticity, and the
puncture test, which is related to the bioyield point. He measured
the deformability and bioyield of eleven different apple varieties.

The deformability was measured as the distance the whole apple deformed
under a 5/16-inch diameter Magness-Taylor punch between 0.5 kg and 2.5 kg
ferce using a universal testing machine. The bioyield point was measured
using the same 5/16-inch diameter Magness-Taylor punch in the universal
testing machine and measuring the force necessary to reach the bio-
yield point (Bourne, 1965). The resulting data showed that there is
no apparent correlation between firmness as measured by deformability
with the firmness as measured by the bioyield point. Both.measurements
are considered to be an index of apple firmness.

, Finney (1969) gave a brief description of methods for measuring the
texture of meats, dairy products, bakery foods, fresh fruits and vege-

tables, and.processed commodities. He also gave new techniques fer ob-

14

jective evaluation of texture of foods. These techniques were based
upon analyses of sound, light transmission, and vibration phenomena.

Howard and Heinz (1970) stated that there were no reports in the
literature on the correlation between objective methods and sensory
analysis for determining the texture of carrots. These investigators
studied the texture of carrots measuring the compression and shear
strength of individual carrots with the Instron Universal Testing
Machine. Based upon the values reported for uncooked carrots (Bourne,
1966b) as Kc = 28.0 kg/cmz, KS = -0.03 kg/cm, and C = 2.18 kg, they
predicted that the compression measurements would be a better indication
of texture than shear measurements.

A compression device was used with the Instron to deform a carrot
in the radial direction (Fig. 3). A typical compression force—distance
move is illustrated in Fig. 4. The compression test was performed
on carrots that were purchased locally and stored in plastic bags at
0 - 2°C. The carrots were removed from the bags prior to testing and
stored on trays in a conditioning room at 21°C and 65 percent relative
humidity for one to twenty- four hours. Carrots were evaluated manually
for compressibility and flexibility by five well-trained judges, using
a nine-point scale. Compressibility was determined by pressing the
middle portion of the carrot with the fingers and evaluating the resis-
tance. Flexibility was determined by gently bending the carrot at the
middle with both hands. The results showed that compressibility
measurements were highly correlated with sensory hardness as judged
by the taste panel. Shear measurements had a low correlation with the
sensory evaluation.

Breene et al. (1972) determined the TPA parameters of cucumbers

using Bourne's (1968a) method of force vs. distance curves. A sliced

15

 

/

«PART A

 

X; y.“ DlAMETER

 

 

 

 

e—PART B

 

 

 

\

S

Fig. 3. Compression device used for compressing carrots.
Part A is attached to the stationary crosshead of
the Instron. Part B is attached to the moving cross-
head. Space C is where the carrots are placed to
be compressed (Howard and Heinz, 1970).

 

 

zs -

l
2.0 . i
h I
(D I
‘2’ l
8 "5 r- l
9.3 0“ I
. a
3 l0 .. r
-J :
I
I
05 _ l

E b

. 1.x.

0 l 2

TIME (MIN)

Fig. 4. Typical compression force-distance curve obtained
for carrots. -(a) Load, with crosshead moving. -(b)
Distance proportional to the change in diameter of the

carrot compressed under a 2.5 pounds load Howard
and Heinz, 1970). (

16

specimen of one- inch diameter and one-centimeter long was used for the
test. The specimen was placed on the load cell of the Instron machine,
and subjected to an axial load. The results of this test indicated

variability in texture from one end of a cucumber to the other.

2.2 Rheological Models

Rheology is defined as "a science devoted to the study of defor-
mation and flow", Mohsenin (1970). Therefore, when the action of forces
result in deformation and flow in the material, the mechanical properties
will be referred to as rheological pr0perties. The rheological behavior
of a material is expressed in terms of the three parameters; force,
deformation, and time. Examples of rheological properties are time-
dependent stress and strain behavior, creep, stress relaxation, and
viscosity. The rheological behavior of linear viscoelastic materials
can be explained and interpreted by use of mechanical models consisting
of springs and dashpots. Based on experimental evidence, agricultural
products are viscoelastic. From the very limited data available in
this area, it appears, however, that the viscoelastic behavior is non—
linear. Since solutions of non-linear problems are very difficult to
obtain, the general procedure has been to make simplifying assumptions
and apply the theories of linear viscoelasticity in an attempt to explain
the rheological behavior of agricultural products.

The two basic mechanical elements used in mechanical models are a
spring which obeys Hooke's law and a dashpot with the pr0perties of a
Newtonian liquid. The two elementary combinations of these elements,
known as the Kelvin Model and Maxwell Model, are used as the rheological
models. Maxwell's model is usually used to represent stress relaxation

under constant strain. Kelvin's model is usually used to represent

17

creep under constant stress. The two models, the stress relaxation and
the creep are illustrated in Fig. 5.

Viscoelastic behavior of agricultural products has been studied
by many investigators.

Barkas (1953) found that the resistance of organic materials to
deformation is mainly dependent on the moisture held by molecular
forces with the capillary water having little effect.

Zoerb (1958) studied small core samples of wheat kernels and found
the response to be more non-linear approaching an elastic-plastic be-
havior with strain hardening tendencies .

Stewart (1964) fomd that an inter-relationship existed between
the wheat kernels moisture coitent and their viscoelastic properties.

Reidy (1970) in an attempt to find relationships between engineering
and texture parameters of pre-cooked freeze-dried beef, developed two
models: a) a four-element linear viscoelastic model of Kelvin and
Maxwell bodies in series (Model 1); and b) an empirical constitutive
equation (Model 2) which contained a probably non-linear term. He con-
cluded that: a) Model I successfully predicted relaxation functions
of the freeze-dried product; b) Model 2 predicted responses to relaxation,
creep, and cyclic tests. This model predicted the texture indices of
hardness and Chewiness; c) water activity had an influence on the
stress-strain behavior of pre-cooked freeze-dried beef. Resistance to
deformation decreased at higher moisture contents; and d) relaxation
stresses decreased with increased water activity.

Herum et a1. (1973) studied the viscoelastic behavior of soybeans
due to temperature and moisture content. They determined the time-
dependent uni-axial moduli of intact soybeans by relaxation tests in

parallel plate compression. Four temperatures and four levels of

18

 

 

Strain e
E m
Time, t
Strain vs. time relationship corresponding to step
function stress history.
Stress o

 

 

 

Time, t

Stress vs. time relationship corresponding to step
function strain history.

Fig. 5. Kelvin and Maxwell models showing creep and stress
relaxation characteristics (Sharma, 1964)

19

moisture content were used in the tests in an effort to determine if the
techniques of time- temperature and time-moisture shift factors could

be applied to describe a relaxation modulus for intact soybeans. Their
goal was to identify the individual and joint contributions of temp-
erature and moisture content upon the overall response. They concluded
that soybeans may be described as thermo-rheologically and hydro-
rheologically simple.

Bashford (1973a) studied creep and relaxation of meat related to
tenderness. In his study Bashford found that rheological parameters
obtained from creep and relaxation investigations correlated to taste
panel evaluation. He concluded that these parameters are potentially
good indicators of meat tenderness.

Chen et a1. (1971) deve10ped a computer program to determine the
coefficient C i and exponential “i in the general relaxation equation:

-a.t’

n
F(t) = 2 Ci e 1
i=1

for the generalized Maxwell relaxation model,

where

3/2 - . - 1/2
1 3/2 a KEi étl e 0‘1“1 T ) I de

0
II

and

“i = 51/"1

Bashford et al. (1973b) developed a computer program to calculate
the elastic and viscous parameters for the generalized Maxwell and Kelvin
models. Maxwell's model is used as a relaxation model. They used the
general form of relaxation equation F(t) =3 C i Eat/Ti to determine

1=1
the coefficient C1 and eXponential t/Ti.

20

2.3 Hertz Contact Problem

Hertz (1896) proposed a solution fer contact stresses in two elastic
isotropic bodies, such as the case of two spheres of the same material
touching each other.

In his problem, Hertz attempted to find answers to such questions
as the shape of the contact surface, the normal pressure distribution
on the surface of contact, the magnitude of the maximum pressure, and
the approach of the centers of the bodies. In developing his theory,
Hertz made some fundamental assumptions.

These assumptions, given by Kosma and Cunningham (1962) are the
following:

1. The material of the contacting bodies is homogeneous.

2. The loads applied are static.

3. HOoke's law holds.

4. Contacting stresses vanish at the Opposite ends of the body

(semi-infinite body).

5. The radii of curvature of the contacting solid are very large

when.compared with the radius of surface of contact.

6. The surfaces of the contacting bodies are sufficiently smooth

so that tangential forces are eliminated.

The first known application of the Hertz solution for contact
stresses in agricultural products was reported by Shpolyanskaya (1952),
who applied it to evaluate modulus of elasticity of wheat kernels.

Morrow and Mohsenin (1966) have applied the results of Hertz's
work to calculate the relaxation modulus and creep compliance of apples.

Fridley et a1. (1968) obtained experimental force-deformation
curves for peaches and pears and compared them to the theoretical curves

as calculated using the Hertz equation fer plate against sphere.

21

Horsfield et a1. (1970) applied theory of elasticity to the design
of fruit harvesting and handling equipment for minimum bruising. They
extended Hertz's contact theory, given by Timoshenko and Goodier (1951),
for the colliding of two spheres, to determine internal shear stresses
generated under impact considering effects of the modulus of elasticity
and radius of both the fruit and the impact surface. They discussed
experimental techniques for measurement of modulus of elasticity under
impact loading. They concluded that the modulus of elasticity and surface
radius of both surfaces, in addition to the impact energy and flesh
strength, permit meaningful prediction of bruising and serve as good
criteria fOr design of machines to reduce bruising.

iMohsenin (1970) reported several applications of Hertz's contact
theory on agricultural products. He reported the formulas to calculate
the modulus of elasticity for a convex body of an agricultural material
being tested under a steel flat plate or under a spherical indenter.

Hoki (1973) studied the mechanical strength and.damage analysis
of navy beans. In his study, Hoki used Hertz's contact theory given
by'Thmoshenko and Goodier (1951) to determine the radius of the contact
surface and the approach of two spheres when compressed together. He
concluded that: a) using the contact theory to predict mechanical
damage of navy beans showed promise; and b) it is appropriate to apply
the contact theory to predict bean deformation under static loading for
beans with low moisture content.

The solution of the viscoelastic counterpart of the Hertz problem
in elasticity can be deduced from the elastic solution (Lee and Radok,
1960).

Foppl (1922) derived an equation to determine modulus of elasticity

of a cylindrical specimen compressed in the direction parallel to the

22
longitudinal axis. His assumption of the contact surface pressure
differs from that used by Hertz. Fbppl used a parabolic distribution

while Hertz used the semi-ellipsoid distribution.

III. PRELIMINARY STUDIES

me to the fact that a very limited amount of information was
available on the texture evaluation of carrots, preliminary studies
were conducted. The Texture Profile Analysis (TPA) parameters , modulus
of elasticity of the two distinct areas of cross-sectional samples of
carrots and Poisson's ratio were measured. The effect of storage time
on these parameters was also investigated.

In looking at a cross-section of a cylindrical sample taken from
a carrot, one can identify two areas, a center core, dark in color, and
outer ring, light in color. In a cylindrical sample of one-inch in
diameter, the center core is approximately one-half—inch in diameter.

Preliminary measurements of the modulus of elasticity for the center
core and the outer ring were made using axial loading of cylindrical
samples. The results showed that the modulus of elasticity, EC for the
center core is approximately twice the modulus of elasticity, Eo for
the outer ring (Table 4).

Moisture content of the whole sample was measured for fresh carrots
and for carrots stored in the Aminco chamber at 70°F and 50 percent
relative humidity. Also, the moisture contents of the center core and
the outer ring of samples, stored at 120°F and 50 percent relative
humidity for 14 hours, were measured (Table 5) . The results of the
moisture content measurements showed no significant changes in the mois-
ture content when measuring for the whole sample or the center core.

But the moisture content changed significantly in the outer ring of
the sample.
23

24

Table 4. JModulus of elasticity for the center core and the outer
ring of six fresh carrot samples.

 

 

 

Sample No. E0 EC
psi psi

1 796 1475

2 847 1000

3 847 1750

4 593 1250

5 847 2000

6 847 1375
Ave. 796 1475

 

Table 5. IMOisture content of fresh carrots and of carrots stored under
different conditions and.time periods.

 

M.C. % of center core
and outer ring of

 

 

Sample M.C. % of M.C. % of carrots carrots stored at
No. fresh carrots stored at 70°F 120°F and 50% R.H.
and 50% R.H. for for 14 hours
24 hours center outer
core ring
1 88.7 85.6 84.0 80.2
2 86.3 84.3 84.8 72.3
3 85.7 83.7 84.8 72.3
4 87.3 84.1 85.2 75.6
5 88.5 83.5 85.5 79.5
6 88.9 85.0 85.5 79.5
7 88.8 83.2 84.8 79.1
8 87.9 84.0 84.4 76.2
9 86.3 84.6 83.6 78.1
10 87.5 84.5 84.6 82.4

 

Ave. 87.6 84.5 84.7 77.5

 

25

A Texture Profile Analysis (TPA) parameters were determined by
the method described by Bourne (1968a) from.force-defbrmation curves
produced.upon twice compressing each sample 0.1-inch on the Instron
Testing Machine .

Cylindrical samples of one-inch in diameter and 3/4-inch long,
were compressed axially. Typical "first bit" and second bit" curves
were as shown in Fig. 6.

The TPA parameters tested were:

1. Hardness

2. Cohesiveness

3. Gumminess

4. Brittleness

.Although there was a noticeable change in the appearance of the
carrots after seven hours of storage, the data showed no significant
change in the values of the measured parameters, especially the hard-
ness parameter. Hardness data fer samples of fresh carrots and carrots
stored at a temperature of 70°F and 60 percent relative humidity for
24 hours are given in Table 6.

It was concluded that axial loading of the cylindrical samples
was not usefu1 in detecting the change in the texture parameters such
as hardness. The nonsignificant results in determining the TPA para-
meters is believed to be due to the difference in the values of modulus
of elasticity of the center core and the outer ring of samples. It.may
also be due to the nonsignificant 10Sses of moisture from the inside
core compared to the outer ring. In defonming the samples axially, the
center core dominates the reaction to the applied ferce at a given
defbrmation, because it is stiffer and contains higher moisture than

the outer ring, leaving the outer ring to fellow the behavior of the

Force, lbs.

 

26

Hardness
i
//
A1

 

 

Fig. 6.

Defbrmation, inches

Typical ferce-defOrmation curve for TPA parameters test.

27

Table 6. Hardness parameter measurements using the TPA.method
of axial defermation. The hardness is measured from
force-deformation curves produced upon twice compressing
cylindrical samples of carrots previously stored in per-
forated plastic bags at 32°F and 95% Relative Homidity
fer three weeks then removed from the bags prior to testing
and stored in a room at 70°F and 60% Relative Humidity for
24 hours

 

Hardness, lbs. obtained at different storage times at
Sample room conditions of 70°F and 60% Relative Humidity
No.

 

 

0 ‘ 4 7 16 21 24

hours hours hours hours hours hours
1 40 38 37 30 33 23
2 35 38 36 34 34 20
3 40 36 37 32 31 21
4 38 39 38 35 30 26
5 35 39 35 33 32 20

 

center core under the applied load. The outer ring from the previous
experiment, appears to be controlling the decrease in the moisture con-
tent. It receives the first action when the carrots are bit or physi-
cally tested for consumer acceptance of fresh products.

This conclusion suggested that the samples be loaded in the radial
(parallel to the longitudinal axis) direction between two flat steel
plates and that the moisture content be measured in the outer portion

of the cylindrical samples.

IV. THEORY

Mechanical properties have been defined as "those properties having
to do with the behavior of the material under applied force".

Mechanical prOperties are widely used to study engineering materials
or non-biological products. As of late, they have also been used to
characterize agricultural products. The modulus of elasticity is one
of the properties which is of considerable interest. The Hertz contact
problem has been one approach to determining the elastic modulus in
agricultural products.

Hertz's problem of contact stress, was originally developed to calcu-
late the stresses resulting fron the contact of common engineering
materials. One result of the theory is an equation giving the modulus
of elasticity, E, for a convex body compressed under a flat steel plate.

This equation is

 

E = 0.338 k3/2 F (1 - v2) (_1_ + l )1/2 [1]
(13/2 R R’
where
o = Total deformation of the body, in.
F = Compression force, lbs.
k = Constant taken from a table (A-l)

R = Minimum radii of curvature, in.
R’= Maximum radii of curvature, in.
The total deformation of the body along the axis of load at the

point of contact is giver by

28

29

/3

Q
ll

1
331W K2 ($112)]
[21

where

The value of k depends on the principal curvatures of the bodies
at the point of contact and the angle 45 between the normal planes con-
taining the principal curvatures. This value can be obtained from
Table A-1 by first calculating cos T, for general case of two bodies,

1 and 2 pressed together, from

 

1 1 2 1 1 1 1 1 1 1 2
cos T = [ (1:1 Rf) "' (Ti-2 fi, ) + Zffil Rf) (1:2 R5 ) C05 24’]
l l l l
(R1+R1+R2+R’) [3]
where

R1 and Rf are minimum and maximum radii of curvature for body
one, and
R2 and R5 for body two.

The preceeding equations are valid when the specimen is a sphere.
When the specimen is a cylindrical (i.e. R1 = co) cos T becomes one and
the value of k is not defined.

To determine the elastic modulus for a cylindrical specimen com-
pressed in the radial direction between two flat plates, an equation has

to be derived based upon Hertz's contact theory.

30

4.1 Derivation of an equation to determine modulus
of elasticity of a cylindrical specimen compressed
radially.
Timoshenko and Goodier (1951) expressed the deformation at the
surface of contact of two bodies as

a = (K1 + K2 )1; q d" + BY2 + (1x? [4]

r
In which B and C are constants depending on the magnitudes of the prin-
cipal curvatures of the surfaces in contact and on the angle between the
planes of principal curvatures of the two surfaces. If R1 and Rf denote
the principal radii of curvature at the point of contact of one of the
bodies, R2 and R5 those of the other body and v the angle between the
normal planes containing the curvatures l/Rl, and l/RQ, then the con-

stants B and C are determined from the equations

_ 1 1 1 1
B + C - 2 ( R1+ Rf + R2 + R5)

_ 1 l 1 2 1 l 2 1 l 1 1 1/2
C - B - 2 [(RI Rf) + (R2 - R5) + 2(R1 - Rf)(R2 — R§)COS 2?]

where

total deformation of the cylinder (in.)

52
ll

q = pressure at any point of load distribution (psi)
dA = element area on the area of contact (in?)
r’= the distance from the center of the area of contact to

the element area dA (in.)

2
Kl_1-v1

 

11E

2
_ 1 ‘ V2

75
N
I

 

11E

31

v = Poisson's Ratio
E = Modulus of Elasticity (psi)
In the case of contact of a cylinder with a plane surface, Figs. 7

and 8, B and C become

B+c=._1._
2R1
C-B=_1_
2R1
whichmeansthatB=0andC=_1__.
2R1

Taking X to be equal to one-half the width of the contact area,
b, at maximum deformation, equation [4] becomes

2
a=(x1+1<2)rrfli‘5+9_ 5
r’ 2R []

The problem now is to find a distribution of pressure to satisfy equation
[5]. Hertz showed that this requirement is satisfied by assuming that
the intensity of pressure q over the surface of contact is represented
by the ordinates of a semi-ellipsoid constructed on the surface of con-

tact as shown in Figs. 9 and 10.

q=§gv’52-xz [6]

The maximum pressure qo is clearly along the center line of the
surface of contact. The magnitude of the maximum pressure is obtained

by summing forces in the y direction, yielding

P = 2 rL/2

b _ b
'L/2 f0 QdA - 2L [0 de [7]

13.4.3513 12’de
L b

32

 

 

 

 

Pl

Fig. 7. Cylindrical sample before compression

> T
‘\ H? 1 0/2

T

 

Fig. 8. Cylindrical sample after compression

33

 

 

      

‘ ioi ; a . .
..l!
flillijl
.n——-—b————ai

1......x

 

 

 

Fig. 9. Semi-ellipsoid pressure distrilartion

 

 

 

 

2
Fig. 10. Pressure distribution over the surface of contact

34

p’ = qob" and qo = 2P’ [8]

2 m:

 

where P is total load and P’ is load per inch of length.

Timoshenko and Goodier give the expression

P’ (K1 + K2) Rle -
b =f 19]
R1 + R2

for the case of contact of two cylinders with parallel axes. R1 and R2

 

 

are the radii of the two cylinders-

When a cylinder contacts a flat surface, R2 becomes 00 and b becomes

 

b = f4P’ (K1 + KZTRI
If the cylinder is a biological material with small modulus of elasticity
E1, and the flat plate is made from steel, E2 = 30 x 106 psi, then
K2 + 0 and b reduces to
b = W [10]
where R is the cylinder radius.

Equation [5] now becomes

2
a=KfffEfi+L [11]

r’ 2R
Utilizing equation [11] and the assumption of semi-elliptic pressure
distribution for the case where the cylinder is loaded with a flat steel
plate, (Fig. 11), an equation for the elastic modulus, E, of the
material can be derived as outlined below.

Starting with [11] and substituting [6] gives

2
a-L =4“?ng 19./”W -x dxdy
2R br’

35

where the integral is over one-fourth of the contact area. Two
changes are now in order if evaluation of the integral is to be simpli-

fied. The first is a change in variables by letting

This change produces

 

2 1 r-Trt——7—2
a-9_=4qu fo /_b____i_)_§. deXdY
2R b 0 ° br
or
2
o-P_=4qubf:1f;v’l-X1dXdY
2R r
where
11:1;
2b

The second involves dividing the contact area (Fig. 11) into two triangles
‘with e and o defined as shown in Figure 12, and then chaning the equation

into the Polor Coordinates. This transformation yields

-1
2 _, Z T
1 (a _ E.) = IEan M fiece //l r cos 6 r dr d6
4Kq b 2R r

-1
tan l/M Msec¢
f f

O O

 

+

 

 

 

“/1 - rYSinze r dr de

1'

 

or

 

2
4K b (C! ' 13—) = II + 12.
qo 2R

36

Considering the individual components 11 and 12 gives

 

-1 - -
_ Z. Z
11 = If)“ M 13°C 6 /i_29_5_9dr d6
1'

 

° 0 c050
t ’111 m d‘

I an _ sec 0 d0
O 4 3.

 

=31. iogem+/T+—M“27 -
4 j
11..

 

and
-1 2.2
12:15:81) l/Mflglseco /1-r51n¢rdrd¢
r

 

-1
= fie-n W 113“” W (N (21¢, Y = r sine
sine

-1
tan l/M 651,14 y + %Y J l - Y2]l:I tam csce d4

0

= f

 

-1
1 _ w
= % [San ”’1 [sin 1 (M tamp) + M tan¢ / 1 — MYtanZ¢]csc¢d¢

sin v

’

 

Substituting sin v = M tamp, sine =
/M2 + sinzv

d¢=Mcosvdv

M2 + sinzv

yields

 

 

 

MA—w- —
u.

 

 

 

Fig. 11. Area of contact

 

 

 

 

Y
I L I -1
M 'Zb. 9 tan M
1 r’
9,
l- a
o " ' 1 K

 

Fig. 12. One-feurth the area of contact divided into
two triangles

 

38

 

 

12 = l fH/z (v + sin v cos V) COt v dv
2 ° / .2 2
1 + (Sin v) /?M
since
M>2,(sinv)/M<—1-,
wecanexpand
.7 -
(1 + srn v) 1/2
M?

by the binomial theorem as

 

 

 

 

sin v -1/2 1 srn v 1 . 3 sin“ v
(1+ 2 ) ‘ [1 "‘ 2 +2— 1" 1+ ..... ]
.M .M M
Then 12 may be written as
0 2 .
I2 = l-fn/Z (v cot v + coszv) (1 — l_srn V + l—' 3_srn“v.. ) dv
2 o 2 M2 2 4 'M“

Integration by parts gives

I“2 v cot v dv = - fn/2 log sin v dv = E.log 2
O 0 e 2 e
and
fg/2 v cot v sin2n v dv = fg/2 v sinzn'1 v cos v dv

. 2n
_ v srn v n/2 l H/2 . 2n
‘[—_2—n—] *23'. 51" “1"

0

II
=— [1 - 1’3'50000 2n " 1)]
4n 20406000211

 

 

 

39

From that 12 becomes

 

 

 

 

II 1 1 1 1 1 11
12=rlfloge2*2‘)'2(z'r*z 2111—12“
1 3 1 1 1,3 1 1 3 1
zrfr‘r 7 1+2 4 63m“
1 3,5(1_1 1.3.5+1,1,3,5)_1]
246662462468M5
12=£[1ogez+%- 3 + 21 -__£9_5___]
4 16 M2 256 M” 6144 M6
12=I-1-[loge2+%-- 3 (l— 7 + 98 ]
4 16 M2 16 M2 384 M"

Since

— _ 2
m“1 X+X .....

 

 

 

 

 

7 98 . . 1

(l - + ) 15 approxmately -———-7———

16 M2 394 M“ 1 +——- .

16 M7-
and 12 becomes

11 l 3 l

12‘111082+7‘ )
e
16 M2 1 + 7
16 M2
3

II 1
4-[10ge2+_2-

40

Then
1 _ b2 _m r—————Z— 1 3
'——-—4qub (a 712') -E[10ge 04+ 1+M) +10g92+7-————16M2+ 7
_II 1 3
-__[log 2(M+/I+MZ)+7- ———————
4 ° 16M2+7
But
2P’
qo TIE
-/T—’Ifii -b2
b” P »K‘1p—m
= L
M rs
Therefore

2

b
a-Q-fi=Hqub[loge(M+/TrMZ)+-%—- 3

16M2+7

b2 1 3
= [log 2(‘M+/-l_+—W)+—-————
.2?- e 2 1m2+7

2
a=%§[logeZ(M+/T_+—M7)+%-__3__
. 16M2+7

Letting W = TI; = 2M yields the following equation

 

01R 1 3_ 3
175‘sz [1°gem+'4+w)+7 4w2+7] “21

 

41

Table A-2 was prepared from equation [12] for values of W at

different values of 9;. Once W is know, E can be calculated as

follows

 

_ L ___ L
W ‘ b" ___“
(4P’KR
L2
K __7—4P’W R
1 - v2 = L2
1113 4 P’WZR
_ 2 e 2
E=4(1 v)PWR [13]

 

mL2

4.2 Fbppl's Equation
An equation which gives the modulus of elasticity for a cylinder
compressed by a flat plate was found after [13] had been derived.
This equatior was developed by Fbppl (1922). There is a major difference
in the assumptions used by Fbppl when compared to the work done by
Hertz. Fbppl used the second power of the semi-elliptic function for

the pressure distirbution. His equation for q was

q = 32 (b2 - x2) [14]
b2

which resulted in the following relationships

1 -v2
TIE

2R

o = 4P’ ( ) (1/3 + loge —b) [15]

 

- Im-

 

 

42

=8 (I-vZ) P’ 22
RD

E

 

[16]

where Z = R/b and values are given in Table (A-3).

4.3 Relaxation test
Stress relaxation experiments have been conducted on several agri-
cultural products. In this rheological test, the specimen is suddenly q
brought to a given deformation (strain), and the stress required to 1

hold the deformation constant is measured as a function of time. Since

 

the deformation of the product under load is held constant, it is usually

'F‘fi .- a x .-
‘Hu_m;

assumed that the loaded area of contact remains constant during the
relaxation test and the recorded force- time is representative of the
stress-time curve.

The generalized Maxwell model can be used to represent force or
stress relaxation. A generalized Maxwell model is composed of n Maxwell
elements with a spring in parallel with the nth element as illustrated
in Fig. 13. When the model is subjected to a constant strain so at

time = 0, the stress can be represented by

0 (t) = so (Eli—2"1t + E2é°2?..+ Ené°nt) [17]

where on = En/nn, E is stiffness or modulus of the spring, m is viscosity
coefficient of the liquid in the dashpot. Then the equation representing

the portion of the curve where t > t1, (Fig. 14) can be written as

1 . éalt’ [18]

"Mb
('3

F(t) = i

43

where C1 = to E1

a3/2 K 1:1 é°1(t1 " ‘)1:2 dr

II
(N
\
N

E
O

t, t'tl
a = rate of deformation in./min.
Factor K is being a function of the geometric parameters of the specimen.

The approximate values of C1 and a1 can be determined by the num-

erical methods which used a computer program developed by Chen (1971).

 

 

 

44

4%m

1

 

 

 

 

 

 

 

El En
n1 ___] m2 LP] mg] I [___] on
F
Fig. 13. Generalized Maxwell model
/
/
I
I n
/ - , _ _ .
I %K 113/2 2 Bi [1:1 e “131 T) :1/2 dr]e “It
/ i=1
8
u. \
.3: 1 \.\
8 *\_
12
0 t1
Time, t
Fig. 14. Graphical representation of equation [18]

’

 

45

V. EXPERIMENTAL INVESTIGATION

5 . 1 Equipment

 

To obtain one-inch diameter samples, one-inch long, a corer (A),
two inches long, was constructed with an internal diameter of one- inch
and mounted on a base (B), (Fig. 15). A holding jug (C), one-inch long

with internal diameter of one inch, and two-bladed lmife (D), (Fig. 16) ,

 

were used to trim the samples to a proper length of one inch for the (1+4..-
radial and Poisson's ratio tests.

To measure the restrained modulus needed to determine Poisson's
ratio, v, a cylindrical die (E) with a one-inch inside diameter mounted
on a base (B) with spacer (F) of outside diameter to exactly fit inside
die (E), (Fig. 17), were used.

Another corer (C), (Fig. 18) , with an outside diameter of 7/8-inch
was used to separate a l/8-inch thick outside ring from the one-inch
diameter sample. This ring was removed after the radial test, for the
purpose of measuring the sample moisture content.

An Aminco-Air unit was used to maintain a constant temperature
and relative humidity in the chamber where the carrots were stored,
(Fig. 19).

An Instron Universal Testing Machine, table model TM, 200 kilograms
capacity, with standard crosshead speeds of 0.2 to 50 in/min. was used

to load the samples, (Fig. 20).

(B)

   

Fig. 15. Sampler, (A) is Corer and (B) is Base.

 

Fig. 16. Trimmer, (C) is holding jug (D) is two-bladed knife.

 

47

(E)
(B)

 

 

Fig. 17. Die to be used in measuring the Poisson's Ratio,
(B) is base, (E) is sample holding die, and (F)
is spacer.

Fig. 18. Corer with 7/8" outside diameter to be used to

separate 1/8" thick outer ring from the 1"
diameter sample.

 

48

   

Fig. 19. Aminco unit with chamber to store carrots at
given temperature and relative humidity.

 

Fig. 20. Instron Universal Machine.

49

5.2 Experimental procedure
The modulus of elasticity is to be calculated using the equation
derived in the previous section. Poisson's ratio and the deformation
force are variables in the equation and therefore must be measured. The
relaxation test is to be performed in order to study the relaxation

behavior under constant strain.

 

Hughes and Segerlind's (1972) method of measuring the Poisson's r—
ratio was used. TWO similar cylindrical samples of one-inch in diameter :3
and one- inch long were removed from the individual carrot (length-wise)
using the corer (A) and the base (B), the holding jug (C), and the
two-bladed knife (D). g

One sample was axially compressed in the Instron Machine, (Fig. 21)
for the unrestrained test. The other sample was placed in the die (E)
as shown in Fig. 22 and then compressed. The two force-deformation
curves were obtained (Fig. 23) and used to determine Poisson's ratio.

Carrots grown in the State of Arizona and purchased through the
Michigan State University food stores were stored in the Aminco unit
chamber at 85°F and 50 percent Relative Humidity for one to 72 hours.
Samples of whole carrots were drawn from the Aminco unit chamber and
cylindrical samples of one- inch diameter and one-inch long were prepared
as described above. The cylindrical samples were placed in the Instron
Machine and compressed radially, parallel to the longitudinal axis,
(Fig. 24), using a crosshead speed of 5 in. /min and a maximum deformation
of 0.1 inches at chart speed of 50 in. /min. A force-displacement curve
similar to the one in Fig. 25 was obtained. The experiment was conducted
using fresh carrot samples and samples drawn from the Aminco lchamber at

time intervals of 4, 6, and 8 hours over a period of 72 hours. Moisture

50

"U

 

 

 

 

Fig. 21. loading for the unrestrained test.

 

mar (F)

   
  
   

 

/
/ I

a

I 7 ,I r
' I, // W
. '1‘
I I
- . '1’ .1 . /
' . ' ‘ x/
’1 . . .
.' I
I ,'
/

I
/
/
//
.l,.

'e (B)

 

37/
.-/

/ /
. - y _I
. / .’
I . ' .‘i I" ’

 

fiample

 

,/

./ ,/..

 

I v f :7 [I /

 

 

e(B)

 

 

80

70

60

50

40

30

20

10

51

/ Restrained

 

 

 

 

 

 

- f/ 51 = 4.75
R - §_%.= 0'2 = 0.041
s1 4.75
R= (1+v) (l-Zv)
: (1 ' V)
b- j!
I v = 0.495
r-
Unrestrained
P /SZ = 0'2
1 L 1 L 1
0 0.02 0.04 0.06 0.08 0.01

Deformation, inches

Fig. 23. Force-deformation curves for a cylindrical sample of
carrots with one-inch in diameter and one-inch long.

 

52

   

Fig. 24. Sample loading, in the radial direction or
parallel to the longitudinal axis.

53

 

 

Force , lbs .

 

 

Deformation, inches or time, sec.

Fig. 25. Typical curve for defamation and relaxation tests.

54

content was determined by removing a 1/8-inch thick outside ring cut
from the cylindrical samples and dried in an oven at 70°C for 20 hours.
Two sets of data were obtained, 138 samples in Set #1 (Table A-4)
and 70 samples in Set #2 (Table A-5) . The moisture content of the
l/8-inch thick outer ring was determined for each sample. The samples
were divided according to the peak compression force into 11 groups
and the average force and moisture content of the outer ring of each
group were obtained (Set #1, Table A-6; Set #2, Table A-7). By
grouping the data according to force, it was possible to select
force-deformation curves with a peak force equal to the group average

force and then use the selected curves to obtain the relation between

 

deformation force, modulus of elasticity and moisture content.

The relaxation behavior was studied by maintaining a constant
defamation (strain) of 0.1-inch on the sample for 30 seconds. A
crosshead speed of 5 in./min and chart speed of 50 in. /min were used.
Welve points on each curve were used to determine the coefficients

and exponential of the generalized Maxwell model.

VI. RESULTS AND DISCUSSION

6 .1 Accuracy of the derived equation

The equation derived for determining the modulus of elasticity
was verified experimentally. The method of examining was a comparison
between the modulus of elasticity values for a homogeneous and isotro-
pic agricultural product as measured by applying an axial load and a
radial load on separate cylindrical specimens. Homogeneous and isotro-
pic materials should have the same modulus of elasticity value in
both axial and radial directions. The potato was used for the speci-
mens since it represents a homogeneous and an isotropic agricultural
product. Specimens with variois combinations of 0.5-inch and 1.0-inch
diameters and lengths were prepared and subjected to a deformation
in the axial or radial direction, of 0.025 inches, 0.05 inches or
0.1 inch. The Instron Testing Machine was used to defonm the cylin-
drical samples. A cross-head speed of S in/min and a chart speed of
50 in/min were used. TWO specimens from the same location of the potato
were prepared. The modulus of elasticity was calculated using the axial
equation, the derived equation, and F6ppl's existing equation.

Assuming Poisson's Ratio, v, for potatoes to be 0.5, the derived
equation [13] and FOppl's existing equation [16] become:

- 2
Derived: E1 =4(1 0-25) 13’sz ___. 0.955 P’W R

H L2 L2

 

[19]

55

 

56

= 8(1 - 0.25) P.22 = 1.91 P’22
IID D

Poppl's: E2 [20]

 

 

The axial loading equation to detenmine the modulus of elasticity
is:

E3:

>l'11
QIL‘"

[21]

A summary of the calculated values of B for different specimen
sizes and defonmations is presented in Table 7. The averaged values of
B were 415 psi, 496 psi, and 419 psi as calculated by the derived
equation, FOppl's equation and the standard axial equation respectively.
The mean, variance, and standard deviation were calculated (Table
.A-8) and an F statistical test was used to determine whether the popula-
tion variances are equal or not. The null hypothesis of = oi was tested
at the 0.05 level. The results showed that there is insufficient evi-
dence to indicate a difference in the population variances. The conclusion.

, 2 2 . .
15 that of = 02 = 03 15 not reJected.

6.2 Poisson's ratio

The Poisson's ratio for fresh carrots was found to be approximately
0.5, The value fer carrots stored in a room temperature of 70°F and
relative humidity of 60 percent for a time period of 24 hours was found
to be essentially the same, 0.50. Twenty-one samples of carrots were
studied in each case. The average values were 0.492 for fresh carrots
and 0.496 fer stored carrots (Table 8). The derived equation [13] reduces

to [19] for carrots since Poisson's ratio is 0.5.

 

57

9}. .-

..'I‘P'

 

 

 

mfie woe mHe name
emm «em Nme AH.m Ne.m omo.o mNo.o m.m N.m m.o m.o mNo.o
Ram ewe was NH.m oe.o~ omo.o mNeoo.o m.e w.~ o.H m.o mNo.o
omm ONm mam mm.h em.o mmc.o omo.o m.e m.mH m.o o.H mmo.o
mam emm ome we.“ mm.mH mmo.o m~ao.o 0.4 m.m o.H o.H mNo.o
«me hoe mNe em.m om.e CH.o omo.o o.HH m.m m.o m.o mo.o
wee com oem em.m mm.ma oa.o mNHo.o R.HH 0.4 o.H m.o mo.o
mew mew Nam afi.m w~.e mo.o o~.o ~.HH 4.0m m.o o.H mo.o
mow see man 5H.m Ne.m mo.o mNo.o H.m w.mH o.H o.H mo.o
«me new man em.m om.~ H.o om. m.HN H.me m.o o.H H.o
flee Hfie mam em.m em.o H.o mo.o m.mH n.4m o.H o.H H.o
emu uma awe N 2 n\5 .wu. :a\na DH seen seen such

mm mm Hm as a m A a a

a a. a N4
.83 35308 880 3 < when: w. m. a mm .MNIM 34 u «m .Immzlum 3.3 u 5.

“momma nomads—poms Ea menwm 553on 3309 30.8.33 pom m mo 833 woumgoamo mo bfiefim .N. 038.

Table 8. Poisson's ratio, v, measurements for fresh carrots and for
carrots stored in a room at 70°F and 60% relative humidity
for 24 hours.

Sample
No.

KOWVO‘UTbQNNF-I'

58

Fresh
carrots

0.493
0.493
0.495
0.492
0.494
0.493
0.494
0.492
0.493
0.489
0.493
0.488
0.483
0.494
0.494
0.485
0.490
0.490
0.486
0.497
0.495

0.492

24 hours
stored carrots

.496
.497
.496
.497
.498
.496
.497
.498
.494
.494
.498
.495
.495
.498
.497
.497
.497
.495
.495
.493
.496

OOCOOOOOOOOOOOOOOOOOO

0.496

 

The relationship between the peak force needed to deform.the sample
0.1 inch in the radial direction and the moisture content in the outer
ring (percent wet basis) was plotted from the results summarized in
Table 9. .A best fit-curve was drawn using a statistical sub-routine
(Fig. 26). A second degree polynomial curve was found to be the best
fit of the plotted data. This curve showed a significant relationship
between the peak force and the moisture content.

content decreased, the force needed to deform the sample, also decreased.

6 . 3 Deformation Test

As the moisture

 

59

A correlation coefficient of 0.937 was obtained for the curve.

The modulus of elasticity, 13, was calculated using the derived
equation, [19] , and the results are plotted as related to the moisture
content (Fig. 27) . The second degree polynomial curve indicated
that there is a significant relationship between the modulus of elas-
ticity and the moisture content. As the moisture content decreased,
the modulus of elasticity decreased. The correlation coefficient was
0.937. The modulus of elasticity at 85 percent moisture content was
725 psi which is almost ten times greater than the value of 70 psi
at 76 percent moisture content. The sharpest drop in the modulus of
elasticity occurs during the first five percent drop of the moisture
content below that of fresh carrots. After this, the rate of decreases
changes less rapidly.

The decrease in the modulus of elasticity is believed to be a
measure of the hardness parameter of carrots . As the carrots lose
moisture from the outer surface of the sample, it becomes less rigid
causing the defamation force to drop.

The results of the defamation indicate that a slight drop in
the moisture content of the outer ring creates a sharp drop in the hard-
ness of carrots. Earlier results had shown no difference in the hardness
of carrots as related to storage time when the measurements were in

the axial direction of cylindrical samples.

6.4 Relaxation tests
The computer program by Chen (1971) was used to calculate the
coefficients C1 and the expanentials 011 using a three element (i=3)

generalized Maxwell model [18] .

 

 

60

 

wwoo.o H5m.o 5o.m vw.o 5N.o mN.o v.H 5N m.N5
owoo.o 5mm.o mo.m mo.H mm.o mm.o w.N em H.m5
omoo.o m5v.o mw.m ow.m oo.H mm.o w.m NHH 5.55
O5oc.o mo¢.o Hm.m wm.m m¢.H NM.H m.w voa m.m5
5moo.o mwm.o om.m H5.w 50.H No.N m.NH HVN w.m5
mvoo.o mmv.o mw.m om.HH OH.N om.N v.0H 5Hm m.Hm
mmoo.o 05m.o mo.N wv.wH mm.N Nm.N o.v~ «ow N.Nw
mmoo.o mom.o mN.m mo.H~ MH.m mm.N m.5N 5mm o.mw
m~oo.o mov.o O5.m wo.5N Nv.m ON.M o.¢m 5mo m.mm
wmoo.o m5v.o em.¢ mm.w~ H5.m em.m v.mm ewe 5.vw
omoo.o va.o mw.m 5H.mm 55.v ow.¢ o.m¢ mow 0.0m

me we :0 mo mu 6 mMH «Emma: .56.:

#0on H3582 woufiehanom 8:53.... 05 Sum wanna—guano mp 335595 98
mucofioflmooo 533309 05 EB .Eumogfi maize Hausafiyaoxa vapoofiom any we auuom Moon
.8333 35.8w 05 Scam Engage mm 3339...? we magmas .3380 0.53on me wonder .m 3an

61

The coefficients C1, C2, C3 and the exponentials a1,'a2, a3,
were determined by taking points from the force-time curve. Table 9
shows the values of C1 and 01 at various moisture contents.

Three relationships between C1, C2, and C3 and the moisture content
'were plotted and the second degree polynomial curves were the best fit
for the plotted data. These relationships are shown in Figures 28, 29

and 30 for C1, C2, and C3, respectively. Each curve shows a signifi-

 

cant relationship between C1, C2, and.C3 and the moisture content with
correlation coefficients of 0.963, 0.965, and 0.927, respectively. The

relaxation coefficients C1, C2, or C3 decreased as the moisture con-

 

i a
a"
i
'0
51
A ". ‘

tent decreased.

The relaxation coefficients C1, C2, and C3 and the expanentials
al, a2, and a3 were used to calculate the farce-time experimental curves
(Table A-9). The experimental and calculated curves of the generalized
Maxwell model, with three elements, are shown in Fig. 31, at sample
moisture contents of 86.6 percent, 83.3 percent, 79.8 percent, and 77.7
percent.

The calculated relaxation curves are identical with the experimental
curves, indicating that the force relaxation curve of a carrot could
be represented by equation [18].

Table 9 shows that there is no significant relationship between the
exponential ml and the moisture content since the data fluctuated with

no behavior pattern.

62

 

 

 

.pmou 539E820 S... Show 3:32 “condom on» no pfiueou 8339: we gamma .3. .mE
w .3: page 05 me 9538 33302
no 00p 0.1m amp 0.0m opp one aim own 0.05.
a
_ i _ n a _ o _ a i _ _ o _ o _ 1 4
L
L
4 l
a 1
i
_ . . . _ _ . _ _ _ _ . _ _

 

OO'T

mfg

W'OT

'3'08 L
°sq1 ‘QDJOd

5'2

5.

.‘OE

E'EE

63

L3

.eoflumsco eo>fihoe one 5: uaumfisuHmu mm mpohhmo mo 5uwuwpmmfio mo measeez. .5N .mwm

w .mewa Hausa can we peoueou ouspmflaz
.mm 0.4m m.on o.pn m.ma o.ma m.mn o.mx-

O
.1
[I]
U)

 

 

L
art

1 l
‘43:
5nI“PQN

J
are

l
'39-.

l l _l 1
'24. . 11?: '63:.
3111/ 5‘11 misuseta JO

'6TB

l
l
'61};

 

64

 

.umou oofimxwfiah 5 poofiuwmmooo umhfl m5 :0 “59:8 8:3on mo uoammm . mm .mfim
» .wfih house 05 mo 95550 830.82

.03 04mm 0.3m 0.3m O.Nm 0.0m O.m5. 0.0.5: 0.35.. O.N5| 0.05.-

 

CO'O

0'“.

0’78 CQ'T

00 E
I 111919155800 ":3

65

 

 

.pmou 83883." 5 uoafiuflmmoou vacuum 05 no 28280 8339: we 903mm .3 .mwm
w .wfiu .830 05 mo 23:00 055302
.00 ohm 0.1m 0mm. 0.0m 01mm 00a 0.15. can con
_
Li _ A i _ _ _ _ a _ _ _ _ o _ _ o o _
17
i
a
a 1
do '1
L
\.. 1

 

0?

09'

(1‘1

03' ’f

' 0-
0

Z 111953133902) ‘ 32)

66

 

.ummp cofiumxgma 5 ucmwummmou v.35 on» :0 33:8 8339: we uuommm .om .mwm
* .3? House 05 mo ucounoo 33302
0.03 0mm ohm 0.1m 9mm 0.0m own OJP Gib 0ND 0.9.
r _
_ _ _ _ a a _ _ _ 1 . _ . _ . _ _ _ _

[1|

I J
I 1
i \ i
.. \ L

\
' \\ IL
\ 4

.l. \m. I;

I. \\ IL
‘\
0 \ J
\

- K 1
n \.. I
. LT

 

 

('1

0'0“

S 11131393903 ‘89

67

 

 

45 -
ll —— Experimental
\‘ ------ Calculated
40 A\
35 '— \~\~§=§\
ll ‘3‘:::~::.-.___ 7.» 86.696 M.C.
h. m:--
‘
a
30 7-\
,fian‘aaflr“; .--.- _ 83.396 M.C.
25 *- -—--—-_--~m
1
20 .—
4
I
15 _
10
_--__________________ 79.8% M.C.

 

  

am 77.7% M.C.

‘- Wis c. -'- do- u d”--- -‘O-‘-----'-~“W'*

   

 

O ‘ l l 1 g l __l_
O 5 10 15 20 25 30
Time, Sec.

Fig. 31. Experimental and theoretical results for relaxation test
at different moisture content. Table (A-9)

 

VII; CONCLUSIONS.AND RECOMMENDATIONS

7.1 Conclusions

The conclusions derived from this study are as follows:

1.

Measuring the change in TPA parameters, using an axial load on F———qg
a sample of carrot, gave no significant results between hard-
ness as a textural parameter and moisture content of the outer

ring. F

 

There are two distinct areas in a cross section of a cylindri- E~ j
cal sample, a center core and an outer ring. The center core 6
has a.modulus of elasticity twice that of the outer ring when

measured in the axial direction of a cylindrical sample.

The moisture content after a period of storing is not unifbrnn

throughout the cylindrical sample. It increases as the center

core is approached.

The Poisson's Ratio of a carrot is approximately 0.5 and is

independent of moisture content.

An equation for calculating the modulus of elasticity, E, of a

cylindrical sample compressed in the radial direction.was de-

rived as
E = fl£l_;_231.p’w2R
nL2
Comparison between the derived equation and Poppl's equation for-
calculating the modulus of elasticity of a radially compressed.
sample gave no significant variation in the modulus values,

indicating that the assumption relative to the pressure

68

10.

11.

12.

13.

69

distribtution may not be a critical factor.

The hardness of texture parameter was found to be an important
parameter as related to moisture content. As the moisture con-
tent decreased, the hardness, as measured to be the force

to deform cylindrical sample compressed radially, decreased.
The force-time relaxation curve can be predicted by using three

elements of the generalized Maxwell Model, i.e.

F (t) = cléalt + czéazt + c3é°‘3t

The coefficients C1, C2, and C3 of the relaxation equation
showed a significant variation when plotted against moisture
content.

The exponential coefficients (:1, a2, (13 showed no significant
variation when plotted against moisture content.

As the moisture content in the outer ring decreases, the cal-
culated modulus of elasticity decreases. The modulus of elas-
ticity is an indication of stiffness.

Moisture content is the critical factor in carrot texture as
indicated by the hardness parameter.

The hardness parameter should be measured as the peak force

applied to deform cylindrical sample radially.

7 . 2 Recomnendat ions

Further study should be made to determine the (TPA) parameters from

force-deformation curves produced by twice compressing a cylindrical

sample in the radial direction.

7“:

 

I
P“. I“.
E‘ v ,— "

:.' La, _

70

A study is recommended on measuring the modulus of elasticity of
carrots stored under different temperatures and relative humidity to find
the range of the modulus of elasticity value as related to the consumer
acceptance of the product. This range could be used as a scale to measure
carrot freshness objectively. The K value (Chen, 1972) in the relaxation
equation should be determined so the modulus of elasticity can be cal-
culated from the relaxation test and compared to the value calculated ”___“.
from the derived equation.

The study of the moisture content distribution from the center to

the outer surface of a cross-section cylindrical sample of carrot is also J .—

 

reconmended . s w..—

 

APPENDIX

71

 

 

72

 

 

Table A-1. values of m, n, and k corresponding to Cos T (After
Kosma and Cunningham, 1962).

Cos T m n k Cos T m n k

0 1.000 1.000 1.3514 0.760 2.111 0.571 1.111
0.020 1.013 0.987 1.3512 0.780 2.195 0.558 1.091
0.040 1.027 0.974 1.3507 0.800 2.292 0.545 1.070
0.060 1.041 0.961 1.3502 0.820 2.401 0.530 1.047
0.080 1.056 0.948 1.3494 0.835 2.494 0.518 1.027
0.100 1.070 0.936 1.3484 0.845 2.564 0.511 1.013
0.120 1.085 0.924 1.3469 0,855 2.638 0.502 0.998
0.140 1.101 0.912 1.3453 0.865 2.722 0.494 0.983
0.160 1.117 0.901 1.343 0.875 2.813 0.485 0.966
0.180 1.133 0.889 1.341 0.885 2.915 0.476 0.948
0.200 1.150 0.878 1.339 0.895 3.029 0.466 0.929
0.220 1.167 0.866 1.336 0.905 3.160 0.455 0.908
0.240 1.185 0.855 1.334 0.912 3.262 0.448 0.892
0.260 1.203 0.844 1.330 0.916 3.326 0.443 0.882
0.280 1.222 0.833 1.327 0.920 3.395 0.438 0.872
0.300 1.242 0.822 1.323 0.924 3.468 0.433 0.862
0.320 1.262 0.812 1.319 0.928 3.547 0.428 0.851
0.340 1.283 0.801 1.315 0.932 3.631 0.423 0.839
0.360 1.305 0.790 1.310 0.936 3.723 0.418 0.828
0.380 1.327 0.780 1.305 0.940 3.825 0.412 0.815
0.400 1.351 0.769 1.300 0.944 3.935 0.406 0.801
0.420 1.375 0.759 1.294 0.948 4.053 0.399 0.787
0.440 1.401 0.748 1.288 0.952 4.187 0.393 0.772
0.460 1.428 0.738 1.281 0.956 4.339 0.386 0.756
0.480 1.456 0.728 1.274 0.960 4.509 0.378 0.739
0.500 1.484 0.717 1.267 0.964 4.700 0.370 0.719
0.520 1.515 0.707 1.259 0.968 4.94 0.361 0.699
0.540 1.548 0.696 1.251 0.972 5.20 0.351 0.676
0.560 1.583 0.685 1.242 0.976 5.52 0.341 0.650
0.580 1.620 0.675 1.232 0.980 5.94 0.328 0.621
0.600 1.660 0.664 1.222 0.984 6.47 0.314 0.586
0.620 1.702 0.653 1.211 0.988 7.25 0.298 0.545
0.640 1.748 0.642 1.200 0.991 8.10 0.281 0.504
0.660 1.796 0.631 1.187 0.993 8.90 0.268 0.472
0.680 1.848 0.619 1.174 *0.995 10.14 0.251 0.432
0.700 1.905 0.608 1.160 *0.997 12.26 0.228 0.376
0.720 1.966 0.596 1.145 *0.999 18.49 0.185 0.278
0.740 2.035 0.584 1.129

 

*These intervals cannot be interpolated.

 

 

73

Table A-2. values of W at different values of %§-of the equation:

 

 

 

01R OR OR
W F W :7 W 1.7
2.00 .36801 4.20 .10322 6.40 .04950
2.05 .35289 4.25 .10113 6.45 .04883
2.10 .33873 4.30 .09910 6.50 .04817
2.15 .32546 4.35 .09713 6.55 .04753
2.20 .31298 4.40 .09522 6.60 .04689
2.25 .30124 4.45 .09337 6.65 .04628
2.30 .29018 4.50 .09158 6.70 .04567
2.35 .27975 4.55 .08984 6.75 .04508
2.40 .26990 4.60 .08815 6.80 .04450
2.45 .26058 4.65 .08651 6.85 .04393
2.50 .25175 4.70 .08491 6.90 .04337
2.55 .24339 4.75 .08337 6.95 .04282
2.60 .23546 4.80 .08186 7.00 .04228
2.65 .22792 4.85 .08040 7.05 .04175
2.70 .22076 4.90 .07897 7.10 .04124
2.75 .21394 4.95 .07759 7.15 .04073
2.80 .20745 5.00 .07624 7.20 .04023
2.85 .20126 5.05 .07493 7.25 .03975
2.90 .19536 5.10 .07365 7.30 .03927
2.95 .18972 5.15 .07241 7.35 .03880
3.00 .18434 5.20 .07120 7.40 .03834
3.05 .17919 5.25 .07002 7.45 .03788
3.10 .17426 5.30 .06887 7.50 .03744
3.15 .16954 5.35 .06775 7.55 .03700
3.20 .16501 5.40 .06666 7.60 .03657
3.25 .16068 5.45 .06560 7.65 .03615
3.30 .15651 5.50 .06456 7.70 .03574
3.35 .15252 5.55 .06354 7.75 .03533
3.40 .14868 5.60 .06255 7.80 .03493
3.45 .14499 5.65 .06159 7.85 .03454
3.50 .14144 5.70 .06065 7.90 .03416
3.55 .13803 5.75 .05973 7.95 .03378
3.60 .13474 5.80 .05883 8.00 .03340
3.65 .13157 5.85 .05795 8.05 .03304
3.70 .12852 5.90 .05709 8.10 .03268
3.75 .12558 5.95 .05625 8.15 .03233
3.80 .12274 6.00 .05544 8.20 .03198
3.85 .12000 6.05 .05463 8.25 .03164
3.90 .11735 6.10 .05385 8.30 .03130
3.95 .11479 6.15 .05309 8.35 .03097
4.00 .11232 6.20 .05234 8.40 .03064
4.05 .10994 6.25 .05160 8.45 .03032
4.10 .10762 6.30 .05089 8.50 .03001
4.15 .10539 6.35 .05019 8.55 .02970

 

 

74

Table A-2 continued.

 

m?

 

“7924703692592592603715937159372615049383837272728383949
8776554322100988776554332210099887765544332211009988776
11111111111111000000000000000099999999999999999998888888
11111111111111111111111111111100000000000000000000000000
00000000000000000000000000000000000000000000000000000000
o ooooooooooooooooooooooooooooo

50505050505050505050505050505050505050505050505050505050
23344556677889900112233445566778899001122334455667788990

coo-eo-oooooooooo00000000000000.00.00.00...0000.00.00.00
444444444444444555555555555555.35555666666666666666666667
11111111111111111111111111111111111111111111111111111111

63963074197421976432109988777778889001234678013578035792
65321087643210876543219876543210987765432109987654432100
7777776666666655S555554444444444333333333332222222222222
11111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000
.0000...OOOOOOOOOOOOOOOOOOOOOO 0 0.0.0.0.... 0000......

50505050505050505050505050505050505050505050505050505050
‘5566778899001122334455667788990011223344556677889900112

Ooooooooooooooooooooooono...cocoooooooooooooooooooo.0000
11111111111222222222222222222223333333333333333333344444
11111111111111111.111111111111111111111111111111111111111

 

999 07 :681 938384073 08 64210999990123579247037159377273940
307752 963185308531864197531864209753197642197642196832098
99888777766665555444433333222221111100000099999988888877
22222222222222222222222222222222222222222211111111111111
00000000000000000000000000000000000000000000000000000000
00......OOOOOOOOOOOOOOOCOOOOOOOOCOOIOOOOOOOOOOOOOO0.0...
05050505050505050505050505050505050505050505050500505050
66778899001122334455667788990011223344556677889901122334
.0...I...OOOOOOOOOOOOOOOOOOOOOO0.0.0.0...I...0.0.0.0....
88888888999999999999999999990000000000000000000011111111

1111111111111111111111111111

Table A-2 continued.

75

 

 

aR aR 0R
W 1'7 W 1.7 W 1:2
17.05 .00865 19.05 .00708—7 21.05 .U039T‘
17.10 .00860 19.10 .00704 21.10 .00588
17.15 .00856 19.15 .00701 21.15 .00586
17.20 .00851 19.20 .00698 21.20 .00583
17.25 .00847 19.25 .00695 21.25 .00581
17.30 .00842 19.30 .00691 21.30 .00578
17.35 .00838 19.35 .00688 21.35 .00576
17.40 .00834 19.40 .00685 21.40 .00573
17.45 .00829 19.45 .00682 21.45 .00571
17.50 .00825 19.50 .00679 21.50 .00569
17.55 .00821 19.55 .00675 21.55 .00566
17.60 .00817 19.60 .00672 21.60 .00564
17.65 .00812 19.65 .00669 21.65 .00562
17.70 .00808 19.70 .00666 21.70 .00559
17.75 .00804 19.75 .00663 21.75 .00557
17.80 .00800 19.80 .00660 21.80 .00555
17.85 .00796 19.85 .00657 21.85 .00552
17.90 .00792 19.90 .00654 21.90 .00550
17.95 .00788 19.95 .00651 21.95 .00548
18.00 .00784 20.00 .00648 22.00 .00545
18.05 .00780 20.05 .00645 22.05 .00543
18.10 .00776 20.10 .00642 22.10 .00541
18.15 .00772 20.15 .00639 22.15 .00539
18.20 .00769 20.20 .00637 22.20 .00537
18.25 .00765 20.25 .00634 22.25 .00534
18.30 .00761 20.30 .00631 22.30 .00532
18.35 .00757 20.35 .00628 22.35 .00530
18.40 .00754 20.40 .00625 22.40 .00528
18.45 .00750 20.45 .00623 22.45 .00526
18.50 .00746 20.50 .00620 22.50 .00524
18.55 .00743 20.55 .00617 22.55 .00522
18.60 .00739 20.60 .00614 22.60 .00519
18.65 .00735 20.65 .00612 22.65 .00517
18.70 .00732 20.70 .00609 22.70 .00515
18.75 .00728 20.75 .00606 22.75 .00513
18.80 .00725 20.80 .00604 22.80 .00511
18.85 .00721 20.85 .00601 22.85 .00509
18.90 .00718 20.90 .00599 22.90 _.00507
18.95 .00715 20.95 .00596 22.95 .00505
19.00 .00711 21.00 .00593 23.00 .00503

 

76

in equation

3
D

Values of 2 at different values of

Table A-3.

(loge 22 + %)

1
227

.u—D

 

00

ED

05

 

_
1
EH I.- Iu.u!!illv|b.

 

988914949642112469372740864333457914826161740853198
05051.6172840628006396296296307418530752075208531964
877665544333221110099988877776665555444433332222111
333333333333333333322222222222222222222222222222222
000000000000000000000000000000000000000000000000000

050505050505050505050505050505050505050505050505050
112233445566778899001122334455667788990011223344556

666666666666666666777777777777777777778888888888888

921796981920176077188176718781754605212483087803841
9.01247049395174297643322233457802479258148259382616
097531086532198754321098765432110987765543321100998
988888877777766666666655555555555444444444444444333
000000000000000000000000000000000000000000000000000

000000000000.O.0.00..OOOOOOOOOOOOOOOOOOOO0000.00.00

505050505050505050505050505050505050505050505050505
566778899001122334455667788990011223344556677889900

333333333444444444444444444445555555555555555555566

45.4227149713889513785302974994209724252361741849943
265978396428243918079558447201495457163101259395310
373099137284209899124703715050506284174185296419753
1864198663109865‘332100988776655‘433322211100009999
544443333333222222222221111111111111111111111110000

on0000000000.coo.00000000000000.0000...000000000000

n505050505050505050505050505050505050505050505050so
001122334455667788990011223344556677889900112233445

111111111111111111112222222222222222222233333333333

77

Table A-3 continued. .

 

00

GE

05

 

rEctiuilil

 

790135680246813681369258148158259
21.11098766503221099876654032210098
222211111111111100000000000000099
1.11111111111111111111111111111100
000000000000000000000000000000000

00000000000000.0000...-oooooooooo

505050505050505050505050505050505
900112233445566778899001122334455

000000000000000000000000000000000
122222222222222222222333333333333
111111111111111111111111111111111

297419753108765432211111112234456
865431098764321098765432109876543
555555544444444433333333332222222
111111111111111111111111111111111
000000000000000000000000000000000

00000000000000.0000...ooooooooooo

050505050505050505050505050505050
334455667788990011223344556677889

0.....0.0.0...’..................
000000000000001111111111111111111
111111111111111111111111111111111

7.76677891357925825938272738406296
208642087531986431986532097653200.
110000099999888888777777766666665
222227..21111111111111.11111111111111..
000000000000000000000000000000000

50505050505050505050505050505 5
67788990011223344556677889900 1
00.00.000.00...0.0000000000000000
8888888999999999999999999990000
1111

0
1

78

Table A-4. Set #1, Experimental data of deformation force and moisture

 

 

content.
1e Force at M.C. % Sample Force at M.C. %
$335. 0.1" def. ‘W;B. No. 0.1" def. 'W.B.
lbs . lbs .

1 50.4 88.3 47 25.0 83.9
2 49.2 87-5 48 25.0 86.4
3 49.0 87.5 49 25.0 87.2
4 47.5 87.5 50 24.2 83.9
5 47.2 88.0 51 23.8 85.4
6 46.3 88.7 52 23.5 86.2
7 44.5 89.1 53 21.4 84.4
8 44.4 89.1 54 21.0 84.7
9 43.0 88.3 55 21.0 86.2
10 43.0 88.3 56 19.4 85.7
11 42.0 87.0 57 19.0 85.4
12 42.0 86.0 58 17.0 85.7
13 41.5 87.9 59 16.0 84.7
14 41.4 89.9 60 15,7 85.5
15 41.0 85.8 61 15.5 84.2
16 40.7 87.9 62 15.4 83.7
17 40.0 89.4 63 15.0 85.7
18 39.0 86-7 64 14.4 85.2
19 38.3 86.6 65 13,0 84.2
20 38.0 87.7 66 13.0 85.2
21 38.0 87-7 67 12.5 85.5
22 38.0 89.6 68 12.5 85.7
23 38.0 86.6 69 11.6 83.7
24 37.5 86.7 70 11.5 82.3
25 37.0 86.7 71 11.3 85.1
26 37.0 86.6 72 11.0 83.4
27 37.0 87.7 73 10.7 85.5
28 36.0 89.2 74 10.3 82.7
29 36.0 86.0 75 10.0 84.7
30 35.0 87.7 76 10.0 85.1
31 34.4 85.3 77 10.0 84.8
32 34.2 87.2 78 9.5 81.1
33 32.0 87.2 79 9.5 82.7
34 32.0 85.3 80 8.5 81.8
35 32.0 86.6 81 8.2 84.0
36 30.0 85.6 82 8.0 82.5
37 30.0 83.9 83 7.5 85.1
38 28.7 85.6 84 7.5 84.3
39 28.6 86.2 85 7.4 79.3
40 28.0 86.2 86 6.6 82.3
41 27 0 87.2 87 6.7 81.8
42 26 4 86.4 88 6.3 81.0
43 26 0 86.4 89 6.0 84.8
44 25 8 86.2 90 5.9 80.4
45 25 3 84.0 91 5.7 81.1
46 25 0 85.3 92 5.4 81.0

79

Table A-4 continued.

 

Sample Force at M.C.% Sample Force at M.C.%
No. 0.1" def. ‘W.B. No. 0.1" def. W;B.
lbs. lbs.

93 5.0 81.1 116 2.9 74.5
94 4.5 81.1 117 2.8 76.6
95 4.5 73.9 118 2.8 79.6
96 4.2 82.4 119 2.5 79.6
97 4.0 80.4 120 2.5 72.3
98 4.0 66.7 121 2.5 78.4
99 4.0 75.4 122 2.4 73.4
100 4.0 79.1 123 2.4 74.1
101 4.0 78.9 124 2.3 75.2
102 4.2 76.1 125 2.3 71.0
103 3.8 84.0 126 2.2 77.8
104 3.7 84.8 127 2.0 73.9
105 3.6 81.1 128 1.8 77.4
106 3.7 75.5 129 1.8 69.8
107 3.5 81.0 130 1.8 78.1
108 3.5 79.0 131 1.6 75.2
109 3.5 71.1 132 1.6 64.7
110 3.3 77.9 133 1.5 75.2
111 3.2 72.4 134 1.5 76.9
112 3.1 75.0 135 1.5 73.9
113 3.0 82.4 136 1.5 77.8
114 3.0 75.0 137 1.5 75.7
115 3.0 79.2 138 1.0 73.0

 

80

Table.A-5. Set #2, Experbnental data of deformation force and moisture

 

content.
Sample Force at ‘M.C.% Sample Force at iM.C.%
No. 0.1" def. 'W.B. No. 0.1" def. ‘W.B.
lbs. lbs.
1 52.0 87.1 36 14.5 79.4
2 47.0 87.9 37 14.0 79.8
3 46.0 86.0 38 13.0 82.5
4 45.0 87.1 39 13.0 80.0
5 45.0 87.6 40 11.5 78.7
6 42.0 85.6 41 11.0 76.2
7 42.0 85.7 42 11.0 81.4
8 40.0 86.1 43 10.0 80.4
9 36.0 84.3 44 9.5 79.8
10 35.0 83.7 45 9.5 81.1
11 35.0 86.2 46 9.0 79.5
12 34.0 83.6 47 9.0 78.1
13 33.0 82.4 48 8.5 78.6
14 32.5 81.9 49 7.0 78.7
15 32.0 85.2 50 6.5 79.5
16 32.0 83.3 51 6.5 79.2
17 28.0 80.7 52 6.5 79.4
18 26.0 85.3 53 6.0 77.0
19 24.0 84.0 54 5.8 79.6
20 23.0 80.2 55 5.6 70.6
21 23.5 81.0 56 5.6 75.3
22 20.0 83.5 57 5.0 79.7
23 19.5 82.1 58 4.5 78.0
24 19.5 84.7 59 4.3 75.4
25 19.0 79.7 60 4.1 79.3
26 18.5 84.6 61 3.7 74.6
27 18.0 82.4 62 3.6 76.6
28 18.0 82.2 63 3.5 75.7
29 17.0 81.0 64 2.8 75.8
30 16.0 80.3 65 2.5 74.0
31 16.0 80.9 66 2.2 74.0
32 16.0 76.8 67 2.1 75.2
33 16.0 81.8 68 1.7 72.9
34 15.5 80.5 69 1.7 70.7
35 15.0 82.5 70 1.4 73.2

 

IT—

81

Table A-6. Set #1, Average deformation force and moisture content
of samples grouped according to force range.

 

Force Range Ave. Force .Ave.iM.C.%
lbs. ‘W.B.
50 - to 40 43.4 88.2
39.9 to 35 37.2 87.4
34.9 to 30 33.3 86.5
29.9 to 25 26.6 85.6
24.9 to 20 21.4 84.9
19.9 to 15 16.8 84.4
14.9 to 10 13.7 83.7
9.9 to 7 8.5 82.4
6.9 to 4 4.8 78.6
3.9 to 2 2.9 75.8
1.9 to 0 1.7 74.4

5.1‘1'fl-w" . .—

 

 

Table A-7. Set #2,.Average deformation ferce and moisture content
of samples grouped according to force range.

 

Force Range Ave. Force Ave. M.C.%
lbs. ‘W.B.
50 - to 40 45.0 86.6
39.9 to 35 35.3 84.7
34.9 to 30 32.7 83.3
29.9 to 25 27.0 83.0
24.9 to 20 22.6 82.2
19.9 to 15 17.2 81.5
14.9 to 10 12.3 79.8
9.9 to 7 8.8 79.3
6.9 to 4 5.5 77.7
3.9 to 2 2.9 75.1
1.9 to 0 1.5 72.5

 

82

rue-

 

 

 

 

 

 

 

mom Ham omm o.oa o.ov

mam Nwm mmm 5H.m mm.e mo.o oa.o e.HH o.Hm m.o o.H

Hem 5mm owe n.HH o.-

mmv mom mom N.n c.5H

mum mwv mow m.m w.vH

m~v ovm mme n.oa 0.0H

mmm mmm mom NH.m Nv.m mo.o mmo.o m.o o.m o.H o.H mo.o
cam .1. o.Hw

mmm mmm mam m.v~ o.om

mmv How «am w.wH N.uo

ewe nee new wm.m om.~ H.o o~.o N.~N H.wo m.o o.H

mom mom wmm c.5H o.m~

omm owm mNm N.n~ N.me

omm mvm NNN m.HH m.n~

emu mmv mac wm.m om.o H.o mo.o m.H~ o.mm o.H o.H H.o
an an «8 a up 5}: £ 65 8.: .05

S. E N z a m8 a m A a 8
8.8 3.8303 80.8 ma < .w H u mm .Ibll 84 u «m .INHII mmmé 1 am
. . A m NNAm MNZXH

"mcowumsvo may

wofims cofiumsuomoo vow meufim ooswooom coupon unmu0MMflw How m we m03Hm> voomH20Hmu may .m.<.w~cm~

83

 

 

 

 

 

.. 666 664 6.6 .1
66m 654 666 6.4 6.4
566 666 666 4.6 6.5
6N6 666 66~ 66.5 66.66 666.6 6666.6 6.6 6.4 6.6 6.6 666.6
664 664 664 6.66 6.6
644 664 664 6.66 6.6
664 564 664 6.66 6.6
664 N64 664 46.6 66.6 6.6 666.6 6.66 6.6 6.6 6.6
664 464 666 6.66 6.4
644 664 666 4.66 4.4
666 464 N66 6.66 6.6
664 466 666 46.6 66.66 6.6 6666.6 5.66 6.4 6.6 6.6
644 i. I. .1. 6.46
666 656 N66 6.66 6.64
644 666 BN4 56.6 66.4 66.6 66.6 6.66 6.66 6.6 6.6 66.6
666 666 666 N 6. .m. .ww _:6\6H 66 60:6 6066 .6066
mm am am 6 .6 a 6 6 4

 

.666666660 6-<.6H646

 

.wouuommu no: m6 66 u m6 n m6 coop .66.6m v 6m 066 6m 00:66
6

6.6 1 66.66 606660 6066

 

 

84

mo
. I m.mom u N I N

6.666 66.666 46.66 1 6 66
66666 66666 6666 u 66 66.6 a _MMhmnI.u MI." 60
664 664 664 "68: 6.666 m6
666 I. I. I. 6.6
666 666 664 4.6 6.6
664 466 666 6.6 6.4
466 664 666 66.6 64.6 666.6 666.6 6.4 6.6 6.6 6.6
666 666 666 6.6 6.6
644 664 644 6.4 6.6
666 664 666 66.6 64.66 666.6 66666.6 6.4 6.6 6.6 6.6

 

How omn awe N 0
com H64 mom o.¢ v.HH
com 66m hem mw.n cm.o mmc.o omo.o N m m.HH m.o o.H mNo.o

 

6mm 6mm 6mg

0 o N z

66. 66\66 66 6066 6066 6066
mm mm 66

do \m m A a a

5“:

 

.ooscauaou mn< oHomH

85

 

 

 

 

 

.‘il" .1234. po‘rL-r. ‘ -‘1.".

 

 

 

 

 

 

 

Table Ar9. Experimental and calculated points for force-time relaxation
curves at different moisture contents.
86.6% M.C. _ 84.7% M.C.'

Point 1119, lbs t’ F t lbs t’

Pipefimental Calculated sec rmenta Ca cu ate sec

1 45.0 44.7 0.0 35.4 35.4 0.0

2 42.7 42.7 0.12 35.0 35.4 0.0
3 41.2 41.4 0.24 33.5 33.5 0.15
4 39.8 39.8 0.48 32.6 32.6 0.27
5 39.0 38.9 0.72 31.6 31.6 0.51
6 37.7 37.8 1.32 30.4 30.5 1.11
7 37.0 37.1 1.92 29.9 29.8 1.71
8 36.1 36.1 3.12 29.0 29.0 2.91
9 35.1 35.0 5.52 28.3 28.3 4.71
10 34.5 34.5 7.92 27.8 27.8 7.11
11 34.0 34.2 10.32 27.2 27.3 10.71
12 33.0 33.0 21.12 26.1 26.6 16.71

83.3% M.C. 83.9% M.C.

Point F(t), lbs t’ F(t). lbs t’
Experifiental ‘Calculated sec Experimental Calculatéaf sec

1 34.0 33.7 0.0 27.4 27.1 0.0
2 33.0 33.2 0.04 26.2 26.2 0.10
3 32.0 32.0 0.16 25.4 25.4 0.22
4 30.7 30.7 0.40 24.5 24.4 0.46
5 30.0 30.0 0.64 23.7 23.6 0.82
6 29.1 29.1 1.24 23.0 23.0 1.42
7 28.1 28.2 2.44 22.6 22.6 2.02
8 27.6 27.6 3.64 22.1 22.0 3.22
9 27.0 26.9 6.04 21.5 21.6 4.42
10 26.4 26.4 9.64 21.2 21.1 6.82
11 25.8 25.9 15.64 20.5 20.5 11.62
12 25.1 25.1 27.64 19.9 19.9 22.47

 

86

Table A-9 continued.

 

82.2% M.C. 81.5% M.C.
Point F(t), lbs F(tj} lbs

Experimental Calculatedf sec Experimental Calculated sec

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 24.0 23.8 0.0 16.3 16.3 0.0
2 23.1 23.1 0.10 15.5 15.4 0.11
3 22.3 22.4 0.22 14.9 14.8 0.23
4 21.6 21.6 0.46 14.1 14.0 0.47
S 21.0 21.0 0.70 13.7 13.6 0.71
6 20.4 20.4 1.06 13.1 13.0 1.31
7 19.6 19.5 2.26 12.8 12.7 1.91
8 19.0 19.0 3.46 12.4 12.3 3.11
9 18.4 18.4 5.86 12.1 12.0 4.61
10 18.0 18.0 8.26 11.6 11.4 9.41
11 17.7 17.7 11.86 11.0 11.0 17.21
12 17.0 17.0 21.46 10.7 10.6 26.81
79.8% M. C. 79.3% M.C.

Point F111; lbs t’ Fit), lbs t’
Experimental calculated sec Experimental Calculatedi sec

1 12.5 12.4 0.0 8.5 8.3 0.0
2 11.5 11.5 0.12 7.9 8.0 0.06
3 11.0 11.0 0.24 7.5 7.5 0.18
4 10.4 10.4 0.48 7.0 7.0 0.42
5 10.0 10.0 0.84 6.5 6.5 0.90
6 9.7 9.7 1.32 6.3 6.3 1.38
7 9.4 9.4 1.92 6.0 6-0 2.22
8 9.0 9.0 3.72 5.8 5.8 3.06
9 8.7 8.7 5.52 5.5 5.5 5.46
10 8.5 8.5 9.12 5.3 5.3 7.86
11 8.3 8.3 13.92 5.1 5.1 12.66
12 8.0 8.0 22.32 4.7 4.7 24.66

 

87

Table.A-9 continued.

75.1%7M.CC’

77.7% NLC.

 

Fit), lbs

Experimental

 

F(t), lbs

 

Point

56C

 

56C

Calculated‘

 

Experimental

, Calculated

 

 

80240888888
002348280462
000000113594014

222222111111

853310987654

4682600

555544443333

863075308754

555544443333

1123456789012
111

 

72.5% M.C.

t ’
56C

F(tjlplbs

Experimental'

 

Point

CalcUlated

 

 

02460488888
012347919955

404077170
432110098877

111111100000

 

REFERENCES

88

 

REFERENCES

Barkas, W. W. , 1953. The mechanical properties of and their relation
to moisture. Part A in Mechanical properties of wood and paper.
R. Meredith, ed. , Interscience Publishers, Inc., New York.

Bashford, L. L. , 1953. Creep and relaxation of meat related to tender-
ness. ASAE Paper No. 73-301, St. Joseph, Michigan.

Bashford, L. L. and R. W. Whitney, 1973. A canputer program for
determining creep and relaxation models. ASAE Paper No. 73-302,
St. Joseph, Michigan.

 

Bourne, M. C., 1965. Studies on punch testing of apples. Food Technology, E.
20; 113—115.

 

Bourne, M. C., 1966a. A clsssification of objective methods for measuring
texture and consistancy of foods. Journal of Food Science, 31;
1011-1115.

 

Bourne, M. C. , 1966b. Measure of shear and canpression components of
puncture tests. Journal of Food Science, 31; 282-291.

 

Bourne, M. C., J. C. Moyer and D. B. Hand, 1966c. Measurement of food
texture by a universal testing machine. Food Technology, 20;
170-174.

 

Bourne, M. C., 1967a. Deformation testing of foods. I. Precise tech-
nique for performing the deformation test. Journal of Food Science,
32; 601-605.

 

Bourne, M. C. and N. Mondy, 1967b. Measurenent of whole potato firmness
with a Universal Testing Machine. Food Technology, 21; 97-100.

 

Bourne, M. C., 1967c. Defamation testing of foods. II. A simple
spring model of deformation. Journal of Food Science, 32, 605-607.

 

Bourne, M. C. , 19683. Texture profile of ripening pears. Journal of
Food Science, 33; 223-226.

 

Bourne, M. C. and J. C. Mayer, 1968b. The extrusion principle in texture
measurement of fresh peas. Food Technology, 22; 1013-1018.

 

Bourne, M. C., 1969. Two kinds of firmness in apples. Food Technolcgy,
23; 59-60.

 

Brandt, M. A., B. Z. Skinner and J. 0. Coleman, 1963. Texture profile
method. Journal of Food Science, 28; 404-409.

 

89

90

Breene, W. M., D. W. Davis and.H. Chou, 1972. Texture profile analysis
of cucumbers. Journal of Food Science, 37; 113-117.

 

Chen, P. and R. B. Fridley, 1971. An analytical method for determining
viscoelastic constants of agricultural materials. .ASAE Paper No.
71-339. St. Joseph, Michigan.

Decker, R. W., J. N. Yeatman, A. Kramer and A. P. Sidwell, 1957.
Modification of the shear press for electrical indicating and
recording. Food Technology, 11; 343.

 

Drake, B., 1962. Automatic recording of vibrational properties of F.-
foodstuffs. Journal of Food Science, 27; 182.

 

FOppl, A., 1922. ‘VOrlesungen fiber Technische Mechanik. 4th ed.,
vol. 5 of Die Wichtigsten Der theren Elastizitatstheorie. B. G.
Teubner, Leipzig and Berlin. p. 342.

Fbppl, A. and L. Fbppl, 1928. Drang und Zwang. 2nd ed., vol. 2. R.
Oldenbourg, Munchen and Berlin. p. 220.

 

Isl”)
A

Finney; E. E., 1969. Objective measurements for texture in foods.
Journal of Texture Studies, 1; 19-37.

 

Fridley, R. B., R. A. Bradley, J. W: Rumsey and P. A. Adrian, 1968.
Some aspects of elastic behavior of selected fruits. Transactions
of the ASAE, 11; 46-49.

 

 

Friedman, H. H., J. E. Whitney and.A. S. Szczesniak, 1963. The
texturometer - a new instrument for objective texture measurement.
Journal of Food Science, 28; 390-396.

 

Herum, F. L. , 1973. Viscoelastic behavior of soybeans due to tanperature
and.moisture content. ASAE Paper No. 73-319. St. Joseph, Michigan.

Hoki,IM. 0., 1973. Mechanical strength and.damage analysis of navy
beans. PhD Thesis, Michigan State University, East Lansing,
lMichigan.

Horsfield, B. C., R. B. Fridley and L. L. Claypool, 1970. Application
of theory of elasticity to the design of fruit harvesting and
handling equipment for minimum bruising. ASAE Paper No. 70-811,
St . Joseph, Michigan.

Howard, P. L. and D. B. Heinz, 1970. Texture of carrots. Journal of
Texture Studies, 1; 185-195.

 

 

Hughes, H. and L. J. Segerlind, 1972. A rapid.mechanical method for
determining Poisson's ratio in biological materials. ASAE Paper
No. 72-310. St. Joseph, Michigan.

Kattan, A6 8., 1957. Changes in color and firmness during ripening of
detached tomatoes, and the use of a new instrument fer measuring

firmness. Proceedings American Society of Horticulture Science,
70; 379.

 

91

Kelvin (Sir W. Thomson), 1891. Popular lectures and addresses. Vbl. I.
Constitution of Matter, 2nd ed.,.MacMillan 8 Company, London,
p. 80.

Kosma,.A. and H. Cunningham, 1962. Tables for calculating the compressive
surface stresses and deflection in the contact of two solid elastic
bodies whose principle planes of curvature do not coincide.

Journal of Industrial Mathematics, 12; 31-40

 

Kramer,.A., K..Aamlid, R. B. Guyer and.H. Rogers, 1951. New shear-
press predicts quality of canned limas. Food Engineering, 23;
112-187. pr-

 

Kramer, A., 1959. Glossary of some terms used in the sensory (panel)
evaluation of foods and beverages. Food Technology, 13; 733.

 

Lee, E. H. and J. R.IM; Radok, 1960. The content problem for visco-
elastic bodies. Transactions of the.ASME, Journal of.Applied
Mechanics, 27; 4381444.

 

 

Magness, F. R. and G. F. Taylor, 1925. An.improved type of pressure
tester for determination of fruit maturity. USDA Circular 350.

Mohsenin, N., C. T. Morrow and L. D. 'I‘ukey, 1965. The "Yield-Point"
non-destructive technique for predicting firmness of golden deli-
cious apples. Proceedings of the American Society_of Horticulture
Science, 86; 70.

 

MOhsenin, N., 1970. Physical properties of plant and animal materials.
V01. 1. Gorden and Breach Science Publishers, New'York.

IMOrrow, C. T. and N. N..Mohsenin, 1966. Consideration of selected
agricultural products as viscoelastic materials. Journal of Food
Science, 31; 686-698.

 

Ourecky, D. K. and M. C. Bourne, 1968. Measurement of strawberry tex-
ture with an Instron Machine. American Society_for Herticulture
Science, 93; 317-325.

 

Pao, Yoh-Han, 1955. Extension of the Hertz theory of impact to the
viscoelasticity case. Journal of Applied Physics, 26; 1083-1088.

 

Parker, R. E., J. H. Levin and H. P. Gaston, 1966. Cherry firmness
and its relationship to pitter loss. U. S. Department of Agriculture,
ARS 42-119.

Reidy, G. A., 1970. Relationships between engineering and texture para-
meters for low and intermediate moisture fOOds. PhD Thesis,
Michigan State university, East Lansing, Michigan.

Roark, R. J., 1954. Formulas for stress and strain. 3rd ed. ‘MCGraw-Hill
Book Canpany, Inc., p. 288.

92

Schaner, H. A., K. L. Olsen and J. N. Yeatman, 1963. A mechanical
thumb for measuring firmness of fruits, U. S. Department of
Agricultm‘e, Marketing Bulletin 25.

Scott-Blair, G. W., 1958. Rheology in food reserach, Advances in Food
Research, 8, l.

Sharma, M. G. , 1964. Viscoelasticity and mechanical properties of poly-
mers. Mimeographed notes, Sumner Institute on Applied Mechanics
and Materials Science, Department of Engineering Mechanics,

The Pennsylvania State University, University Park, PA.

Shpolyanskaya, A. L., 1952. Structural-mechanical properties of wheat
grain. Colloid Journal, 14; 137-148.

 

Stewart, B. R. , 1964. Effect of moisture content and specific weight
on the internal friction properties of sorghum grain. ASAE Paper
No. 64-804. St. Joseph, Michigan.

Szczesniak, A. S., M. A. Brandt and H. H. Friedman, 1963. Developnent
of standard rating scales for mechanical parameters of texture
and correlation between the objective and sensory methods of tex-
ture evaluation. Journal o_f__Food Science, 28; 397-403.

 

Szczesniak, A. S., 1966. Texture measurements. Food Technology, 20;
1292-1298.

 

Thanas, H. R. and V. A. Hoersch, 1930. Stress due to the pressure of
one elastic solid upon another, Engineering Experiment Station,
University of Illinois, Bulletin 21.

Timoshenko, S. P. and J. N. Goodier, 1951. Theory of elasticity.
McGraw-Hill Book Canpany, Inc. , New York.

Zoerb, G. C., 1958. Mechanical and rheological properties of grain.
PhD Thesis, Michigan State University, East Lansing, Michigan.