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This is to certify that the
thesis entitled
THE QUASI-STATIC CONTACT PROBLEM
FOR NEARLY-INCOMPRESSIBLE AGRICULTURAL PRODUCTS

presented by
SHARAFELDIN M. SHERIF

has been accepted towards fulfillment

of the requirements for _
Agr1cultural

Ph. D. degree in Engineering

fiéte February 26, 1976

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ABSTRACT

THE QUASI-STATIC CONTACT PROBLEM FOR NEARLY-
INCOMPRESSIBLE AGRICULTURAL PRODUCTS

BY
Sharafeldin M. Sherif

The objective of this work was to implement the
finite element method for studying the mechanical behavior
of nearly-incompressible agricultural materials subjected
to a large deformation and a quasi-static loading. The
geometric nonlinearity was formulated using the Lagrangian
strain tensor and the resulting nonlinear equations were
solved using an incremental displacement procedure. The
properties for peaches, potatoes and apples were used in
the numerical model to calculate the stress components
and plot isostress lines for the two dimension plane
strain case of diametrical loading and the axisymmetric
case of a loading perpendicular to the axis of symmetry.
Flat plate loadings were analyzed in each case. The
material was considered to be elastic, isotrOpic, and
homogeneous.

Cylindrical samples of white potato and apple
flesh were compressed diametrically to failure. Inspec-

tion showed that the white potato samples split, with a

Sharafeldin M. Sherif

crack initiated at or near the center. The stress com-
ponents calculated using the finite element analysis
indicate this failure may be due to tension stress or a
combination of tension and maximum shear stress. The
apple samples were bruised in the vicinity of the contact
surface. The calculated stress components indicate that
this failure is probably a result of the maximum shear
stress.

Loading semispherical samples of white potato and
apple flesh to failure was performed. The failure crack
in the potato occurred at the center and the calculated
stress components indicate that this failure may be due
to tension stresses or a combination of tension and
maximum shear stresses. Inspection of the resulting
bruise shape which occurred in apples indicated a shape
which passes through the point of maximum shear stress.
The numerical results showed that the maximum shear
stress in peaches occurred on the axis of symmetry and
the bruise shape passes through that point which indicate
that the failure is probably a result of the maximum
shear stress.

Approve
Maj rof ssor

Approved , .
Department Chairman

THE QUASI-STATIC CONTACT PROBLEM FOR NEARLY-

INCOMPRESSIBLE AGRICULTURAL PRODUCTS

BY

-L'»
’v\.-
’\
\

Sharafeldin MT Sherif

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Agricultural Engineering

1976

ACKNOWLEDGMENTS

Throughout the course of this graduate program
the author has been especially appreciative for the
counsel, guidance and encouragement provided by his major
professor, Dr. Larry J. Segerlind (Agricultural Engineering).
To the other members of the guidance committee,
Dr. C. J. Mackson (Agricultural Engineering), Dr. G. E.
Mase (Metallurgy, Mechanics and Material Science), and
Dr. R. T. Hinkle (Mechanical Engineering), the author
expresses his deepest gratitude for their time, professional
interest and constructive suggestions.
The author is particularly indebted to the
University of Tripoli (Libya) for the scholarship which
made this work possible.
The author wishes to express his gratitude to
Dr. D. R. Heldman (Chairman, Agricultural Engineering)
and to the faculty, staff and fellow graduate students
of the Agricultural Engineering Department for their
hospitality and friendship during his stay in the United
States.
Special sincere gratitude to my host family in

the United States, the Wilkinson's, Bill, Caroline,

ii

Leslie, Ricky and Greta for their hospitality and encour-
agement.

The author dedicates this work to his family,
especially to his uncle, Ahmed T. Sherif for his foresight
and support to undertake a career in Agricultural Engineer-
ing and his wife Nadia and their children, Nabela, Usama

and Sumeia for their encouragement and love.

iii

TABLE OF CONTENTS
Page

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . Vi

I. INTRODUCTION . . . . . . . . . . . . . . . . . 1
II. LITERATURE REVIEW . . . . . . . . . . . . . . . 4
2.1 Mechanical Damage . . . . . . . . . . . . 4
2.2 Strength of Fruits and Vegetables . . . . 6
2.3 Contact Stresses . . . . . . . . . . . . . 9

2.4 Stress Analysis in Fruits and
Vegetables . . . . . . . . . . . . . . . 15
2.5 Summary . . . . . . . . . . . . . . . . . 17

III. BASIC THEORY . . . . . . . . . . . . . . . . . 19

3.1 Non-Linearities . . . . . . . . . . . . . 20
3.2 Stress and Strain . . . . . . . . . . . . 21
3.3 Green's Strain Tensor . . . . . . . . . . 26
3.4 Finite Plane Strain . . . . . . . . . . . 29
3.5 Axisymmetric Strains . . . . . . . . 31
3.6 A General Formulation for Elastic

Bodies . . . . . . . . . . . . . . . 34
3.7 A Variational Principle . . . . . . . . . 38

IV. FINITE ELEMENT FORMULATION . . . . . . . . . . 42

4.1 Plane Strain . . . . . . . . . . . . . . . 44

4.2 The Axisymmetric . . . . . . . . . . . . . 52
4.3 Element Stresses . . . . . . . . . . . . . 59
4.4 Nodal Stresses . . . . . . . . . . . . . . 59
4. 5 Other Finite Element Formulations . . . . 59
4. 6 Summary . . . . . . . . . . . . . . . . . 62

V. COMPUTER IMPLEMENTATION . . . . . . . . . . . . 63
5.1 Iterative Procedures . . . . . . . . . . . 64

5.2 Incremental Procedure . . . . . . . . . . 67
5.3 Solution of the Contact Problem . . . . . 70

iv

VI. VERIFICATION OF THE FINITE ELEMENT MODEL

6.1 Experimental and Finite Element

Results . . . . . .
6.2 Summary . . . . . . .

VII. APPLICATION TO AGRICULTURAL PRODUCTS

7.1 Two-Dimensional Analysis--

Brazilian Test . . .

7.1.1 Potato . . . .
7.1.2 Apples . . . .

7.2 Spherical Shapes . . .
7.2.1 Potato . . . .
7.2.2 Apples . . . .
7.2.3 Peaches . . . .

7.3 Summary . . . . . . .

VIII. SUMMARY AND CONCLUSIONS . .

IX. SUGGESTIONS FOR FUTURE STUDY

BIBLIOGRAPHY . . . . . . . . . . .

APPENDIX . . . . . . . . . . . . .

Page
73
73
83

87

87

87
97

103
103
113
116
129
131

133

135

144

Figure

2-1.

3-1.

3-2.

LIST OF FIGURES

Page
Relation of yield force, yield stress,
and yield deformation for various
fruits . . . . . . . . . . . . . . . . . . . 8

Pressure distribution on the contact
surface of a sphere loaded with a
flat plate . . . . . . . . . . . . . . . . . 10

Pressure distribution on the contact
surface of a cylinder loaded with a
flat plate . . . . . . . . . . . . . . . . . 11

Theoretical stress distribution under
a rigid loading die . . . . . . . . . . . . 13

Comparison between using flat plate
and plunger test with different
product . . . . . . . . . . . . . . . . . . 16

Relationship between several definitions
of stress and strain . . . . . . . . . . . . 24

Force-displacement curve of a cylindrical
specimen of potato under uniaxial com-
pression (L = D = 25.4) . . . . . . . . . . 27

Finite plane strain deformation of a
small element . . . . . . . . . . . . . . . 3O

Triangular element in plane strain
and nodal displacement . . . . . . . . . . . 45

Triangular axisymmetric element and nodal
deformations . . . . . . . . . . . . . . . . 53

Flow chart for Finite Element computer
program for iterative procedures . . . . . . 65

Flow chart for Finite Element computer
program for incremental procedures . . . . . 68

vi

Figure Page

5-3. Spherical sample in contact with rigid

flat plate: Prescribed displacement of

the contact nodes . . . . . . . . . . . . . 71
6-1. Sample preparation . . . . . . . . . . . . . . 74
6-2. Sample cutting machine . . . . . . . . . . . . 74
6-3. Cylindrical sample compressed dia-

metrically between two flat plates . . . . . 76

6-4. Finite Element Grid of one fourth of a
cylindrical sample . . . . . . . . . . . . . 78

6-5. Value of stresses using different formu—
lation for u = 0.49 for cylindrical

sample . . . . . . . . . . . . . . . . . . . 79
6-6. Comparison between different formu-

lations for complete incompressibility

for a cylinder in simple tension . . . . . . 81
6-7. Hydrostatic pressure as a function of

displacement for cylindrical sample . . . . 82
6-8. Effect of strain rate on the resultant

stress compared with different

formulation for cylindrical sample . . . . . 84

6-9. The effect of elastic modulus on o
with constant Poisson's ratio
(u = 0.49) . . . . . . . . . . . . . . . . . 85

22

7-1. Crack just initiated at the center of a
potato sample . . . . . . . . . . . . . . . 89

7-2. Crack propagated outwards of a potato
sample . . . . . . . . . . . . . . . . . . . 89

7-3. Crack propagated outwards of a potato
sample . . . . . . . . . . . . . . . . . . . 90

7-4. Crack propagated outwards of a potato

sample . . . . . . . . . . . . . . . . . . . 90
7-5. Finite element grid of a spherical shape . . . 91
7-6. A deformed shape of potato sample com-

pressed diametrically . . . . . . . . . . . 92

vii

Figure Page

7-7. Stresses along the axes of symmetry in
the deformed shape (potato) . . . . . . . . 93

7-8. Lines of constant stress in X-direction
of a cylindrical potato sample com-
pressed diametrically in N/cm2 . . . . . . . 94

7-9. Lines of constant stress in Y-direction
of a cylindrical potato sample com-
pressed diametrically in N/cm2 . . . . . . . 94

7-10. Lines of constant maximum principal
stress of a cylindrical potato
sample compressed diametrically
in N/cm2 . . . . . . . . . . . . . . . . . . 95

7-11. Lines of constant minimum principal
stress of a cylindrical potato
sample compressed diametrically
in N/cm . . . . . . . . . . . . . . . . . . 95

7-12. Lines of constant maximum shear
stress of a cylindrical potato
sample compressed diametrically
in N/cm2 . . . . . . . . . . . . . . . . . . 96

7-13. The normal stresses on a small element
at the center of the cylindrical
shape compressed diametrically . . . . . . . 96

7-14. A deformed shape of apple sample com-
pressed diametrically . . . . . . . . . . . 98

7-15. Stresses along the axes of symmetry in
the deformed shape (apple) . . . . . . . . . 99

7-16. Lines of constant stress in X-direction
of a cylindrical apple sample com-
pressed diametrically in N/cm . . . . . . . 100

7-17. Lines of constant stress in Y-direction
of a cylindrical apple sample compressed
diametrically in N/cm2 . . . . . . . . . . . 100

7-18. Lines of constant maximum principal

stress of a cylindrical apple sample
compressed diametrically in N/cm . . . . . 101

viii

Figure

7-19.

7-20.

7-23.

7~24.

7-25.

7-26.

7.28.

7-29.

7-31.

Lines of constant minimum principal
stress of a cylindrical apple
sample compressed diametrically
in N/cm2 . . . . . . . . . . . . . . .

Lines of constant maximum shear
stress of a cylindrical apple
sample compressed diametrically
in N/cm2 . . . . . . . . . . . . . . .

Crack propagation in a semispherical
potato sample . . . . . . . . . . . .

Finite element grid of a spherical
shape . . . . . . . . . . . . . . . .

A deformed shape of finite element of a
semi-spherical potato sample . . . . .

Lines of constant stress in the Z-
direction £05 a spherical potato
sample, N/cm . . . . . . . . . . . .

Lines of constant maximum shear stress
of a spherical potato sample, N/cm .

Lines of constant minimum principal
stress of spherical sample of
potato, N/cm . . . . . . . . . . . .

Lines of constant maximum principal
stress of a spherical potato specimen,
N/sz O O O O O O O O O O O I O I O 0

Lines of constant stress in R-direction
of a spherical potato specimen, N/cm2

Stresses along the axes of symmetry of
a spherical sample in deformed shape
(potato) . . . . . . . . . . . . . . .

The applied stresses on a small element
at the center of spherical potato
sample, n = 0.48 . . . . . . . . . . .

A deformed shape of finite element
spherical sample of apple . . . . . .

ix

Page

101

102

105

106

107

108

109

110

111

112

114

115

117

Figure

7-32.

7-33.

7-35.

7-36.

7-37.
7-38.
7-39.

7-40.

7-41.

7-42.

7-43.

Lines of constant stress in the Z—
direction for a spherical apple
sample in N/cm2 . . . . . . . . . . .

Lines of constant maximum shear
stress of a spherical apple sample
in N/cm2 . . . . . . . . . . . . . . .

Lines of constant maximum principal
stress of a spherical apple sample,
N/cm2 . . . . . . . . . . . . . . . .

Lines of constant minimum principal
stress of a spherical apple sample,
N/cm2 . . . . . . . . . . . . . . . .

Stresses along the z-axis of a spherical
apple sample . . . . . . . . . . . . .

One quarter of a peach . . . . . . . . .
Finite element grid for peach . . . . .

A deformed shape of finite element of a
peach . . . . . . . . . . . . . . . .

Lines of constant stress in the Z-
direction for a peach, N/cm2 . . . . .

Lines of constant maximum shear stress
in a peach, N/cm . . . . . . . . . .

Lines of maximum principal stresses
for a peach, N/cm2 . . . . . . . . . .

Lines of minimum principal stress for a
peach, N/cm2 . . . . . . . . . . . . .

Page

118

119

120

121

122
123

125

126

127

127

128

128

I . INTRODUCTION

Fruits and vegetables are subjected to different
types of mechanical treatments in harvesting that can
damage the product. Widespread mechanization has generated
a major concern about the effect mechanical harvesting and
handling has on the quality of the final product. In
order to prevent or minimize the mechanical damage to
mechanically harvested fruits and vegetables it is neces-
sary to determine the maximum permissible load that these
products can support before failing. Since the failure
is most likely related to the stresses in the material, it
is necessary to know the intensity and distribution of
the stresses in the fruit or vegetable for various loadings.

Many techniques used in the engineering sciences
to study the behavior of engineering materials have also
been used to obtain the stresses which occur in agri-
cultural products during a contact loading. For example,
‘Hamann (1967) considered the viscoelastic boundary value
€13 study the bruising of apples during impact. A defi-

r“ition of a failure criterion was attempted by Miles and
REJdkugler (1971) along the lines of the failure theories

de\Ieloped for non-biological materials.

A stress analysis of three dimensional bodies is
very difficult, particularly when the irregular shapes of
agricultural products are involved. The finite element
~method is a powerful tool for analyzing irregular shaped
bodies and for evaluating the stresses and displacements
in three dimensional bodies, provided computer facilities
are available. The finite element has been used to
analyze contact stress distribution in agricultural pro-
ducts for infinitesimal strains by Apaclla (1973), Rumsey
and Fridley (1974), and De Baerdemaeker (1975).

Contact loadings occur repeatedly during harvest-
ing and handling operations. Present design of cushion-
ing materials is based on the contact theory of elasticity.
This theory holds for small displacements and a Poisson's
ratio in the range of 0.3 - 0.4 but is not valid for
large deformation failures of nearly-incompressible
materials such as peaches or potatoes which have a
Poisson's ratio in the vicinity of 0.48 (Hughes and
Segerlind, 1972).

The overall objective of this study was to apply
the finite element technique to the study of nearly-
incompressible materials subjected to a quasi-static con-
‘tact loading. Specific objectives included:

1. To implement the finite element method for the

solution of problems involving incompressible and

nearly-incompressible materials which experience

either small or large displacements.

2. To study the stress distribution in semi-spherical
samples of potatoes and apples and semi-elliptical
samples of peaches.

3. To investigate the failure modes of white potatoes
and apples using a diametrically loaded cylin-
drical sample.

It is important to note that this work was based
on the assumption that the damaged fruit and vegetable
tissue can be considered homogeneous, isotropic and
elastic. Recent research has also approached the behavior
of the tissue by considering its basic composition as a
mixture of solids, liquids and gas (Brusewitz, 1969;

Akyurt, 1969; Gustafson, 1974).

II . LITERATURE REVIEW

The importance of the physical and mechanical
properties of agricultural products and the need for study
and research in this area was emphasized by Mohsenin
(1971). Some of this work relates to the material prop-
erties needed for the evaluation of stresses in fruits
and vegetables subjected to static and dynamic loads and

in the study of bruise susceptibility of the product.

2.1 Mechanical Damage

 

Mechanical harvesting, bulk storage and the han-
dling of fruit and vegetable products has indicated a
need for basic information on material properties.
Bruising and skinning of mechanically harvested potatoes,
distortion of onion bulbs at the bottom of storage piles,
and mechanical damage to fruits and vegetables by com-
pression, impact, and vibration have lowered the grade
Of these products, with a consequential loss to the
Eirower. Mechanical injury to agricultural commodities
F168 resulted from excessive stresses during mechanical
heirvesting and handling operations. As a result, many
irIvestigations have been conducted to determine the
me=<:hanica1 behavior of such agricultural products as

4

apples, onions, potatoes, peaches and grains when subjected
to various types of external forces.

Because of the increasing emphasis on the mechan-
ization of harvesting and handling of fruits and vege-
tables, it is even more essential to determine the
engineering and physical properties of these commodities
if bruising and mechanical damage are to be minimized
(Finney, 1967). Mattus 3E 21. (1960) showed that drOp
heights exceeding six inches on a hard surface produced
internal bruises in pears which developed into brown
spots. Mohsenin and G6hlich (1962) evaluated the resist-
ance of apples to injury by obtaining the yield and the
rupture parameters for apple fruit. They observed, for
example, that the force-deformation curve for an apple
was very similar to the stress-strain curve of steel;
that is, the force deformation relationship was approxi-
mately linear up to an apparent yield point after which
the force suddenly decreased and then continued to
increase up to some point of rupture. The yield point
in the apple fruit was significant in that it corresponded
to the point where the cells of the fruit were sufficiently
damaged to cause discoloration and deterioration of the

fruit. Unless the yield point was exhibited, no bruising
0f the fruit was indicated.

Zoerb (1958) indicated that the strain rate effect

fol: biological materials may be influenced by the moisture

content. He stated, for example, that the energy required
for shearing high moisture grain under impact loading was
greater than the static shearing energy; the reverse was
true for low moisture grain.

Lamp (1959) made an extensive study of the load
bearing capacity of the potato as influenced by climatic
conditions, variety, position of the tuber in the soil,
cultivated practices and storage. Variation of climate
from year to year influenced the resistance of the potatoes
to injury. Other significant factors were depth of
planting, soil and storage conditions.

Finney (1963) reported the influence of variety
and time on the potato's resistance to mechanical damage.
Park (1963) studied the effect of impact forces upon the
potato tuber and its resistance to mechanical damage.

The damage was evaluated in terms of the number of tubers
split and bruised.

2.2 Strength of Fruits and
Vegetables

 

 

The response of fruits and vegetables to a loading
can be used to define mechanical prOperties. The relation-
Ship between stress and strain indicates how a material
<ieforms under the load, and provides data related to yield
:EOrce, deformation at yield or failure and the modulus

of elasticity.

 

(II
(D

' v
.tl

.61

up
’( 9

“VA
's
. a

'6'.

‘5‘
u...

s“ {

 

in w
5....

'i;
3"
P"'

Mohsenin and G5hlich (1962) and Mohsenin 2E.2l'
(1965) repeated data on the maximum allowable load for
several varieties of apples at harvest maturity. Resist-
ance to bruising under dead load and quasi-static loading
in terms of allowable surface pressure and deformation was
given. Fridley and Adrian (1966) reported data on the
resistance to mechanical injuries for apples, pears,
apricots and peaches by employing compression tests and
impact tests which were basically similar to those used
by Mohsenin and Géhlich. Pears can withstand large com-
pression forces before bruising and thus present little
problem when bulk handled in deep containers. On the
other hand, apricots bruise under a relatively small
compressive force and these fruits are more susceptible
to damage during bulk handling than pears (Fig. 2-1).

Nelson and Mohsenin (1968) reported that bruises
in apples caused by dynamic loads are considerably larger
than those caused by equivalent quasi-static loads.
However, Wright and Splinter (1968) found that lower
energy inputs are required to damage sweet potatoes under
impact than under quasi-static conditions.

Location of the bruise has suggested that the
Inaximum shear stress can be a possible failure parameter
CFridley and Adrian, 1966). Miles and Rehkugler (1971)
cOncluded that shear stress is the most significant

failure parameter in apples.

in N

LD FORCE

q
.Q
U

COMPRESSION YI'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

90 180
I!
I'
72 .’ 14b.
4?
I
Pears
54 108
36 Api:es 72
i, Peaches
I
18 ; 36
"d’
’D
/‘_,a’ Apricots
‘___1 fl
0 l O
O .05 01 .15 02 025 '3

YIELD DEFOBMATION in cm.

Fig. 2-1 Relation of yield force, yield stress, and
yield deformation for various fruits.
(Fridley and Adrian, 1966).

SSION YIELD STRESS in N/cm2

fl
.4
find

COMPR

Huff (1967) attempted to explain the mechanism
causing the cracking of potatoes in handling. Data was
obtained on tensile stress-strain prOperties of the
potato's skin and flesh. He found the tensile strength
of the potato varied throughout the potato tuber being
higher at the center than under the skin. The tensile
strength also varied from year to year. The failure
stress for the skin decreased after the storage. The
failure modulus was higher for specimens taken from the
center than when the specimen was obtained from flesh

immediately below the skin.

2.3 Contact Stresses

 

The general method for determining the distribution
of stresses in the zone of the contact area of two elastic
bodies was introduced by Hertz (1896). The common types
of contact stresses are caused by the pressure of two
bodies initially having a point contact: as two spheres,
or a sphere and a plane (Fig. 2-2), or a line contact: as
two cylinders, or a cylinder and a plane as shown in
Fig. 2-3. Contact stresses in biological products pro-
duces bruising in apples, and peaches and internal splitting
.in potatoes.

The solution of a problem of a concentrated force
Eic‘ting on the boundary of a semi-infinite body was solved

by Boussinesq (1885). Timoshenko and Goodier (1970) dis-

Cuass the application of the Boussinesq solution to determine

10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2-2 Pressure distribution on the contact
surface of a sphere loaded with a
flat plate.

11

.mumHQ pmHH.m spa: Godmoa hmcflwaho m mo
oomhucm pomucoo on» :0 coapsnwppmac enammmpm

 

 

 

 

 

 

mam .msm

12

the relationship between a load distributed uniformly
over the area of a circular radius and the resulting
stresses. Finney (1963) suggested using a solution
derived by Boussinesq for analysis of a potato tuber sub-
jected to plunger loading. It is necessary to assume
homogeneity and isotropy and to also assume that a half-
space exists. However, it is realized that because of
the small radius of the curvature of certain products, the
latter assumption may not be completely valid.

The normal stress under a punch as shown by

Timoshenko and Goodier (1970) takes the form:

 

P
0' I =
zzzuo 2na/g2-r2

where the origin of the Z-direction, the surface of the
half-space and a closed form of the pressure distribution
are shown in Fig. 2-4. Finney (1963) reported a signifi-
cant difference existing between certain potato varieties
in their response to applied stresses of surface pressure.
Mohsenin and Gohlich (1962) applied the same technique

to apples, potatoes, pears and tomatoes concluding that
the compression test appeared to offer the most promise
(bf evaluation of mechanical behavior as related to
bruising.

Evaluation of the stress-strain relation for

CCnIvex bodies subjected to uniaxial compression by means

13

 

 

 

 

 

 

 

 

 

 

 

 

Gzz
0'
WP |rr
I 622'
I l
I I P
0‘2 =
I Z I Zfla‘{a2 - r2
I I z — 0
I 7 .
V ‘
l
I I
dig cLo

Fig. 2-4 Theoretical stress distribution under
a rigid loading die.

14

of flat plates has also been investigated relative to its
application to agricultural products.

When a cylinder is compressed diametrically between
two flat plates, the conventional two-dimensional elas-
ticity theory predicts that the failure is initiated at
some point within the cylinder, with the exact position of
the critical zone dependent on the material under test.
This test is used to determine the failure under tensile
strength of such materials as rock, glass, and concrete,
and is an accepted standard test for brittle materials.
This test is occasionally called the Brazilian test.
Isenberge (1965) used this procedure to investigate the
effect of the amount of moisture content on the strength
of concrete. He concluded that failure occurs under lower
loads as the concrete becomes saturated. Brown and
Trollope (1967) stated that the failure in concrete was
initiated somewhere near the center of the disc and propa-
gated outwards to the loading point. ColbaCk (1966)
argued that the Brazilian test was valid for tensile
strength only if failure was initiated at the center of
the disc. Thaulow (1957) loaded a cylindrical specimen
to failure, summarizing that the tensile splitting strength
(appeared largely independent of the length and diameter
(bf the specimen. Durelli and Mulzet (1965) used photo-
elasticity to determine strain and stress distributions

irl a linear material subjected to large deformation. More

15

detailed information for stress distribution between two
cylinders in contact are given by Radzimousky (1953),
DePater (1960), D6rr (1955), Poritsky (1950), and Smith
and Liu (1953).

The contact stresses in a sphere for the case of
infinitesimal strain can be examined in detail as given by
Hertz (1896), with later works consolidated by Love (1944)
and Timoshenko and Goodier (1970). Durelli and Chen
(1973) experimentally determined the displacements and
strains in a solid sphere diametrically loaded. Chen and
Durelli gave the stress distribution of the case described
above using the strain-energy function and the concept of
natural stress.

Fridley 33 31. (1968) applied the plunger and flat
plate tests on pears and peaches, concluding that the
theory of elasticity gives reasonable predictions of
stress distribution and bruising. The flat plate results
were more reliable and useable than the plunger test.
Comparison between the two tests are represented in
Fig. 2-5 as given by Fridley e: 31.

2.4 Stress Analysis in Fruits
and Vegetables

 

 

Limited work has been completed in analyzing
=3tresses in fruits and vegetables. Hamann (1970) solved
the contact problem involving a viscoelastic spherical

body falling onto another. In later studies, the finite

16

10—
O-I-IOPlunger Test
1- fi—‘Flat Plate Test
8—
z 6—
£1
.4
5:] -
3%
c>l+—
[1'4
2L
0 1 l l I

 

 

1
O .127 .254 .38 .508 .635
DEFOHMATION in mm
Fig. 2-5 Comparison between using flat plate and

plunger test with different product.
(Fridley e§_§1. 1968)

17

element method has been used to determine the stresses

in apples resulting from contact with a flat plate.
Apaclla (1973) assumed an elastic material. Rumsey and
Fridley (1974) used a material with a constant bulk
modulus and time dependent shear relaxation. De Baerde-
maeker (1975) used a material with time dependent bulk
modulus and shear modulus while studying the behavior of
a sphere in contact with a flat rigid plate to obtain the
creep deformation and the stress distribution. Gustafson
(1974) obtained a numerical solution to the axisymmetric

boundary value problem for the gas-solid-liquid medium.

2.5 Summary

The resistance of fruits and vegetables to the
applied forces is important in view of mechanical and
handling injuries to the product. Damage in the form of
skin removal, bruises or cuts may be increased during
digging, shaking, storage, grading and shipping. Knowledge
of the stress distribution in fruits and vegetables under
static and impact loads is limited because of the diffi-
culty involved in determining material properties and the
lack of analytical solutions valid for the irregular shapes
involved.

The finite element technique has potential when
analyzing agricultural products because of its ability to
handle irregular shapes. The stress distribution within a

product is required knowledge before failure criteria can

18

be established. The finite element analysis of agricul-
tural products to obtain stress distributions has been

initiated but the final results are far from complete.

III . BASIC THEORY

The solution of problems in the classical theory
of elasticity are generally obtained by assuming infini-
tesimal deformation. The strain is evaluated by con-
sidering only the first order terms in the displacement
gradient; the second order terms are neglected. Both sets
of terms must be taken into consideration when calculating
the strains which occur during large deformation.
Sokolnikoff (1956) stated that many technically important
problems in elasticity call for consideration of finite
deformation; deformation in which the displacements
together with their derivatives are no longer small.

Numerous papers considering the large deformation
of elastic solids have been published by Rivlin (1948a,
1948b, 1956, 1960, 1970) and by Green and Adkins (1960).
Ericksen and Rivlin (1954) treated the case of large
elastic deformation of homogeneous anisotropic materials.

The behavior of most materials including metals,
rubber-like materials and biological products are not
linearly elastic, except within specified limits. The

stress analysis of rubber-like and biological materials

19

20

possess two rather unique features as compared to the
analysis of more conventional structural materials:
(a) The materials are nearly-incompressible (i.e.,
their bulk moduli are much larger than their
shear moduli). The potato, for example, can be
placed in this range of nearly—incompressible
materials. Finney (1963) reported the bulk moduli
of potatoes as K = 7791 N/cmz, the shear moduli
as G = 124.8 N/cmz, and a Poisson's ratio of
u = 0.492.
(b) They are capable of experiencing large deformation

before any type of failure occurs.

3.1 Non-Linearities

 

The large change in geometry experienced by the
body as a load is applied produces a non-linearity, as
described by Durelli and Mulzet (1965). Because of these
changes:

(a) The higher order terms in the strain displacement
equation can no longer be neglected.

(b) The stress-strain relationship becomes considerably
more complicated.

(c) The resulting state of strain may depend on the
order of application of the load.

The non-linearities can be classified in three

categories, as defined by Desai and Abel (1972)

and Biot (1965).

21

Material (physical) Non-Linearity: The stresses
are not linearly prOportional to the strain, though small
displacement and small strain are considered. Evans and
Pister (1966) developed a constitutive equation for elastic
solids sustaining deformation for which displacement
gradients were small and material non-linearity was per-
mitted.

Geometric Non-Linearity: Where linear stress—
strain equations are assumed to hold, the geometric non-
linearity arises both from a non-linear strain displace-
ment relation and from a finite change in the geometry of
a deformed medium. In other words, this category encom—
passes large displacement and large strains. This non—
linearity is introduced into the theory of elasticity
through the equilibrium equation and by inclusion in the
strain-displacement relation of higher order terms.

Material and Geometric Non-Linearities: The most
general category of non-linear problems is the combination
of the first two categories involving non-linear con-
stitutive behavior as well as large strain and finite

displacement.

3.2 Stress and Strain

 

The common definitions of strain in simple tension

or compression are:

(a)

(b)

(C)

22

Conventional Strain (Lagrangian): Commonly referred
to as the strain in the coordinate system of the

undeformed body.

= Total Change in Length _ 2f-Q'o

original length l

 

0

where 20 the original length

if = the final length
Natural Strain: Natural strain is introduced to
describe the large change in geometry due to the
behavior of material subjected to large strain and

defined as the integral of instantaneous or

increment change in length.

2
E—If%£

2
o

= 1n(1 + E)

or in another form:

 

R
E _ z Instantaneous change in length
2 Instantaneous length

Final Strain (Eulerian): Commonly associated with

a coordinate system of the deformed body.

Total change in length = 2f-£o

e = Final length if

 

23

The final strain is related to the conventional

strain by

(d) Green Strain: The strain tensor, known as Green's
Strain (Finite Strain), is often referred to as
the strain relative to the undeformed body, and

is defined as:

(e) Eulerian Strain Tensor: The strain tensor related

to the deformed body and defined as

2 2
es = if ‘ £0 = 6 _ 152
2 2
22f

The general definition of Green's and Eulerian Strain
tensor are valid, whether small or large displacements
exist. The five definitions of strains, as given by
Parks and Durelli (1969), are shown in Fig. 3-1.

All these definitions give the same value of
strains for small deformations. Parks and Durelli (1969)
pointed out that with a large change in geometry, the
stress acting on a specific element area should be com-

puted taking into account the change in the size of that

24

 

LAGRANGIAN NATURAL. EULERIAN

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

STRESS STRESS STRESS
(CONVENTIONAL) (TRUE)
L F — dF c F
0' I -— = —— l—
A; 0' Am 0' A1
I 2- I-2
LAGRANGIAN CH...“ ' 2'
STRAIN W 2 li
”ENSOR
coumutm) L
L- (“T—
e - 2: ol -I
T ‘ J1:
LAGRANGIAN
(CONVENTIONAL) "-0 II
(ENGINEERING) 1
————-LE . In (“t")
I 1%
NATURAL ,- . .n L
STRAIN '*
l
T- E
»————1< z-In (I-c)
T 91(-
EULERIAN (E. hm "_"L
STRAN 'I-0 'I
l 1‘
——{£[ a I " ‘20:"
T
EULemAN , ,
STRAIN a: : m h_—'_.
(TENSOR [,0 2 I
coueoururI ' '

 

 

 

 

 

 

*Iumz (my; u-Jm new In: SAME SLOPE If ms untRIAL Is INCOWRESSIBLE.

Fig. 3-1 Relationship between several definitions
of stress and strain (Parks and Durelli, 1969)

25

area to improve the linearity of the stress-strain rela-
tion for a large strain.

The actual (true) stress on the deformed body can
be expressed in the Eulerian definition of strain. The
stress-strain relation obtained using true stresses and
natural strain is, for some materials, more nearly linear
than the one obtained using conventional stress and
natural strain.

The three definitions of the stress in simple
compression or tension are:

(a) Conventional Stress

L _ Applied Load =
o T Original Cross-Sectional Area

3""!
O

(b) True Stress

E _ Applied Load
0 _ Final Cross-Sectional Area

:ulu:
H}

(c) Natural Stress

A
6=f
A
0

fee
A

where A0 and Af are the original and final areas, reSpec-
tively, and dF is the increase in the load acting on an

area, A.

26

The stress-strain curve for a cylindrical section
of tissue removed from a white potato tuber, Fig. 3-2,
closely approximates a linear relationship. The modulus

of elasticity can thus be defined as:

EM

u
(we

where o is the uniaxial principal stress and s is the

associated principal strain.

3.3 Green's Strain Tensor

 

Green's strain tensor (finite strain tensor) in
Lagrangian coordinates (Xi' i = l, 2, 3) (Fung, 1965),
describes the deformation of a body relative to the unde-

formed state. The indicial form of this equation is:

_ 1 Bxk axk
Ei‘ "2‘ T T-Gi"
3 i j 3i

i
3
l
I

where the indices take values of l, 2, 3 and dij is the

Kroneker delta. Green's strain tensor is:

1 8Ui 3U. 3Uk BUk

when written in terms of the displacements (u, v, w) of

a rectangular cartesian coordinate system (X1 = X, X2 = Y,

X3 = Z). The strains in matrix form as given by Hughes

and Gaylord (1964) are:

27

400“-

300~

200—

JOO~—

 

0 l l l l l l
O l 2 p 3 4 5 6
DISPLACEMENT in mm

Fig. 3-2 Force-displacement curve of a cylindrical
specimen of potato under uniaxial
compression. (L = D = 25.4)

28

 

 

 

1
6n 1 v+ an I dug + dII
01 2 (TI fly 2 III Hz
1 I1 II 0 U 0 v I I7 w 0?.)
— ._ -_.. _- .1... + - _.
Eij _ 2<JU+OI) 6U 2<Ily (72 302
1 (I'm 610) 1 2(d:+ +010) 0w
.- L..- + . _ .‘ -
‘2 dz 0.; 0y dz
L .J
P 1
. ., I '
éfi\2 + 9.3 \2 + IIII) '1 .317". {I}: + {luau + 9!) ‘3'“1 Jim} dI_¢ + «TIE (:12 + (HI "'01
31' 0.1-) "x I l 0x 01/ 0.1: I?” 01: a” I [fix (I: «I: dz "I a: J
l flu Em (Iv {In (In) aw‘l 1 I’m ‘2 (III \ 2 (”U ‘ I If": flu {III 0v {III LN"
+ . ':~ '+':‘-" 1' .". l‘f‘ »-) '1" ) ’T‘T“'1'."Z—+
2 (Ir 0” 0.1: III] le (Iy J l I)” , ‘01, (I), J l d” uz My «12 (I), a; J

flu flu av 81) {no Ow 1 [an JII 0v «1v 0w aw Im :10 fl '1
+"-.--'+ ‘.':‘+—*.—+-- -- t 3. '
L (it '7: a: "Z (71‘ I): [ 1”” dz I)” «I: (7U dz «'2 \ dz " J

 

-J

The principal strain values are the unit relative
dI'LSE>1acements (normal strain) that occur in the principal
direction and are denoted by El, E2 and E3, and are given
by the determinatal equation:

IE..-E6..I=0 3.3
13 13

The roots of the characteristic cubic equation as given

by Eringen (1962) :

-E3 + IE2 - IIE + III = 0

where the three strain invariants I, II and III are:

kk 3.4—a

II

2[EiiE jj - EijEij] 3.4-b

III

II
(D
["1

ijk E11 EjZ k3 3°4'C

29

The Latin indices take on values 1, 2 and 3, and eijk is
three dimensional permutation symbol.
For an incompressible material, the volume in

the deformed and undeformed state are equal and the con-

dition of incompressibility is:
III = l 3.4-d

3 - 4 Finite Plane Strain

 

An undeformed body loaded in the X1, X2 plane
undergoes deformation in this plane plus the body may be
subjected to a uniform extension parallel to the X3 axis

as shown in Fig. 3-3. Equation (3.1) yields the following

st rain equations:

30 au 30 an
E = 1 a + B + Y Y 3 5-a
a8 7 5X 3X 5X 5X '
B o: B a
Ea3 = o 3.5-b
E = lIAZ - 1) = Constant 3 5-c
33 2 °

where the indices on, B, y take values of l, 2 and A is
the ratio of the thickness of the deformed to that unde-
formed state normal to the plane of deformation.

Oden (1967) stated that the problem of finite
Plane strain superimposed on uniform finite extension is

also encompassed by equation (3.5-a, 3-5-C) if: instead

30

 

 

 

 

 

 

 

 

13

Fig. 3-3 Finite plane strain deformation of a small
element

31

of setting E33 equal to zero, the strain normal to the

plane is computed by using equation (3.5-c).

Eringen (1962) described plane strain by having
itientical deformation in a family of parallel planes and
zero deformation in the direction of their normals.

Substitution of the strains from (3.5-a) into

(:3..3) yields the strain invariants for finite plane strain

problem which are :

I - E11 + E22 + A - Il + A 3.6 a
II ER BB +A2(E +E)=
ll 22 21 12 ll 22
3.6-b
2
12 + A II
ar1<fl
2
III = A 12 3.6-c
3-5

‘Axisymmetric Strains

 

It is convenient to express the axisymmetric

prc>lblem in terms of cylindrical coordinates (R, 6, Z) as
SI1§J€J€33ted by Timoshenko and Goodier (1970) with corres-
pon<3ing displacement components (11, V. W) where u and V

are the radial and tangential direction and w is parallel

t" tithe Z-direction. The component v vanishes for axi-

sflqnnnertric bodies with axisymmetric loadings, and u, w are

lndeE>endent of angular (6) coordinates. All derivatives

32

with respect to 6 vanish and the shear strain components
Yre' 726' and Yea are zero. The non-zero strains are

E E Y 'and E

66'

Equation (3.1) can be written for purely axi-

rr' zz' rz'

gsynmmetric deformation drawn from the theory of finite
eajuasticity, Green and Zerna (1968) and Green and Adkins
(.1960). Green's strain tensor can be expressed as a

function of displacement:

l aUn 3Um BUR. 3UP.
Enm=2f+§r+fif “‘3
m n n m
En3 = 0 3.7-b
_ l 2 _ _
E33 - —2-()\ 1) 3.7 C

where indices (n, m, IL) take the values of l and 2.

The function A = A (r, z) is the extension ratio
in the circumferential direction, i.e., A is the ratio of
t1r1€3 length of a circumferential fiber in the deformed
k3C3C1)' to its original length in the reference configuration,

and takes the value:

Equation (3.7-a) yields a set of strain equations
which can be used during the formulation of the finite

eleluent method. These equations are:

33

2 2
_ Bu 1 Eu 3w _
Err—7+: K’SE) + ("5? 3-9a
~2
_u l u _
EGG—f+§- (E) 3.9b
2 2
_3w 1 au\ aw _
Ezz‘a'a+‘§ 5‘2/ *6?) ”C

The strain equations for the infinitesimal theory
of elasticity are obtained by neglecting the second order
terms in the strain-displacement relationship in (3.9-a)

through (3.9-d) .

The strain invariants for large deformation

Strain relative to Lagrangian coordinates are:

34

and
u 2 ‘F Bu 3w
III=(1+E) ‘l+fi)<l+§§)+
- L 3.10-c
EB _3_w_ 2
32 Sr
For incompressible materials, III = l.

3 - 6 A General Formulation
For Elastic Bodies

The near-incompressibility of the material pre—
sents difficulties because analyses based upon the con-
ventional displacement formulation (Navier's equation of

equilibrium) may be greatly in error when Poisson's ratio
is equal to one half (mechanically incompressible material).
The usual displacement formulation is no longer valid,

Green and Zerna (1968). Glauz (1962) stated that when

Poisson's ratio approaches u = 0.5, the solution of these

equations by numerical techniques results in large errors.
It is desirable, therefore, to have a formulation in terms

of displacements valid for all admissible values of

Pc‘l.l-Sson's. ratio. The elastic field equation for an incom—

Pressible material as given by Herrmann and Toms (1964)

1‘3 Valid for any admissible value of Poisson's ratio

(0 i u i 0.5). This formulation starts by considering

N - .
av; er ' 3 equation:

35

Tij = A Ekk 6ij + 26 Eij - (3A + 2G) 3.11

HAT 8..
3]

udncrv A and G are the lamé constant

a = thermal expansion
AT = temperature difference
u = Poisson's ratio

The mean effective pressure expressed in terms of

t:11<2 normal stresses is:

o=-1-T
3 kk

vvlfiLjLZLe it is
(3A + ZG)

O: 3 E

 

kk
Vtkleaun expressed in terms of the normal strains. Substituting
ZGH/(l - 2H)

13(31? A yields the mean effective pressure in terms of the

Shear modulus and Poisson's ratio

 

_ 2G(l + u)
- 3(1 ’2U) Ekk 3.12

11 ' . .
avler's equation, 3.11, can now be written as:

36

Ban
I y = ——_—_— o o + o o .
T1] (1 + u) 51] ZG E13 3 13

where the thermal effects have been dropped.

We now define a mean pressure parameter

T
_ 3o _ kk
H ’ 2(1 + u)G ‘ EM 3'14

 

which is a hydrostatic pressure. It is also referred to
as a Lagrangian multiplier in the case of incompressible
Inaterials.

Oden (1967) stated that, in general, the hydro-
231:atic pressure must be determined from an equilibrium
eaqzuation or a static boundary condition in the case of

jL11130mpressibility to satisfy (III = l) and to determine
t:11<e total stress.

With constant material preperties, the equation

can be rewritten as,

Tij = G[2Eij - Ekk Gij] + G H Si. 3.15

J

The total stress can be expressed in the form:

Tij = 2G Eij + 2p G H 6ij 3.16

37

For incompressible material (i.e., u = 0.5) the

stress-strain relation is

II
N
C
["1
+
Q

T.. .. 6..
13 1] 1]

while the condition of incompressibility is III

In matrix form,

{T} = [D] {E}
F“
T 1 l 0 0 0 0
XX
T 0 1 0 0 0
YY
Tzz 0 0 l 0 0
T = 26 0 0 0 k 0
XY
.Tyz 0 0 0 0 5
“r 0 0 0 0 0
X2
<3 1) u u u 0 0

 

To accommodate geometric non-linearity and large
deflection, the Green strain tensor Eij is divided into

tszc> parts (a comma denotes a differentiation)

1 1
‘ 7 [Ui,j + Uj,i] + f (Uk,i Uk.j)

0 u
0 u
0 u
0 0
0 0
k 0

0 -u(1-2u)

 

3.17

— 1.

 

3.19

The first part is a linear strain tensor, the

S o o '
eCOnd part 15 non-linear or quadratic

3.20

3.18-b

38

II
th
F‘I
c:
+
c:
L-J

e.. 3.21-a
1]

and

NIH

n.. = U ) 3.21-b

1] (Um m
where the indices take the value 1, 2 and 3.

3.7 A Variational Principle

 

In the theory of elasticity, variational principles
aare used as a means of deriving the governing differential
euguations and to obtain approximation solutions. These

trsiriational principles may be classified into three types
(Washizu, 1968):

l. The Theorem of Minimum Potential Energy (in terms
of displacements): i.e., among any possible dis-
placement fields satisfying the required displace-
ment boundary conditions, the actual one minimizes
the potential energy of the system.

2. The Theorem of Minimum Complimentary Energy (in
terms of stresses): i.e., among all possible
stress fields satisfying the stress boundary
conditions and the equilibrium equations, the
actual one minimizes the Complimentary Energy.

3. The Hellinger-Reissner Variational Theorem (in
terms of displacements and stresses). This theorem

can be derived from either the potential energy

39

or the complimentary energy principle by applying

suitable constrain conditions.

In both the Theorem of Minimum Potential Energy
and the Theorem of Minimum Complimentary Energy, it is
difficult to satisfy the boundary conditions and the
equilibrium equations. Some numerical solutions, using
the finite element method, Melosh (1963), have been
developed incorporating the Theorem of Minimum Potential
Energy in conjunction with the Ritz procedure; these
solutions are inaccurate when applied to nearly-

incompressible materials and furthermore, are completely

LJIisatisfactory for incompressible materials, Hwang gt a1.

(1969) . The Hellinger-Reissner Variational Theorem,
£2<3clssner (1950), is more general and includes the previous
tzlleaorems as special cases. The large number of unknowns
lhjrnnits its application for approximate numerical solutions.

Finite element analysis of incompressible and

Ileeaixlybincompressible materials commenced with a paper by
IiEBJE‘rmann (1965). Herrmann presented a modification of
Reissner's Variational Principle for isotropic materials

based on the elastic field equation. This variational

Principle is

40

HH = f G [I2 - 2II + 2pHI - u(l — 2p) H2 -
v 3.22

{¢}T {2}] av - f {¢}T {T} asl
S

l

where the thermal effect has been eliminated.

The symbols I, II, ¢, H, F, T, G and u respectively,

denote the first and second invariants, displacement, mean
pressure parameter, body forces, applied surface forces,

.shear modulus and Poisson's ratio.

In rectangular cartesian coordinates, equation

(23.22) takes on the following form:

_ 2 2 2 1 2 2 2
HH - é G [Exx + Eyy + Ezz + 5 (ny + sz + sz) +
2
ZuH (Exx + EYY + Ezz) - u(l - Zu)H - 3.23

{¢}T {F}]dV - r {¢}T{T}dSl
S
1

It must be noted that there is an arithmetic error
21!) sequation (3.23) as given by Herrmann; the value of 8
appeared in print as a 2.

Taylor gt 21. (1968) extended the above equation
to Orthotropic materials. Reissner (1953) formulated a
variational principle for hyperelasticity. Both stress
and displacement are varied, and the principle yields both

the condition of equilibrium of forces and the stress-

3 - .
tra 1n relation .

41

Tong (1969) presented a variational principle,
based on the assumed stress hybrid method that was suitable
for incompressible and nearly-incompressible material
solids. Tong and Pian (1969) reformulated the variational
principle for the finite element method based in an
assumed stress distribution. Hughes and Allik (1969)
used Herrmann's work for the case of the plane strain.

Key (1969) derived a system of equations as a special
form of Reissner's variational principle which was suitable
:for anisotropic incompressible and nearly-incompressible

materials.

IV. FIN ITE ELEMENT FORMULATION

The finite element method is a numerical procedure
for solving differential equations and can be used in con-
junction with the variational formulation for an incom-

pressible body to calculate stresses in a body of arbi-

trary shape. The stiffness matrices used when solving

ea geometrically non-linear problem while employing the
finite element method are discussed by Martin (1966) ,

Oden (1969) and Sticklin e_t_ _a_1. (1971). Oden (1967) and

Oden and Sato (1967) investigated the large deformation
for non-linear elasticity problems and analyzed the large
(11 splacement and finite strain using Green's strain

tensor, assuming the hydrostatic pressure constant over

the element. Oden (1968) formulated the finite plane

strain problem for incompressible solids with the hydro-
static pressure appearing as a Lagrange multiplier to

Satisfy the condition of constraint (III = 1).

Oden and Key (1970, 1971) applied the finite
eleInent method to the problem of finite axisymmetric
deformations of incompressible elastic solids. Hibbitt

3 fl. (1970) formulated the finite element equations for

42

43

large displacements and large strains with a particular
reference to the elastic-plastic behavior of solids.

Argyris gt gt. (1974) employed combined natural

strains with the finite element method. Difficulties
arose in the application of this formulation to incom-
pressible or nearly-incompressible materials. The results
were sensitive to the boundary conditions and to the ori-
entation of the elements. Solution for absolute incom-
pressibility, the equivalent of allowing the compressibility
inodulus to become infinite, did not converge to the true
ssolution as the element size was reduced. Iding gt gt.
(£1974) analyzed experimental data to characterize the
ssizress constitutive function for non-linear elastic solids
151:3 an inverse boundary value problem.

Many soil mechanics problems have been solved
ulsszing finite elements. Naylor (1974) analyzed porous
‘nmeaciia for both linear and non-linear materials by
sseez>arating the stiffness matrix into "effective" and "pore
fluid" components, allowing excess pore pressure to be
calculated explicitly. Yokoo gt fl° (1971) applied a
Variational principle equivalent to the governing equation
11" IBiot's Consolidation Theory assuming the soil to be
non“homogeneous, anisotropic, elastic, and saturated by
incOIhpressible water. The deformation of the soil was not
dependent on pore water pressure but on effective stress.

Thomas gt gt. (1972) assumed the soil homogeneous,

44

isotropic and saturated and assumed plane strain to
evaluate the displacement and the pore pressure in soft
soils.
A detailed discussion of the general theory of
the finite element method is given in Zienkiewicz (1971),
Oden (1972), Desai and Abel (1972), Martin and Carey
(1973), Cook (1974) and Segerlind (1975). The region
under consideration is divided into small elements con-
nected at node points. The unknown displacements and
.hydrostatic pressure are approximated over each element by
Iqolynomials using three parameters at each node, two

(irisplacements and one hydrostatic pressure.

4 - 1 Plane Strain

 

The displacements in each subregion or element are
approximated by linear polynomials expressed in terms of

tLIIEB displacements of element nodal points (Zienkiewicz,

JLSB'7JJ.

u = [N] {U} 4.1

Where [N] is the matrix of shape functions (interpolation
functions) relating the element displacements, u and v, to
the nodal displacements {U}. An example for plane strain

elenient is given in Fig. 4-1. The horizontal displacement

u = a1 + a2X + d3Y

<
ll

81 + 82X + B3Y

45

 

 

 

 

 

131g. 4.: Triangular element in plane strain and nodal
diSplacement.

46

Solving the equations for the coefficients, using
the nodal values of the displacements allows u and v to be

written as

u = N. U . + N. U . + N
1 3

21-1 23-1 k UZk-l

<
H

Ni U2i + Nj U2j + Nk 02k

where the shape functions (interpolation functions) are

Ni = (ai + bix + CiY)/2AO

Nj = (aj + ij + CjY)/2AO

Nk = (ak + ka + CkY)/2AO
and

a1 = Xi Yk - Xk Yj

b1 = Yi — Yk

c1 = Xk - XJ

The other constants aj, bj, etc. are cyclic per-
nntrtations of subscripts and A0 is the area of the element
<ill the undeformed state.

The hydrostatic pressure (mean pressure), H, as
given in equation (3.14) can be written in terms of the

Shape functions as

h = [N] {H} 4.5-a

47
where {H} is the vector of the nodal values of h .

{H}T = {Hi H. H

II
M
Z

[N] i N. N

The general formulation of Green's strain tensor
which is valid for either large or small strains, is

defined in equation (3.20) for two-dimensional plane

 

strain
E = e.. + n 4.6
1] 13 13
The infinitesimal strain in matrix form is
{e} = [BO] {g} 4.7
where
T _ T
{e} - {eXX yy ny} 4.8
PEN. 3N. 3N -
_i _l __15
3x 0 3x 0 3x 0
E BNi 3N. BNk
‘80] - 0 5;“ 0 5y; 0 5y_ 4.9
EN EN. EN EN. 8N 3N
__1L_ 1 _i __l ._k ___k_
LPY 3x 3y 8x 3y 3x

 

 

48
and

U}T

T _
{q} ‘ {UZi-l UZi U2j-1 U2j U2k-l 2k

The large displacement component for plane strain

as adapted from Zienkiewicz (1971) is

m

 

 

 

 

 

- .. 5E
Bu 3v
5i‘ 5; 0 0
’22
3x
1 Bu 3v
=_ 4.
fig
22 9.! 12 8_v 3y
L8y 3y 3x 8x
.._J
av
337
or in matrix form
— 1 4 12
{n} — f [A] {e} .

The components of {6} can be written in terms of
the shape functions [N] and the nodal parameter {q} by

differentiating (4.3) which yields

49

 

 

3Ni 0 SN. 0 BNk 0
5x x 5x
0 BNi 0 EN. 0 aNk
8x 3x 3x
{8} = {q} 4.13
BN1 0 SN. 0 aNk 0
3y 3y 3y
0 aNi 0 EN. 0 aNk
5y 5y 3y
L- .3
or
{e} = [G] {q} 4.14

when written in matrix form.

Substitution of (4.14) into (4.12) gives

{n} = % [A] [a] {q} = [BG] {q} 4.15
where

[36] = % [A] [G] 4.16

The subscript G is used to denote the geometric matrix
which is a function of the displacements.

Green's strain tensor in matrix form becomes

{E} = [BO] {q} + [BG] {q} 4.17

or by combining the infinitesimal and geometric parts

{E} = [B] {q} 4.18

50

where
[B] = [Bo] + [86]

The variational H is for the region which is sub-

divided into a number of elements, therefore,
N
n = 2 1(9)

where N is the total number of elements. Equation (3.23)
for one element can be written in terms of displacements

and the mean-effective pressure as

(e) _ 2 2 t 2
H — £(e) G [EXX + Eyy + 2 ny + 2pH(EXX + Eyy)
2 T
u(1 - 2p) H ]dV - 1(e) [N] {F}dV -
V 4.19
T
t(e) [N] {T}ds

The above equation is valid for either large and small
displacements. The strain components can be expressed

in terms of the displacements as follows

33X + Eiy + % yiy = {q}T [BJT [I] [a] {q} 4.20—a
EXX + Eyy — {q} [B] {J} 4.20 b

and

h = [u] {H} 4.20-c

51

The diagonal matrix [I] and the vector {J} are

defined as

'1 o 6} /1
[I] = 0 1 o {J} 1
{90: o

 

 

Substituting equations (4.18a, b, c) into Herrmann's

functional and obtaining its stationary value gives

_ an an _
5H - ETET {Sq} +‘5TET {6H} - 0 4.21

The governing finite element equations for a

single element take the form

[K] {¢} = {P} 4.22

which can be partitioned into

 

1 ‘1
[K11] ;[K12] {q} Q
—-— —f7 _._ — — — — — 4.23
LtKlz] '[K22{_ {H} o

 

where [K], {¢}, and {P} are the global stiffness matrix,
the column vector containing the unknown displacement and
hydrostatic pressures, and the force vector, respectively.

The submatrices in equation (4.23) are defined by

[K11] = 2G I [BJT [I] [B]dV 4.24
V

52

[K12] = 215 £ [B]T {J} [NJdv = [K21] 4.25
[K22] = -2uo(1 - 2p) i [u]T [NJdV 4.26

and the element force matrix
Q = f [n1T {F}dV + f [u]T {T}dSl 4.27

V 81.

The global matrices are assembled using standard techniques

(Segerlind, 1975).

4.2 The Axisymmetric

 

The axisymmetric element is quite similar to the
two-dimensional plane strain element. Using the same
formulation for displacements and the same shape functions,
the variable x is replaced by r, and the variable y is
replaced by 2. An example of an axisymmetric element is
presented in Fig. 4-2. The strain displacement relation-
ship, which is valid for small and large displacement, is
given in equation (3.9). The strain can be divided into
two parts: one representing the small (infinitesimal)
strain and the other for second order terms. This can be

written in the form:

53

 

 

 

--B.

 

 

 

Fig. 4-2 Triangular axisymmetric element and
nodal deformations.

54

The infinitesimal part is

 

{e} = [C0] {q} 4.28
where
T _ T
{e} — {err e66 ezz er} 4.29
3Ni 0 3N. 0 3Nk 0
3r 3r 3r
N_i 0 51 0 .NJ: 0
- r r r
[Co] — 4.30
0 3Ni 0 3N. 0 3Nk
32 32 32
3N. 3N. 3N. 3N. 3N 3N
_}_ __1. i J k __k
32 3r 32 3r 32 3r__

 

 

and {q} is defined by the equation (4.10).
The second order terms in the strain can be
treated in the same manner as in the plane strain case,

Producing :

{n} = % [D] {8}

\

 

 

 

9.9.

l. .. 3r

3u 3w

4'; 51: ° ° 0 23—;

{n} =1 0 0 g 0 E 431
r .
3u 3W

0 ° 0 32 3—2 3u
3_2'

32 3! 0 32. 3w 3w

32 32 3r E k32—

55

The components of {8} can be related to the shape

functions by differentiating the displacement equations

 

 

giving
7' _.
1 3Ni 0 3N. 0 3Nk 0
3r 3r 3r
0 3Ni 0 3N. 0 3Nk
3r 3r 3r
{0} = E; 0 El 0 it o {q} 4.32
r r
3Nl 0 3N. 0 3Nk 0
32 32 32
0 3Ni 0 3N 0 3Nk
__ 32 z 32_t
01:
{e} = [L] {g}

Substitution into the equation for n yields

{n} = § [D] [L1 {4} = [CG] {q}

and the strain tensor can be written as

{E}

[CO] {q} + [CG] {q} 4.34

{E}

[C] {q} 4.35

56

where [C] is the sum of the infinitesimal and geometric

matrices

[C] = [C0] + [CG] 4.36

The hydrostatic pressure can be defined in terms

of each nodal value by
h = [N] {H}

[N] is the shape function as defined in equation (4.5-c)
aiud {H} is a vector of the nodal values of {h}, as defined
iri equation (4.5-b).
The incompressibility condition III = 1, must hold
{at: every point in continuum. This means that

I (III - 1)dVO = 0 4.37-a

V
O

Oden and Key (1970, 1971) suggested a procedure for
1“Handling the incompressibility condition suitable for
ftiriite element formulation. They compared the volume of
tile: element in the undeformed (V0) and undeformed (Vf)

Shapes as:

4.37-b

III
<
I
<
I)
o

h

i n which

57
V =21rRA Vf=21T(R-{-U)Af

_ 1 ‘ _ i
’ 3(R1 + Rj + Rk) U ' 3‘021-1 + UZj-l + UZk-l)

”I

In the above, E is the radial distance to the

centroid of the undeformed triangular (cross-section)
area A0, 6 is the average radial displacements of the

j and k and Af is the cross-sectional

area of the element after deformation. Either the incom-

element nodes i,
Lxressibility condition (4.37) or (III = 1) can be used,
though (4.37) is more convenient in deriving the stiffness
realation of the element. Equation (4.37-a) was not used

explicitly. The incompressibility condition is satisfied
When the variational formulation is a minimum.

The variational equation for the axisymmetrical

Problem is given below.

n‘e) = [(e)G [{q}T [C]T [I] [C] {q} +
V

26m] {11} {J}T [g] [c] - 4.38
u(1 - 2n) [an {H}T {J} [N]]dV -

T T
£(e)[N] {F}dV - £(e)[N] {T}ds

58

The minimizing of H reduces this to

4n‘e’ = c 1(e)([c]T [I] [c] {q} + “[c1T {a} [u] {a}

V
+ p[N]T {J}T [c] {g} + u(l - 2U) [u]T [N]

T T
{H})dv - e)[N] {F}dv - t(e)[N] {T}

f
v(
d5 = 0

where the diagonal matrix [I]and the vector {J} take the

fomnu
[1 o o 6] 1
o 1 0 o! 1
[I]: ‘, {J}:
0 0 1 05 1
0 0 0 %| 0

 

The submatrices for the element stiffness matrix

and the element force vector take the form

26 x [ch [I] [C]dv 4.39
V

[K11]

[K12] = 2G0 i [c1T {J} [NJdv = [K21]

aljél

[K22]: -2u(1 - 2u)G f [n]T [N]dv 4.41
V

firkle element force vector is the same as defined in

E2(Juation (4.27).

59

4.3 Element Stresses

 

The stresses in each element can be calculated

from the stress-strain relationship
{T} = 2G [I] {E} + 2 u G h {J} 4.42

This equation can be written in terms of the computed
nodal displacements {q} and the nodal hydrostatic pressures

{H} giving
{T} = 2G [I] [E] {q} + 2pG [N] {H} {J} 4.43

For the axisymmetric the matrix [C] replaces

matrix [B] in equation (4.43).

4..4 Nodal Stresses

 

Nodal values of the stress components are needed
.iri order to plot isostress lines. The nodal values can be
<3krtained from the element stresses using the conjugate
Stress idea developed by Oden and Brauchli (1971) and
discussed by Gallagher (1975) . An existing two-dimension
cOmputer program was used to calculate the nodal stress
Components for plane strain. This program was modified
£131: the axisymmetric case and used to calculate the nodal
values in the axisymmetric bodies analyzed.
4 - 5 Other Finite Element

Formulations
Many unsuccessful attempts were made to program

‘zlle large displacement problem for incompressible and

60

nearly-incompressible, homogeneous, isotropic, other
formulations which were tried and rejected are discussed
in this section.

Rivlin (1948a) proposed a potential energy equation
using a simplex triangle element with two unknown param—

eters at each node. This equation was

 

U = % i [(A + 2G) (Exx + Eyy + E22)2 +
G(Y:y + Viz + Yiz) _ 4G (Eny22 + EzzExx
EXXEyy)]dv
ivtmere
G = EM A = uEM
771:3T (1+u)(1-2u)

3Fc>xran incompressible material, the equation reduces to

_EM 2 2 2
U - g— £ [(YXY + sz + sz) 4(Enyzz

E E + E E )]dv
22 xx xx yy

The hydrostatic pressure does not enter into the formu-
lation. It can take an arbitrary value. The results
Obtained using this formulation gave unsatisfactory values
‘Vthen.a specimen (of dimensions 5 x 4 x 1 cm) was compressed
‘1rliaxially up to 10 percent; a negative displacement

(DCZCurred perpendicular to the load.

61

Martin and Carey (1973) introduced a modified
strain energy expressed in terms of Green's strain tensor
as

+ 2C

_ 1
U ‘ 2 £ (Cijkl ek1 eij ijkl ekl nij +

Cijkl ”k1 “ij’dv

where eij and nij are as defined in equation (3.21) and
Cijkl is a fourth-order tensor. An approximation to the
displacement field obtained by dropping the last part of
the strain energy and solving for the displacement as
suggested by Martin and Carey (1973), gave unsatisfactory
results for geometric non-linearity. Carey (1974) stated
that dropping the last term in the potential energy
induced an error and must be taken into consideration for
obtaining accurate results.

Herrmann's modified equation was formulated using
the linear displacement triangle and a constant mean
effective pressure function (H). Each node has two dis-
placements. The hydrostatic pressure is evaluated at the
centroid of the triangle. The question arose as to which
was a more desirable approach, a linear displacement and
constant mean pressure model, or linear displacement and
linear mean pressure model. The linear displacement and

{Constant hydrostatic pressure could be described as a

JJDgical and consistent assumption, but the linear

62

displacement and linear hydrostatic pressure model allows
a more logical program development because it greatly
decreases the band width of the final system of equations.
Two programs based on equation (3.23) were devel-
oped and compared for small and large displacements using
Poisson's ratios up to 0.5. The displacements and mean
pressure values obtained using the linear displacement and
constant hydrostatic pressure model were the same as those
obtained using the linear displacement and linear hydro-
static pressure model. This was true for both small
displacements and large displacements. A cylinder
(L = D = 25.4 mm) was compressed uniaxially (25 percent)
using both models to verify the axisymmetric formulation.
Also a specimen (dimensions 5 x 4 x 1 cm) was compressed
up to 20 percent to verify the formulation of the two

dimensional problem.

4.6 Summary

A finite element method was formulated for nearly-
incompressible materials using a simplex triangle element.
This numerical technique is now available for calculating
the displacements and stresses for either small or large
displacements and any shaped body which satisfies the

conditions of two-dimensional plane strain or axisymmetry.

V. COMPUTER IMPLEMENTATION

The solution of geometrically non-linear problems
resulting from large deformation was considered by Argyris
(1965). He used a procedure to account for non—linear
effects when the displacement became large. Incremental
stiffness relations were discussed by Oden (1969) for
quasi-static behavior of a compressible material with no
memory. Oden and Key (1970) considered general incremental
forms of the equations of motion for both compressible and
incompressible finite elements subjected to finite defor-
mations. They suggested that such incremental forms are
particularly useful in problems of static and dynamic
stability and static and quasi-static behavior of elastic
solids. Stricklin gt gt. (1971) concluded that the
solution of the geometrically non-linear problem as an
initial value problem is inferior to either the modified
Newton-Raphson or the modified incremental stiffness
approach.

The two generally accepted techniques for solving
geometric non-linearity finite element problems are

(a) Iterative solution

(b) Incremental application of a load or a displacement.

63

64

5.1 Iterative Procedures

 

The iterative method for large displacement is
relatively simple. The total load is applied and the
calculated displacements are used to revise the coor-
dinates of the nodal points after each iteration (Desai
and Abel, 1972). The new geometry is used to recompute
the stiffness matrix and the nodal loads or displacements.
The solution is obtained for the total load and only one
load vector (displacement vector) may be considered at a
time. The stiffness matrix [K] is updated after each
iteration and the new and old displacements compared until
there is no significant change. The flow diagram for this
procedure is given in Fig. 5-1. The iterational procedure

in symbolic notation is

Step Stiffness Matrix Déigézgzgggts
1 [Ko(0) + KG(0)] U1 - o = U1
2 [KO(U1) + KG(U1)] 02 - Ul = AU
N [KO(UN_1) + KG(UN_1)] UN - UN_l s o

where [K0] is the stiffness matrix related to eij and [KG]

is the geometric stiffness matrix related to nij' It

should be noted that [KG(0)] = 0 since the geometrical
stiffness matrix is proportional to the nodal displace-

ments which are zero at the start of Step 1.

65

IX)

 

 

1 to NE

 

 

 

CalcUlato second order diSplaccment terms

 

 

{

 

 

 

1 NO

Un—U = O I

 

Yes

 

 

Call STRESS

Calculate strains, stresses at the centroid

 

 

 

 

I

 

 

(2:11 I. CUNS'I‘R

Calculate stresses at Nodal Points

 

 

 

 

V

 

Ca 11 DEFO R N
Plotting the deformed shape

)

/0U'I‘PU:/
( END

Fig. 5-1 Flow chart [or Finite Element
computer program for iterative procedures.

 

 

 

 

 

 

 

66

( START )

)

 

 

 

READ
— number of degrees of freedom (NP) - NBN

- number of elements (NH) material preperties

 

V

 

 

 

Initialization
set size of global stiffness matrix SM - SETEL‘

 

 

 

 

D0
1 to N v

 

 

Initialize global stiffness matrix — 2 NP

 

 

IJU
1 to N

 

 

Calculate Element Stiffness Matrix

 

(r

 

 

 

 

Insert element stiffness into global SN

 

 

l

 

 

Call Subroutines

0018 Head Boundary Conditions

iXHHWH) .
SLVBD to find the displacements

 

 

 

 

 

 

67

The iterative method is similar to the Newton-

Raphson method of solving non-linear equations.

5.2 Incremental Procedure

 

The incremental procedure is the preferred approach

if a solution is needed at different load or displacement
values. The load or specified displacement acting on the
deformable body is considered to be applied in increments,
AP or AU. These increments are taken sufficiently small
so that a linear response occurs during each increment.
At the end of each load or displacement increment, a new
updated stiffness relation is calculated and another
increment of load (or displacement) is applied. The cal-
culated displacements must be added to the preceding
results before the new stiffness matrices are calculated
for the next step. The flow diagram for this procedure
is given in Fig. 5-2.

The incremental step procedure, in a symbolic

notation, is given by Przemieniecki (1968):

Step Stiffness Matrix DiggIzgzgzzt
1 [Ko(0) + KG(0)] AUl
2 [KO(U1) + KG(Ul)] AU2
N [KO(UN_1) + KG(UN_1)] AUN

68

 

Calculate second order displacement terms

 

 

 

 

 

Call STRESS

Calculate strains, stresses at the centroid

 

 

 

 

I

 

~ ()zi'l.l ()(1IJ£5]3{i
Calculate stresses at Nodal Points

 

 

 

 

 

 

 

Call DEFORM
Plotting the deformed shape

a 1
/ OU'I‘ PUT]

END

 

 

 

 

 

Fig. 5-2.--Flow chart for Finite Element computer
program for incremental procedures.

69

< {Slfiili'l >

?

 

 

‘ READ
- number of dearees of freedom (NP) - NBN
'- number of elements (NB) material properties

 

 

 

I

 

 

Initialization
set size of aloha] stiffness matrix SH'~ SETEL

 

 

 

 

‘ I)()
1 to N ~

 

Initialize global stiffness matrix - 2 NP

 

 

 

 

U0
1 to N

 

 

Calculate Element Stiffness Matrix

 

 

 

l

 

 

 

insert element stiffness into global SM

 

 

 

V

 

Call Subroutines

DDIS 'Head Boundary Conditions}

xzr ‘n
{SM/1:31)]? to “”d ”‘0 disulaccments

 

 

 

 

 

Addinfi the resultant ASU to the previous ones

 

 

 

 

70

The total displacement

The variables [K0] and [KG] are the same as those

defined in the previous section.

5.3 Solution of the Contact
Problem

 

The solution of the contact problem using finite
elements requires special care in determining which nodes
are in contact with the flat plate. The flat plate loading
was perpendicular to the axis of symmetry and each incre-
ment of loading was determined such that there was a node
at the end of the contact region as shown in Fig. 5-3.

The calculation of the resultant contact force
between the flat plate and the specimen also required
knowledge of which nodes were in contact with the flat
plate. They determined over which elements the stress
had to be integrated to obtain the resultant contact
force. The integration was done by multiplying the element

stress 022 by the surface area of those elements in contact

with the plate.

An extremely fine grid in the region of the flat
plate was not attainable in this study because of the
storage limitations of the computer. The large deformation

analysis requires the storage of several displacement

71

 

 

 

 

 

 

 

*1

Fig. 5-3 Spherical sample in contact with rigid.f1at plate:
Prescribed displacement of the contact nodes

72

vectors. There were also three unknowns at a node instead
of the usual two encountered when the material has a lower
value for Poisson's ratio.

The results obtained using the two procedures
differ for the nodes in contact with the flat plate. As
an example the node on the axis of symmetry in contact with

a flat plate has a stress value of -155 N/cm2

for the
iterative procedure and -l60 N/cm2 for incremental dis-
placement method when considering a cylindrical potato
sample compressed diametrically. The stress (eyy) along
the axis of symmetry were different in a range 10 N/cmz.
The iterative procedure had higher values. The results
at the center differ completely -181 N/cm2 for iterative
and -106 N/cm2 for incremental displacement.

The results given in Chapter VIII are based on
the displacement incremental procedure. The iterative
procedure required more time for high values of Poisson's
ratio because the diagonal value of the hydrostatic
pressure parameter approaches 0 as u approaches 0.5 and
the convergence to AU 5 0 is slower. For an example it
took 16 iterations for the system of equations to become
in equilibrium required 552 seconds on M.S.U. computer
CDC 6500 while 9 displacement increments required 212

seconds for the same amount of the deformation using the

same semi-spherical grid.

VI. VERIFICATION OF THE FINITE

ELEMENT MODEL

A finite element formulation for the solution of
the boundary value problem for incompressible and nearly-
incompressible materials was presented in Chapters III and
IV. Geometrical non-linearity can be formulated in two
different coordinates, Lagrangian coordinates, where the
stresses are calculated over the original area as sug-
gested by Oden (1969) and Oden and Key (1971), or in
Eulerian coordinates where the stresses are calculated
over the final area as suggested by Chen and Durelli
(1973). A question arose as to which approach would give
the most agreeable results when using the finite element
method. This chapter is devoted to a discussion comparing
the results obtained by using both definitions of strain.

6.1 Experimental and Finite
Element Results

 

 

Cylindrical samples with a diameter of 25.4 mm
were cut by driving a corkborer into a white potato tuber
from different positions of the potato tuber, as shown
in Fig. 6-1. The samples were then placed into a hole

of a plexiglass plate of 25.4 mm thickness and the ends

73

74

 

 

Fir. 6-1 Sample preparation

 

Fig. 6-2 Sample cuttinp machine

75

were cut parallel to the plate using a sectioning machine
(Fig. 6-2). The final length of the specimen was measured
to a tenth of a millimeter. Two cylindrical samples were
cut from each potato, one for the determination of a
uniaxial test and the other to determine the failure load
of a sample compressed diametrically as shown in Fig. 6-3.
The elastic modulus (EM) was determined from the
force-deformation curve of a uniaxially compressed sample.

The elastic modulus was calculated by using

where F = compression force
L = length of the sample
A = cross-sectional area
a = deformation

The average value of 22 samples was 306 N/cm2 ranging from
259 to 346 N/cmz.
An alternate method for calculating the elastic

modulus as given by Sherif et al. (1976) is to compress

a cylinder diametrically. The elastic modulus is given by

 

2 2
EM=8(l-U)FZ 6.2
_ nD
where u = Poisson's ratio
D = diameter of the sample
F = compression force

76

 

Fig. 6-3 Cylindrical sample compressed diametrically
be+ween two flat plates

77

and the parameter Z is obtained by solving

a/D = —£§ [1n 2Z + %]‘ 6.3
2Z
where a is the total deformation.

The values of Z for various a/D are given in
Appendix 1. The average value of elastic modulus based
on 22 samples was 310 N/emz.

A finite element grid for one-fourth of a
cylindrical sample with a 12.7 mm radius and a length of
25.4 mm is shown in Fig. 6-4. The strains in the unde-

formed coordinate system (Lagrangian) and deformed coor-

dinate system (Eulerian), are defined as

L 2
E = 3w 1 3w

E SW 1 3w 2

E22 = 55 ' 2 (52> 6-4‘b

The finite element values for a cylinder subjected to a
displacement level of 25 percent are shown in Fig. 6-5.

A value of 0.49 was used for Poisson's ratio and the
elastic modulus was 310 N/cmz. The analytical values
were calculated by evaluating 6.4-a and b for the various
strain levels and multiplying them by the elastic

modulus.

Z-AXIS mm

78

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0
0.5
0.0 1 1
0 0.5 1.0
R-AXIS mm

Fig. 6-4 Finite Element Grid of
one fourth of a cylindrical sample.

2

NORMAL STRESS, Cr"

2’ N/em

‘-

9O

80

7O

60

50

no

30

20

10

 

79

 

 

o-——--—438ulerian Sq. 6.4b
(D ~“~”“(3Lagrangian Sq. 6.43
o ————— o F LC Lagrangian ,u/ = 0.149
-+—————~+F E Eulerian EH = 310 2
N/cm
A———AAverage stress, 0': F/A
_. //3
H / »
JD
1 1 J l l
O 5 10 15 20 £5

PE HG lflN‘l‘ DE F0 HHAT I OH

Fig. 6—5 Value of stresses-using different formulation

for/U>= 0.49 for cylindrical sample.

80

The finite element values and results calculated
using 6.4-a differ after 15 percent deformation. The
normal stress calculated using c = F/A, where F is the force
obtained experimentally from uniaxial test and A is the

original area, agree with the finite element values until

approximately 20 percent deformation. The finite element
values and the values calculated using 6.4-b (Eulerian
coordinates) are higher in value than those formulated in
Lagrangian coordinates.

The equation for 022 given by Rivlin (1948a) for
a completely incompressible material in simple extension
is

022 = g5 (A2 - %) 6.5
where A is the extension ratio and its value A = /l—:_2E:z.
The finite element stress values given by this equation
agree with the finite element results until the deformation
approaches 15 percent, Fig. 6-6. They disagree by approxi-
mately 5 percent for 25 percent deformation. The finite
element stresses calculated using u = 0.49 and u = 0.5
differed in the second decimal place.

The finite element hydrostatic pressure was
linearly proportional to displacement (Fig. 6-7) and
equal to one-third the stress. This agrees with the

theoretical solution 022/3.

90

80

Dis 70
O
t:

. 60
N
N
k)

50
m.
E/l

& to
E4
U)

$3.. 3 0
E
92

20

10

 

81

 

 

:SH 2 1
I
/'
o————o F.E. Lagrangian //

4P’=0J5 2 /

EN = 310 N/Cm /
1 1 I l I _~
5 10 15 20 25

PEBC ENT DISI'OHMAT 1 ON

Fio. 6—6 Comparison between different formulations

’ _)

for complete incompressibility for a
cylinder in simple tension.

HYDROSTATIC PRESSURE N/cm2

30

20

10

 

82

1 l l I J
5 10 15 20 25

PERCENT DEFORMATION

Fig. 6-7 Hydrostatic pressure as a function of

displacement for cylindrical sample.

83

The values of oz calculated by using the finite

2
element method are compared with the stresses calculated
from two experimental force deformation curves obtained at
deformation rates of 25.4 mm/min and 12.7 mm/min, Fig. 6-8.
The lower strain rate agrees more closely with the
theoretical solutions because it more closely satisfies
the quasi-static loading assumption.

Calculations were made to determine the effect of
changes in Poisson's ratio on the normal stress in an
axially loaded cylinder when the elastic modulus was held
constant. The difference in the values between u = 0.49

and u = 0.3 is 1.325 N/cm2

for the maximum deformation of
25 percent. The change of the elastic modulus values,
keeping the Poisson's ratio constant, produces noticeable
changes in the resultant value of the stresses, Fig. 6-9.
This is in agreement with 6.5 as expected. Since the
elastic modulus is affected by temperature (Finney, 1963),

variety, storage (Huff, 1967) and stage of maturity, it

is an important factor when considering failure loads.

6.2 Summary

Experimental loading of a cylindrical sample of a
white potato compressed uniaxially up to 25 percent was
conducted. An evaluation of the effect of the loading
rate, the change in Poisson's ratio and elastic modulus
were studied. Comparison between the theoretical formula-

tion in two coordinates, Lagrangian and Eulerian, for the

2
NORMAL STRESS, 6;z, N/cm

84

 

 

 

90 ~ D----Dihflerian
+ ........ + Lagrangian ID
80 —- Fm—‘Finite Element ,’,0
O—--O 0': F/A Strain rate // /‘
70 __ 25.4 mm/m'in / I! ‘
.—. g: m ,n _/ ,,..+
Stain rate / O/
60 — 12.7 mm/min / / ,"
/ N
50,—
#0 ~
30 —
20 —
10 —
0 l I I l l
0 5 10 15 20 25

PERCENT DEFORMATION

Fig. 6-8 Effect of strain rate on the resultant stress
compared with different formulation for
cylindrical sample.

NORMAL s'msss', 0'22, N/cm2

90—

80——

70—

60-

ao~

30r-

20L-

 

Fig.

85

._....-. 13m = 310 N/cm2
+—-—+F.‘M = 350 N/cmz .4-
o—Osm = 400 N/cm2 ./

 

l l l l l
5 10 15 20 25

PERC ENT DEFORMATION

6-9 The effect of elastic modulus on 6'22 with
constant Poisson's ratio (ILL: 0.1+9)

86

finite element method were performed. The formulation of
the finite element method in terms of Lagrangian coor-

dinates gives more acceptable results.

VII. APPLICATION TO AGRICULTURAL PRODUCTS

A finite element analysis of cylindrical samples
of white potatoes and apples compressed diametrically by
a flat plate was performed. The behavior of spherical
specimens of white potatoes and apples in contact with a
rigid flat plate was also investigated, as was the contact
problem for whole peaches. This chapter contains a dis-
cussion of the numerical results of these analyses and
how they relate to tissue failure reported in the litera-
ture.

7.1 Two-Dimensional Analysis—-
Brazilian Test

 

 

7.1.1 Potato

Twenty four potato samples were loaded dia-
metrically till failure to examine failure under tension.
The diameter and the length of the specimen equaled 25.4
mm. The elastic modulus and Poisson's ratio were 310 N/
cm2 and 0.49, respectively. The average failure load was
366 N, and the total displacement 7.41316 mm (29.185

percent). The failure, a crack, initiated at or near the

87

88

center of the cylinder and then propagating outwards,
as shown in Fig. 7-1 through 7-4.

Nine displacement increments were applied to the
finite element grid shown in Fig. 7-5. The total defor-
mation was 7.41316 mm. Elastic modulus of 310 N/cm2 and
a Poisson's ratio of 0.49 were used in the calculations.
The final volume decreased by 1.3 percent from the initial
volume of 12.84 cm3 and the radius along the X-axis
increased by 21.1 percent. The final deformed shape is
shown in Fig. 7-6.

The stress components along the Y and X axes of
symmetry in the deformed shape are shown in Fig. 7-7.
The isostress lines are shown in Fig. 7—8 through 7-12.
The stresses in the Y-direction and the minimum principal
stress appeared largest under the initial point of con-
tact (-l60 N/cmz) and decreased with increasing distance
from the contact point. The stresses in the x-direction
and the maximum principal stress had a largest negative
value under the initial point of contact (-111 N/cmz)
and maximum positive value near the center (+73.6 N/cmz).
The maximum shear stress, +88.3 N/cm2 occurred at the
same point as the maximum oxx’ The applied stresses
on a small element at the center subjected to compressive
and tensile stress is shown in Fig. 7-13.

The maximum tensile strength for potato flesh, as

reported by Huff (1967), found a mean value of 73 N/cm2

89

 

Fig. 7-1 Crack just initiated at the center of a
potato sample

 

“in. 7—2 Crack hrojavated outwards of a potato sample

90

 

Fig. 7- 3 Crack propagated outwards of a potato
sample

 

Fig. 7—4 Crack propagated outwards of a potato
sample

91

E; Number of elements (NE) = 330

E3 Number of nodes (ND) = 194
Bandwidth (NBW) = 39

C)

9

co

   
  
 

C)
C)

(33.00 4.00 8.00 12 00 15.00 20 so
R-RXIS MM

Fig. 7-5.--Finite element grid of a spherical shape.

92

L)

(\J

L3 e Location of Max shear stress and tensile stress

(0 #- 1.051 cm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C)‘

 

 

O

D
C’0300 0340 piss 1.20 1.60 2.0
X-HXIS CM

Fig. 7-6.--A deformed shape of potato sample compressed
diametrically.

93
+120 +80 +110 0 4&0 -80 ~120 ~160 -200

I I Y I j I I

 

0' 0" .
(max XX yy

 

 

 

STRESSES ALONG Y-AXIS

+80 +60 +40 +20 0 --20 4+0 -60 -80 -100 -120
T I I l l I l I T T

 

max xx yy

 

STRESSES ALONG X-AXIS

Fig. 7-7 Stresses along the axes of symmetry in
’ the deformed shape (potato).

94

-9111

-70
.40
-10
+10
+30
+40
+50

+60

 

 

 

 

 

+70
+73.6

 

 

 

 

Fig. 7-8 Lines of constant stress in X-direction
of a cylindrical potago sample compressed
diametrically in N/cm

-160

-1u0L—————~’//I

-130

 

 

-120
-110

 

-106

 

-100 -50 -20

Fig. 7-9 Lines of constant stress in I-direction
of a cylindrical potato sample compressed
diametrically in N/cm2 .

95

-111
“my

_uo

-10

+10

+40

+50 7“\

+60

+70

+73.6 \ i f

Fig. 7-10 Lines of constant maximum principal
stress of a cylindrical potato 33mple
compressed diametrically in N/cm .

 

  

 

 

 

 

 

' v

 

 

 

-160 '

-1u0_,/’////7

—13o ‘

—120

-110

-106 _
-100 -50 -20

Fig. 7-1] Lines of constant minimum principal
stress of a cylindrical potato sample
compressed diametrically in N/cmd.

96

 

20~.__

60
70

 

80

88?§~\\\\ ]

Fig. 7-12 Lines of constant maximum shear stress
' of a cylindrical potato sample compressed
diametrically in N/cmZ.

 

 

_ 2
ny — -106 N/cm

 

 

‘— '—_"" 6xx = 70 N/cm2

 

 

 

Fig. 7-13 The normal stresses on a small element at
the center of the cylindrical shape
compressed diametrically.

97

with a range of 22 N/cm2 to 184 N/cmz. The calculated

results indicate that the potato sample fails under
tension or under a combination of both tension and maximum
shear stress since both maximum values occurred at the

same point, 0.4 mm from the center.

7.1.2 Apples

Ten cylindrical apple samples were loaded dia-
metrically till failure. The diameter and length of the
samples equaled to 25.4 mm. The elastic modulus and
Poisson's ratio were determined as 350 N/cm2 and 0.3,
respectively. The specimens failed under an average load
of 60 N with a total deformation of 2.413 mm (9.5 percent).

Three displacement increments were applied to the
finite element grid shown in Fig. 7-5. The total
deformation 2.413 mm or 9.5 percent. An elastic modulus
of 350 N/cm2 and a Poisson's ratio of 0.3 were used in
the calculations. The final volume decreased by 1.076
percent from the initial volume 12.84 cm3. This change
in volume compares with a 0.06 percent change when the

2 and Poisson's ratio 0.49.

elastic modulus is 310 N/cm
The final deformed shape is shown in Fig. 7-14.

Stresses along Y and X axes of symmetry in the
deformed shape are shown in Fig. 7-15. These values are
in agreement with the elastic contact theory described

by Timoshenko and Goodier (1970). The isostress lines

are shown in Fig. 7-16 through 7-20. The stresses in

98

l e Location of Max. shear stress

 

LO
C?
00.03

20 1.50 2-00

0-40 0~50

1.
X~HXI§ CM

]?5143. 7-14.--A deformed shape of apple sample compressed
diametrically.

99

3o 20 10 o -10 -20 -30 410 -5o -60
1 I l

 

’Cmax

 

 

 

STRESSES ALONG I-AXIS

29 10 o -10 -20 -30

I I I j
030:

max (7&y

 

’E

 

 

STRESSES ALONG X-AXIS

Fig. 7-15 Stresses along the axes of symmetry in the
deformed shape (apple).

100

 

~55

-10
.+5
+8

 

:‘2‘8 //

 

+9
Fig. 7-16

Lines of constant stress in
X-direction of a cylindrical
apple sample compresssd
diametrically in N/cm

 

 

Figs 7‘17

Lines of constant stress in
Y-direction of a cylindrical
apple :sample compressed
diametrically in N/cm2 .

101

 

-52

~40 /

~20
~10

+5
+8

 

+9

Fig. 7—18 Lines of constant maximum
principal stress of a cylindrical
apple sample compressed diametrically
in N/cm .

 

~60

-5o_,,/////
-uo '
-30
-25 '
~20 ~1 0 I

Fig. 7-19 Lines of constant minimum
principal stress of a cylindrical
apple sample compressed diametrically
in N/cm2.

 

 

102

 

10
15

23.1b
23

20

17

 

 

Fig. 7-20 Lines of constant maximum shear
stress of a cylindrical apple
sample compressed diametrically
in N/cmz.

103

the Y-direction and the minimum principal stress appeared
largest under the initial point of contact (-57.6 N/cmz)
and decreased with increasing the distance from the contact
point. The stresses in the X-direction and the maximum
principal stress had a largest negative value under the
initial point of contact (-55 N/cmz). The maximum shear
stress occurred at a distance of 8.55 mm from the center
(at a distance 0.6 of half the contact width in the
deformed shape) with a maximum value of 23.1 N/cmz.
Miles and Rehkugler (1971) indicate that the
shear stress is the most significant failure parameter
and the average value of the shear stress is 26 N/cm2

of the apple flesh at failure for a uniaxial stress of a

cylindrical specimen. This value agrees with the finite

element results of 23.1 N/cmz.

7.2 Spherical Shapes

7.2.1 Potato

Twenty four semispherical potato specimens, with
a diameter 35.56 mm, were removed from white potatoes.
{These samples were used to determine the failure mode of a
spherical potato flesh in contact with a flat rigid
[plate. The elastic modulus was calculated from a uniaxial
compression of a cylindrical specimen of diameter and
length equaling 25.4 mm and value equal to 310 N/cmz.

Thee average failure load equaled 458 N and the

104

displacement was 6.3764 mm (35.863 percent) for the semi-
sphere. The crack initiated at the center or near center
and propagating outwards are pictured in Fig. 7-21.

Nine displacement increments were applied to the
finite element grid shown in Fig. 7-22. The total

deformation was 6.3764 mm. The elastic modulus of 310 N/

cm2 and a Poisson's ratio of 0.48 were used in the calcula-

tions. The final volume decreased from the initial volume

by 1.921 percent for Poisson's ratio 0.49, and 2.1 and
2.25 percent for Poisson's ratio 0.48 and 0.47, respec-
tively. The deformed shape is illustrated in Fig. 7-23
for u = 0.48. The radius of the semi-sphere increased by
7.49 percent in the R-axis.

The results for Poisson's ratio 0.48 are shown in
Fig. 7-24 through 7-28. The stresses in the Z-direction
and the minimum principal stress have the largest value
at the initial point of contact and decreases with
increasing distance from the contact area. The maximum

2

values of 022 are -237, —221, and -215 N/cm for Poisson's

ratio 0.49, 0.48 and 0.47 respectively. The maximum shear
stress had a maximum value of 63.2, 62.5 and 61.8 N/cm2
.for Poisson's ratio 0.49, 0.48 and 0.47, respectively,
aand occurred near the farthest end of the contact point.
(Wther large values of the maximum shear stress were 61.5,

59.6 and 58.8 N/cmz, for three respective values of

105

 

Fig. '7-2‘3 Crack propagation in a semispherical
potato sample

Z-RXIS CM

1.20
1

0:80
1

106

Number of elements (NE)
Number of nodes (ND)

A

 

 

 

0 00

Bandwidth (NBW) =

= 256
= 153

54

 

0.00 e. e (3' so C20 {.60
R-HXIS CM

Fig. 7-22.--Finite element grid of a spherica

107

 

 

 

 

 

 

 

C)
c3
(‘0
0 Location of Max. shear stress
L3
(.0
Li
.J
F‘—** 1.498 cm
C)
N
Z ' 4
L4"
(0
F—4
X0
CId)
| "-4
ND
C)
V
C)"
C)
c:
’4 I I I r I
Cbps 0.40 0-80 1.20 1.50 2.00

R-HXIS CM

Fig. 7-23.—-A deformed shape of finite element of a
semi-spherical potato sample.

108

~221

-200L::::”””/’,~’~////7
~160
~130
~110
-90

~81

 

 

 

Fig. 7-24 Lines of constant stress in the
Z-direction for a spherical potato
sample, N/cmz.

109

 

 

 

 

30 —-—————_____
“0” a» 30

50
59

59.6
58
57

\175

 

 

55

Fig. 7-25 Lines of constant maximum shear stress
of a spherical potato sample, N/cmz.

110

 

-234
-200

~150

 

~100

 

~82

 

-70 -50

Fig. 7-26 Lines of constant minimum principal
stress of spherical sample of potato.
N/cmz.

111

~175
-140
-100

 

-501

0.
+10

 

 

+20

 

+24

 

 

\

 

 

 

+29

Fig. 7-27 Lines of constant maximum principal
stress of a spherical potato
specimen, N/cmz.

112

.-187

 

 

-10

-30
-6
+10

 

+20

+24

 

 

 

+29

Fig. 7-28 Lines of constant stress in R-direction
of a spherical potato Specimen, N/cmz.

113

Poisson's ratio. These values occur on the axis of sym-
metry. The maximum for u = 0.48 occurred 5.25 mm from the
center of the sphere (0.41 of half the contact width of
the deformed shape). The maximum shear stress at the
center was 55.6, 55.5, and 55.2 N/cm2 for u = 0.49, 0.48,
and 0.47, respectively. The maximum principal stresses
have their largest negative value under the initial point
(of contact with values of -204, -187 and -180 N/cm2 for
11 = 0.49, 0.48 and 0.47, respectively. The maximum
Ipositive values at the center were 38, 29 and 26.5 N/cm2
for u = 0.49, 0.48 and 0.47 respectively. The stresses
Eilong the Z and R axis of symmetry in the deformed shape
Eire shown in Fig. 7-29. The applied stresses on a small

eelement at the center for u = 0.48 of a semispherical

shape is shown in Fig. 7-30.

'7 - 2. 2 Apples

Ten semispherical apple specimens with a diameter
<31? 35.56 mm were removed from golden delicious. These
£3aunples were loaded to failure which occurred at an average
lead of 39 N and a displacement of 1.892 m (10.64 per—
<=enat).

Two displacement increments were applied to the
finite element grid shown in Fig. 7-1. The elastic
mlOdulus of 350 N/cm2 and a Poisson's ratio of 0.3 were
used in calculations. The final volume decreased from

the initial volume by 0.5 percent. The dimensions of

114

+80 +40 0 -'+0 ~80 420 -16O -200 -260
T I A- l 1 I |

(R
”0x
*1
N9
N

max

 

 

 

l 1

S’I‘HLSSSES ALONG Z—DI RECT ION

+60 +uo +20 0 -20 -40 -60 -80 —100
I I I

 

‘
—i

1 I I

 

 

 

 

STRESSE'IS ALONG R-DIHECTION

Fig. 7-29 Stresses along the axes of symmetry of a
spherical sample in deformed shape (potato).

115

6‘
22

-81
Orr = +29

96

0.99 = +29

Fig.-7-30 The applied stresses on a small element
at the center of spherical potato
sample,I/L = 0.48.

116

the deformed shape is shown in Fig. 7-31. The radius of
the semispherical shape increased by 0.72 percent along
the R axes. The isostress lines are shown in Fig. 7-32
through 7-35. The stresses in the Z-direction and the
minimum principal stress had a largest value under the
initial point of contact (-82 N/cmz) and decreased with
increasing distance from the contact area. The maximum
principal stress had its largest absolute magnitude under
the load (-63 N/cmz) and smallest absolute magnitude

(3 N/cmz) at the center. The maximum shear stress was
largest near the contact point farthest from the axis
of symmetry. A value of 34.5 N/cm2 was obtained. The
other largest value occurs on the axis of symmetry and

its value of 28.3 N/cm2 at a distance 12.63 mm from the
center (at a distance 0.41 of half the contact width in
the deformed shape). The stresses along the Y—axis of the
deformed shape is shown in Fig. 7-36. The radial tensile
stress at the circular boundary of the surface of con-

tact is +13.6 N/cmz.

7.2.3 Peaches

The shape of the peach was determined averaging
the dimensions of fifteen peaches. The final shape is
shown in Fig. 7-37. Fifteen whole peaches were subjected
‘to a flat plate load to failure. An average failure load
<>f 73.4 N and total displacement of 9.92 mm (19.1 percent)

was observed .

C"

C3

C3

CW

Fig.

 

7“ rr‘ n N ('5‘\
-J:JJ \J-‘AO (J J'J
"W

117

 
  
  

 

        

I‘ J
D
L)

_ 1.20 1.50
a HXIS CM

7-3l.--A deformed shape of finite element spherical
sample of apple.

-82

118

 

-80 /

-70
~60

-40

-30

~20

 

 

 

Fig. 7-32

-10

Lines of constant stress in the
Z-direction for a spherical apple
sample in N/cmz.

119

 

2%

2

28.3:
28

27
20

20

15

10

 

 

Fig. 7-33 Lines of constant maximum shear
stress of a spgerical apple
sample in N/cm .

120

 

38%

-15

 

 

+3

Fig. 7-30 Lines of constant maximum principal
stress of a spherical apple sample,
N/cmz.

121

 

 

 

 

 

Fig. 7-35 Lines of constant minimum principal
strefis of a spherical apple sample.
N/cm .

122

30 20 10 0 ~10 -20 -30 -40 -50 -60 -70 -80 ~90

 

I I

max

I l I r r I I 1*

La

 

 

 

 

i l

 

-2a

Fig. 7-36 Stresses along the z-axis of a
spherical apple sample.

123

 

 

 

 

I PIT

, //// ' . [13:38”

_._____18 mm___._l

~-——-— 28 mm *

 

 

 

 

Fig. 7- 37 One quarter of a peach.

124

Eight displacement increments were applied on a
finite element grid (Fig. 7-38). The elastic modulus was
taken as 100 N/cm2 (Fridley 32 31., 1968) for the flesh
and assumed 2500 N/cm2 for the pit. The Poisson's ratio
of the flesh is 0.485 and 0.3 for the pit. The final
shape is shown in Fig. 7-39. The isostress lines are
shown in Fig. 7-40 through 7-43. The stresses in the
Z-direction and the minimum principal stress reached a
maximum value (—112 N/cmz) under the initial point of
contact and decreased with increasing distance from the
contact point. The largest value of the principal stress
was -101 N/cm2 under the initial point of contact. The
maximum shear stress had its maximum value of 20 N/cm2
along the axis of symmetry, at a distance 16.23 mm from
the center (at a distance 0.363 of half the contact width
in the deformed shape). The radial tensile stress at the
circular boundary of the surface of contact is +18.7 N/cmz.

Fridley at 31. (1968) reported that the maximum
shear stress at failure, calculated using contact theory,

2 2

to 40 N/cm

ranged from 20 N/cm (T = 0.27 oz for

max 2
u = 0.49) considering the whole peach as one material.
frhe final diameter of contact is between 7.6 mm to 12.7
rum, and the bruises were observed at depths of 1.5 mm to
12.54 mm (about 0.4 times the half contact width).
Iiorsfield 35 31. (1972) reported that the range of values

lior the maximum shear stress during an impact test

20.00

16.00

4.00

 

125

Number of elements (NE) =
Number of nodes (ND) =
Bandwidth (NBW) =

c:

O

C0.00 4.00 3.00 12.00
R—HXIS CM

Fig. 7-38.--Finite element grid for peach.

286
171
48

20.00

126

4 00

3.20

.4

 

e Location of maximum shear stress

9 1.32 cm

 

(j

0 00 0‘

'0

\l

5 1.80 2.40 3.20 4.00
R—QXIS CM

Fig. 7-39.--A deformed shape of finite element of a
peach.

127

 

~112
-60:::::://///r
-00

-30

~20

, //
/

Fig. 7-40 Lines of constant stress in the
Z-direction for a peach, N/cmz.

 

 

 

20
18

17
15
14

10
/ 5

/7

Fig. 7-01 Lines of constant maximgm shear
stress in a peach, N/cm

 

128

~10}
-50’//
~20

~10

 

 

fl“ //”

Fig. 7-42 Lines of maximum principal stresses
for a peach, N/cm .

 

-121
-60 Q
-40

~20

//‘° /7°

Fig. 7-43 Lines of minimum principal stress
for a peach, N/cm .

 

129

depended on the variety. For example, a bruise occurred
in the Klampt variety at a maximum shear stress between
7 to 21 N/cm2 while it occurred between 14 and 24 N/cm2

for the Gaume variety.

7.3 Summary

Cylindrical samples of the white potato and apple
flesh were compressed diametrically to failure. Inspection
showed that the white potato samples split, with the crack
initiated at or near the center. The stress components
calculated using the finite element analysis indicate
this failure may be due to tension stresses or a com-
bination of tension and maximum shear stress. The apple
samples bruised in the vicinity of the contact surface.
The calculated stress components indicate that this
failure is probably a result of the maximum shear stress.

The loading of semispherical samples of white
potato and apple flesh to failure was performed. The
failure crack in the white potato occurred at the center.
The calculated results indicate this crack may be due to
tension stresses or a combination of tension and maximum
shear stresses. Inspection of the resulting bruise shape
which occurred in apples indicated a shape which passes
through the points of maximum shear stress.

The numerical results showed that maximum shear
stress in peaches occurred on the axis of symmetry and

the bruise shape passes through that point indicating

130

the failure is probably a result of the maximum shear

stress .

VIII. SUMMARY AND CONCLUSIONS

A numerical analysis technique, the finite element
method, was used to calculate the stresses in selected
fruits and vegetables. Two-dimensional and axisymmetric
computer programs valid for all admissible values of
Poisson's ratio and for small and large displacements were
developed. These programs were used to calculate the
stress components in fruits and vegetables under quasi-
static loading. The displacement increment method was
used to solve the resulting nonlinear equations. The
following conclusions can be drawn from this study:

1. White potatoes and peaches do not fail until
large displacements have occurred.

2. The formulation of the finite element model in
Lagrangian coordinates gives more acceptable
results than the formulation in Eulerian coor-
dinates.

3. Solutions for elastic nearly-incompressible and
incompressible materials may be used to evaluate
the probable stress-strain behavior related to
bruise or crack problems in some agricultural

products.

131

4.

132

A tensile stress exists at the circular boundary
of the contact region and the maximum shear
stress exists near the end of the contact region
for semispherical apple and potato samples sub-
jected to a flat plate loading. A high value of
the maximum shear stress also exists on the axis
of symmetry.

White potato splits at or near the center. This
may be due to maximum tensile stress or a com-
bination of both tensile and shear stresses.
Bruises in peaches may occur due to the maximum
shear stress which occurs on the axis of symmetry
less than a quarter of the contact width below the

surface.

IX. SUGGESTIONS FOR FUTURE STUDY

The research reported in this dissertation is a
part of Michigan's contribution in the development of
failure criteria related to harvesting and handling of
fruits and vegetables (NE - 93). This dissertation is a
step focusing on the determination of the stress compon-
ent within incompressible and nearly-incompressible
materials when subjected to static loads.

1. This numerical method can be expanded by intro-
ducing the special "crack elements" as described
by Cook (1974) to study further details of the
shape of bruises. This method can be modeled by
disconnecting nodes along element boundaries to
be separated by a crack, to evaluate the critical
stresses at that point.

2. Expanding the finite element programs for incom-
pressible and nearly-incompressible materials
for impact loading and comparing the results with
quasi—static loading. Different conclusions may
be reached to explain some of the modes of failure

of agricultural products.

133

134

More research is needed to study the variation of
the elastic modulus in different locations of the
same fruit or vegetable as reported by Huff (1967)
for the potato. This variation in mechanical
properties may give additional information about
the location and values of maximum tensile and
shear stress components. The magnitude of the
elastic modulus is affected by maturity, storage,
and temperature. These changes will affect the
resultant stresses for static and impact loading.
The maximum shear theory and the maximum stress
theory should be investigated relative to the
application for predicting the failure of peaches,

apples and potatoes.

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1973

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Biot, M. A.
1965

Boussinesq,
1885

Brown, E. T.
1967

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APPENDIX

APPENDIX

Table A-l.--Values of 2 at different values of a/D in the
equation a/D = 1/222[1n 22 + %].

 

z d/D z a/D z a/D

 

.500000 1.000000 1.350000 .409671 2-200000 .204711
.550000 .983983 1.400000 .390209 2.250000 .197933
.600000 .947668 1.450000 .372107 2.300000 .191498
.650000 .902206 1.500000 .355247 2.350000 .185383
.700000 .853543 1.550000 .339521 2.400000 .179567
.750000 .804857 1.600000 .324834 2.450000 .174030
.800000 .757815 1.650000 .311096 2.500000 .168755
.850000 .713237 1.700000 .298231 2.550000 .163724
.900000 .671473 1.750000 .286165 2.600000 .158924
.950000 .632606 1.800000 .274835 2.650000 .154340
1.000000 .596573 1.850000 .264183 2.700000 .149958
1.050000 .563236 1.900000 .254155 2.750000 .145768
1.100000 .532420 1.950000 .244704 2.800000 .141758
1.150000 .503935 2.000000 .235786 2.850000 .137917
1.200000 .477593 2.050000 .227363 2.900000 .134236
1.250000 .453213 2.100000 .219397 2.950000 .130706

1.300000 .430624 2.150000 .211856 3.000000 .127319

144

145

Table A-1.--Continued.

 

Z d/D Z a/D Z a/D

 

3.050000 .124068 4.050000 .079008 5.050000 .055142
3.100000 .120944 4.100000 .077457 5.100000 .054255
3.150000 .117941 4.150000 .075954 5.150000 .053391
3.200000 .115053 4.182550 .075000 5.200000 .052548
3.250000 .112274 4.200000 .074496 5.250000 .051725
3.300000 .109599 4.250000 .073081 5.300000 .050922
3.350000 .107021 4.300000 .071708 5.350000 .050139
3.400000 .104538 4.350000 .070374 5.359040 .050000
3.450000 .102143 4.400000 .069079 5.400000 .049374
3.496320 .100000 4.450000 .067821 5.450000 .048628
3.500000 .099833 4.497160 .066666 5.500000 .047899
3.550000 .097603 4.500000 .066598 5.550000 .047186
3.600000 .095450 4.550000 .065409 5.600000 .046490
3.650000 .093371 4.600000 .064253 5.650000 .045810
3.700000 .091361 4.650000 .063129 5.700000 .045146
3.750000 .089418 4.678580 .062500 5.750000 .044496
3.800000 .087539 4.700000 .062035 5.800000 .043861
3.801070 .087500 4.750000 .060970 5.850000 .043240
3.850000 .085721 4.800000 .059934 5.900000 .042632
3.900000 .083961 4.850000 .058924 5.950000 .042038
3.918260 .083333 4.900000 .057942 6.000000 .041457
3.950000 .082258 4.950000 .056984 6.050000 .040887

4.000000 .080607 5.000000 .056051 6.100000 .040331

146

Table A-l.--Continued.

 

Z a/D Z d/D Z a/D

 

6.150000 .039785 7.200000 .030548 8.300000 .024019
6.200000 .039252 7.250000 .030194 8.350000 .023775
6.250000 .038729 7.300000 .029846 8.400000 .023535
6.300000 .038217 7.350000 .029504 8.450000 .023299
6.350000 .037715 7.400000 .029169 8.500000 .023067
6.371850 .037500 7.450000 .028839 8.550000 .022838
6.400000 .037224 7.500000 .028516 8.600000 .022612
6.450000 .036743 7.550000 .028197 8.650000 .022391
6.500000 .036271 7.600000 .027885 8.700000 .022172
6.550000 .035809 7.650000 .027577 8.750000 .021957
6.600000 .035356 7.700000 .027275 8.800000 .021745
6.650000 .034911 7.750000 .026978 8.850000 .021536
6.700000 .034475 7.800000 .026686 8.900000 .021330
6.750000 .034048 7.850000 .026399 8.950000 .021127
6.800000 .033629 7.900000 .026117 9.000000 .020928
6.835940 .033333 7.950000 .025840 9.050000 ..020731
6.850000 .033218 8.000000 .025567 9.100000 .020537
6.900000 .032815 8.050000 .025298 9.150000 .020346
6.950000 .032419 8.100000 .025034 9.200000 .020158
7.000000 .032031 8.106550 .025000 9.250000 .019972
7.050000 .031650 8.150000 .024774 9.300000 .019789
7.100000 .031275 8.200000 .024518 9.350000 .019608

7.150000 .030908 8.250000 .024267 9.400000 .019431

147

Table A-l.--Continued.

 

 

Z a/D Z a/D Z a/D
9.450000 .019255 10.550000 .015944 11.700000 .013341
9.500000 .019082 10.600000 .015815 11.750000 .013244
9.550000 .018912 10.650000 .015687 11.800000 .013147
9.600000 .018744 10.700000 .015562 11.850000 .013051
9.650000 .018578 10.750000 .015437 11.900000 .012957
9.700000 .018414 10.800000 .015315 11.950000 .012863
9.750000 .018253 10.850000 .015193 12.000000 .012771
9.800000 .018094 10.900000 .015074 12.050000 .012679
9.850000 .017937 10.950000 .014955 12.100000 .012589
9.900000 .017782 11.000000 .014839 12.149830 .012500
9.950000 .017629 11.050000 .014723 12.150000 .012499'

10.000000 .017478 11.100000 .014609 12.200000 .012411
10.050000 .017329 11.150000 .014496 12.250000 .012323
10.100000 .017183 11.200000 .014385 12.300000 .012237
10.150000 .017038 11.250000 .014275 12.350000 .012151
10.200000 .016895 11.300000 .014166 12.400000 .012066
10.250000 .016753 11.350000 .014059 12.450000 .011983
10.281240 .016666 11.400000 .013953 12.500000 .011900
10.300000 .016614 11.450000 .013848 12.550000 .011818
10.350000 .016477 11.500000 .013744 12.600000 .011737
10.400000 .016341 11.550000 .013642 12.650000 .011657
10.450000 .016207 11.600000 .013540 12.700000 .011577
10.500000 .016074 11.650000 .013440 12.750000 .011499

148

Table A-l.--Continued.

 

 

Z o/D Z a/D Z o/D
12.800000 .011421 14.200000 .009537 16.500000 .007339
12.850000 .011344 14.300000 .009421 16.600000 .007262
12.900000 .011268 14.400000 .009308 16.700000 .007186
12.950000 .011193 14.500000 .009196 16.800000 .007111
13.000000 .011118 14.600000 .009087 16.900000 .007038
13.050000 .011044 14.700000 .008980 17.000000 .006966
13.100000 .010971 14.800000 .008874 17.100000 .006894
13.150000 .010899 14.900000 .008771 17.200000 .006824
13.200000 .010828 15.000000 .008669 17.300000 .006755
13.250000 .010757 15.100000 .008569 17.400000 .006687
13.300000 .010687 15.200000 .008471 17.500000 .006620
13.350000 .010617 15.300000 .008374 17.600000 .006555
13.400000 .010549 15.400000 .008280 17.700000 .006490
13.450000 .010481 15.500000 .008187 17.800000 .006426
13.500000 .010413 15.600000 .008095 17.900000 .006363
13.550000 .010347 15.700000 .008006 18.000000 .006301
13.600000 .010281 15.800000 .007917 18.100000 .006240
13.650000 .010215 15.900000 .007830 18.200000 .006180
13.700000 .010151 16.000000 .007745 18.300000 .006121
13.800000 .010023 16.100000 .007661 18.400000 .006063
13.900000 .009898 16.200000 .007579 18.500000 .006005
14.000000 .009776 16.300000 .007498 18.600000 .005949
14.100000 .009655 16.400000 .007418 18.700000 .005893

149

Table A-l.--Continued.

 

Z o/D Z a/D Z a/D

 

18.800000 .005838 21.500000 .004609 25.000000 .003529
18.900000 .005784 22.000000 .004425 25.500000 .003407
19.000000 .005730 22.500000 .004253 26.000000 .003292
19.500000 .005474 23.000000 .004091 26.500000 .003182
20.000000 .005236 23.500000 .003938 27.000000 .003078
20.500000 .005013 24.000000 .003794

21.000000 .004804 24.500000 .003658