EXPERIMENTAL AND NUMERICAL TECHNIQUES
RELATED TO THE STRESS ANALYSIS
OF APPLES UNDER STATIC LOADS

I ASDIueriaflon‘ I
for TEN Degree of DEL D.
MICHIGAN STATE UNIVERSITY
Josse G. De Baerdemaeker
~ 1975 V

2x04255970

' LIBRARY

      

  
   
   

Ill sill/77111 llflfllil/“mllm 2L Michigan 5m '*

This is to certify that the

I". _ thesis entitled

Experimental and Numerical Techniques
Related to the Stress Analysis of
Apples under Static Loads

presented by

Josse G. De Baerdemaeker

has been accepted towards fulfillment
of the requirements for

Ph. D. dpgfiwin Agricultural Engineering

 

 

.- l b 1 75
"Date 9 Fe ruary 9

0-7 639

ABSTRACT

EXPERIMENTAL AND NUMERICAL TECHNIQUES
RELATED TO THE STRESS ANALYSIS
OF APPLES UNDER STATIC LOADS

By

Josse G. De Baerdemaeker

The objective of this work was to develop a technique
for studying the mechanical behavior of apples under
static loading conditions. The apple flesh was assumed
to have linear viscoelastic properties and isotropic
constitutive equations were experimentally determined.
These material properties were used in a numerical model
which described the behavior of a spherical body under

static loads.

An experimental procedure for determination of bulk
and shear relaxation functions and a time dependent
Poisson's ratio were described. The bulk and shear
relaxation functions and Poisson's ratio for Red
Delicious apples exhibit time dependence. Suggestions
were made for the determination of dynamic material
properties and for the inclusion of a ripeness factor

in the constitutive equations.

The finite element method was used to obtain
numerical solutions to the viscoelastic boundary value
problem of a viscoelastic sphere loaded by a rigid flat
plate. The experimental relaxation functions were used.
Experimental and calculated force—deformation curves at
low deformation rates were compared and the differences

were discussed.

An iterative procedure was developed for the study
of creep behavior of a sphere under a constant load. The
stress distribution in the sphere at the initial loading
and at a later time were compared. The location of
maximum compressive stresses was shown to be at the center
of the contact area while maximum shear and tensile
stresses Were found near the circular boundary of the

surface of contact.

This thesis concludes with suggestions for the
development of a failure criterion which should also
include the proven effect of time in storage on the

failure strength.

 

 

Department Chairman

EXPERIMENTAL AND NUMERICAL TECHNIQUES
RELATED TO THE STRESS ANALYSIS
OF APPLES UNDER STATIC LOADS

By

Josse G. De Baerdemaeker

A DISSERTATION

Submitted to
Michigan State University

in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1975

ACKNOWLEDGMENTS

The author sincerely appreciates the guidance and
counsel he received during his graduate work. The close
cooperation with Dr. Larry J. Segerlind, his major
professor, was fruitful and agreeable. Discussions with
Drs; D. H. Dewey (Horticulture), J. B. Gerrish (Agricul-
tural Engineering) and W. N. Sharpe (Metallurgy, Mechanics,
and Material Science) were very inspiring and constructive.

Appreciation is extended to the Agricultural Engineer—
ing Department for a research assistantship, and to the
Commission for Educational Exchange between the United
States of America, Belgium and Luxembourg for the scholar—

ship at the beginning of this graduate work.

ii

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

1.

2.

INTRODUCTION

LITERATURE REVIEW

2.1 Constitutive Equations or Stress—Strain
Relations

2.1.1 Viscoelastic Behavior of Agricultural
Products

2.1.2 Viscoelastic Constitutive Equations

2.1.3 Constitutive Equations of
Agricultural Products

2.1.4 Some Other Experiments
2.2 Analysis of Stress in Fruits Under Loading
2.3 Criteria for Maximum Allowable Load

2.4 Comments and Conclusions

CONSTITUTIVE EQUATIONS OF APPLE FLESH

3.1 Equations of the Linear Theory of Visco—
elasticity

3.1.1 Laplace Transform of Viscoelastic
Equations

3.1.2 Complex Modulus Representation
3.2 Analysis of a Relaxation Experiment

3.2.1 Experimental Determination of the
Relaxation Functions E(t) and X(t)

iii

Page

ix

11

13

15

18

18

18
20

24

28

3.2.2 Numerical Interconversions of the
Relaxation Functions: Approximate
Laplace Transform Inversion

3.2.3 Accuracy of the Method

3.2.4 Summary of the Procedure

3.3 Determination of Dynamic Properties

3.4 Closure

EXPERIMENTAL PROCEDURE
4.1 Apple Selection and Storage
4.2 Sample Preparation

4.3 Relaxation Tests

EXPERIMENTAL RELAXATION FUNCTIONS
5.1 Relaxation Force Curves
5.2 Relaxation Functions

5.3 Average Values of Relaxation Functions

FINITE ELEMENT FORMULATION IN VISCOELASTICITY

6.l Viscoelasticity Boundary Value Problem:
Variational Theorem

6.2 Axisymmetric Viscoelastic Solids

6.3 Discretization of a Region: Finite Element
Formulation

6.3.1 Nodal Displacements

6.3.2 Element Stresses
NUMERICAL ANALYSIS OF MECHANICAL BEHAVIOR OF
APPLES UNDER LOADS . . . .

7.1 Division into Elements

iv

Page

34
38
42
43

45

47
47
47

49

53
53
62

68

75

75

78

81
81

87

88

89

7.2 Special Techniques for the Solution of
Contact Problems

7.3 Material Properties

7.4 Specimen Subjected to a Constant
Deformation Rate

7.5 Behavior of Fruit Under a Constant Load

7.5.1 Creep Behavior of a Cylindrical
Specimen . . . .

7.5.2 Creep Behavior of a Spherical
Specimen . . . .

7.6 Closure
8. SUMMARY AND CONCLUSIONS

‘9. FURTHER DEVELOPMENTS AND SUGGESTIONS FOR FUTURE
RESEARCH

9.1 Change of Material Properties During
Ripening

9.2 The Effect of the Skin
9.3 Dynamic Mechanical Properties

9.4 DeveIOpment of a Failure Criterion
BIBLIOGRAPHY

APPENDIX A

Page

89

94

.100

.108

.108

.110

.124

.125

.127

.127
.129
.130

.130

.134

.142

LIST OF FIGURES

Uniaxial Compression and Constrained
Compression of Cylindrical Samples

Ramp—Step Deformation Function

Force on a Specimen Subjected to a Ramp—Step

Deformation Function
Sample Cutting Machine

Force and Deformation Measuring and
Recording Equipment

Schematic Diagram of Equipment for a
Relaxation Test .

Analysis of Relaxation Test Data to Obtain
Relaxation Functions and Poisson’s Ratio

Uniaxial Relaxation Force; Model Not
Through First Data Point

Constrained Relaxation Force; Model Not
Through First Data Point

Uniaxial Relaxation Force; Model Through
First Data Point

Constrained Relaxation Force; Model Through
First Data Point

Uniaxial Relaxation Function

Constrained Relaxation Function

Bulk Relaxation Function

Shear Relaxation Function . . .

Time Dependent Poisson's Ratio

Average and Standard Deviation of Uniaxial

Relaxation Function for 24 Red Delicious

Apples .
vi

Page

25

31

31

48

48

50

52

54

55

59

60
63
64
65
66

67

70

q «a -q q

.11

.12

.13

.14

coooqovow

.10

Page

Average and Standard Deviation of the
Constrained Relaxation Function of 24
Red Delicious Apples . . . . . . 71

Average and Standard Deviation of the Bulk
Relaxation Function of 24 Red Delicious
Apples . . . . . . . . . 72

Average and Standard Deviation of the Shear
Relaxation Function of 24 Red Delicious
Apples . . . . . . . . . 73

Average and Standard Deviation of the
Poisson's Ratio of 24 Red Delicious Apples 74

Stresses in Axially Symmetric Problems . 79

Triangular Axisymmetric Element and Nodal
Displacements . . . . . . . . 83

Finite Element Grid for a Cylindrical
Specimen . . . . . . . . . 90

Finite Element Grid for a Spherical
Specimen . . . . . . . . . 91

Spherical Sample in Contact with Rigid Flat
Plate: Prescribed Displacement of the

Contact Nodes . . . . . . . . 93
Iterative Procedure for Determining the

Deformation Under a Constant Load . . . 95
Uniaxial Relaxation Function for Sample I . 96

Constrained Relaxation Function for Sample I 97
Bulk Relaxation Function for Sample I . . 98
Shear Relaxation Function for Sample I . 99

Comparison of Analytical and Numerical

Force Versus Time Curves of a Cylindrical
Specimen Under Constant Deformation Rate of
2.54 mm/min . . . . . . . . lOl

Experimental and Calculated Force—Time
Curves of a Cylindrical Specimen, Sample I.
Deformation Rate = 2.54 mm/min . . . 103

.ll

.12

.13

.14

.15

.16

.17

.18

.19

.20
.21

.22

.23

.24

.25

.26

Page

Experimental and Calculated Force—Time
Curves of a Cylindrical Specimen, Sample II.
Deformation Rate = 2.54 mm/min . . . 104

Experimental and Calculated Force-Time
Curves of a Spherical Specimen, Sample I.
Deformation Rate = 2.54 mm/min . . . 106

Experimental and Calculated Force-Time
Curves of a Spherical Specimen, Sample II.
Deformation Rate = 2.54 mm/min . . . 107

Uniaxial Compliance (Sample I) . . . 109
Creep Response of a Cylindrical Specimen of
19.1 mm Length Subjected to a Constant

Compressive Force of 80 N . . . . . lll

Calculated Creep Response of a Sphere with
a 17.8 mm Radius Under Flat Plate Compressive

Load of 50 N . . . . . . . . 112
Lines of Constant Stress in the z—Direction
at Time t=0 . . . . . . . . 113
Lines of Constant Maximum Principal Stress
at Time t=0 . . . . . . . . 114
Lines of Constant Minimum Principal Stress
at Time t=0 . . . . . . . . 115
Lines of Maximum Shear Stress at Time t=0 . 116

Lines of Constant Stress in the z—Direction
at Time t=0.76 . . . . . . . 118

Lines of Constant Maximum Principal Stress
at Time t=0.76 . . . . . . . 119

Lines of Constant Minimum Principal Stress
at Time t=0.76 . . . . . . . 120

Lines of Maximum Shear Stress at Time t—0.76 121
Maximum Compressive and Shear Stress in a
Spherical Sample Under a 50 N Creep Load
(Sample Diameter = 17.8 mm) . . . . 122

Deformed Finite Element Grid of the Spherical
Sample at Time t=0.76 . . . . . . 123

viii

Page
Failure Locus of Parenchyma of Northern

Spy Apples at Two Strain Rates (Calculated
From Data by Miles and Rehkugler, 1971) . 131

ix

LIST OF TABLES

Table Page

3.1 Influence of the measurement errors on the

values of the bulk and shear relaxation

functions . . . . . . . . . 41
5.1 Variance-covariance matrices of parameters

in a two-term exponential series . . . 57
5.2 ' Variance-covariance matrices of parameters

in a two—term exponential series subjected

to the linear restraint C1 = F(t'=0)—C2 . 61
5.3 Average values and standard deviations of the

relaxation functions of 24 Red Delicious

apples . . . . . . . . . . 69

1. INTRODUCTION

Mechanization of fruit harvesting and handling
operations has become widespread in recent years. A
major design concern is the effect mechanical harvesting
and handling has on the quality of the products, parti-
cularly the level of bruise damage that occurs during
these operations.

One phase of the fruit handling process that is
undergoing some major changes is the storage. A lumber
shortage has stirred an interest in the bulk storage of
apples, thereby eliminating the need for a large number of
bulk bins.

Fresh market apples have to be (almost) completely
bruise free, but bruise damage in apples for the process-
ing industry is acceptable as long as most of the bruised
tissue can be removed during the peeling operation.
Larger bruises require bleaching and increase processing
costs.

The increased interest in the bulk storage of apples
for processing has raised the question of the allowable
depth to which apples can be piled before excessive
bruising occurs. The answer to this problem can be sought
along two lines. First, experiments can be conducted to
determine the maximum allowable drop height, the

1

2
relationship between drop height and bruise size, and the
relationship between stack height and bruise size. The
results of these experiments are valuable only for the
practical applications they simulate because they do not
furnish enough basic information for possible use in
other applications. An alternate method is to approach
apple bruising from a mechanics of deformable bodies
point of View, using the laws that govern the static or
dynamic behavior of the material.

The latter approach has become more acceptable in
recent years. Most research in this area, however, has
been limited either to attempts to formulate the consti—
tutive equations of the tissue (Mohsenin and Goehlich,
1962), to solve viscoelastic boundary value problems
(Hamann, 1967), or else to define a failure criterion
(Miles and Rehkugler, 1971). While these are essential
in the study of bruising of fruits, they are, by themselves,
insufficient to completely describe the behavior of fruit
during bruising. The study of bruising is hindered by
two primary items: the difficulty in completely defining
the time dependent material properties and the difficulty
in calculating the stresses in the fruit which result from
a contact type of loading.

The objective of this work was to develop some
experimental and analytical techniques required in study-
ing the behavior of fruit during loading. The specific

objectives were:

3
(1) To establish an experimental technique for
determining the constitutive equations governing

the mechanical behavior of apple tissue.

(2) To use the experimental values of the consti—
tutive equations in a finite element model for the
solution of viscoelastic boundary value problems

and to verify the numerical technique by comparing

experimental and numerical force-deformation curves.

(3) To study the stress distribution in apples
under loading and gain some insight in the formation

of a bruise.

It is important to note that this work was based on
the assumption that the undamaged fruit tissue can be
considered homogeneous and isotropic, a good macroscopic
approximation. Other 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).

2. LITERATURE REVIEW

In a review of a decade of research on mechanical
properties of fruits and vegetables, Mohsenin (1971)
cited some immediate applications, among which were:
characterizations of the material, optimum time to
harvest, quality evaluation, damage in collecting, hand—
ling and storage. Mohsenin (1971) cites ample literature
on the work done toward these applications. Some of this
work relates to the characterization of the material and
the use of the material properties in evaluating stresses
in fruit under static and dynamic loading and the study of
the bruise susceptibility of the product. The pertinent

topics are summarized below.

2.1 Constitutive Equations or Stress-Strain Relations

2.1.1 Viscoelastic Behavior of Agricultural Products

During the early experiments on mechanical behavior
of fruits and vegetables, it was observed that force
deformation relations include time effects (Finney, 1963;
Mohsenin, 1963; Timbers ct a2., 1966). Morrow and
Mohsenin (1966) review some of these early investigations
and conducted a study of their own. Their study led them
to conclude that McIntosh apples can be considered as

4

5
viscoelastic bodies, with a nearly linear behavior.
Chappell and Hamann (1968) also studied the viscoelastic
behavior of apple flesh, but they found the material
properties to be somewhat stress dependent and thus could
not be exactly characterized as linear. Hamann (1967 and
1970) also noted the non—linear properties in apple flesh,
but he made the assumption of linearity to facilitate his
analysis. Clevenger and Hamann (1968) concluded from
uniaxial tensile tests that the apple skin also has a

viscoelastic behavior.

2.1.2 Viscoelastic Constitutive Equations

The theory of Viscoelasticity is adequately described
by several authors such as Flugge (1967), Christensen
(1971), and Ferry (1970). There are several equivalent
forms of the constitutive equations of viscoelastic
materials: hereditary integral forms, differential
operator form, complex modulus form. The integral form
of the constitutive equations can be written as

(Christensen, 1971)1

 

1The standard indicial system for a rectangular Cartesian
reference frame is employed whenever applicable: Repeating
the subscripts i, j, k or 1 implies summation, Kronecker's
delta is denoted by 61-, differentation with respect to
space is indicated by Subscripts preceded by a comma.

 

de-.(T)
, t 1
8-13 =f 91””) J dT (2'1)
1: d e: (T)
_ kk
Okk — [m G2(t-T) dT (2.2)
_ dT

where G1(t) is a relaxation function appropriate to
states of shear and G2(t) is a relaxation function
appropriate to states of dilatation. The deviatoric

stress and strain tensors are

l
— 0,, — 3 dij Okk , Sii — 0 (2.3)

and

l
.. = .. — — = 2.
913 €13 3 5ij Ekk ’ 8ii 0 ( 4)

stress tensor

respectively with Oij

8,, = strain tensor

13

Sij = Kronecker delta, zero for i f j

.. = 6 + 6 + 6 = 3

and 611 11 22 33

Ckk = first invariant of the stress
tensor

Ekk = first invariant of the strain

tensor

7
In order to use the same notation as in elasticity, the

relaxation functions in simple shear and dilation are

taken as
G(t) = G1(t)/2 (2.5)
K(t) = G2(t)/3 (2.6)

These relaxation functions are equivalent to the elastic
shear and bulk moduli, respectively.

An alternate form of the stress—strain relation is
obtained by using creep functions to represent the current
strain as determined by current value and past history of

stress (Christensen, 1971)

t d 813(T)
.. = - x 2.
e13 [m J1(t T, ——7fF———- dT ( 7)
dT (T)
c = ft J (t-T) ___393___ dT (2.8)
kk _m 2 dT

where J1(t) and J2(t) are creep functions for states of
shear and dilatation. They can be related to the
relaxation functions by use of Laplace Transforms or
other interconversion techniques (Knoff and Hopkins, 1972).
Shear and bulk modulus are two independent constants
characterizing a homogeneous elastic solid, and the
relations between these and more commonly used engineering

parameters like Young's modulus and Poisson's ratio have

8
been established (see Sokolnikoff, 1956). Similar
relations exist between the Laplace transforms of the
viscoelastic relaxation functions (Christensen, 1971).
The commonly used relaxation function E(t) which
characterizes a state of uniaxial extension and a visco—
elastic Poisson's ratio v(t) can be defined as the
viscoelastic equivalents to Young's modulus and Poisson's

ratio in elasticity.
2.1.3 Constitutive Equations of Agricultural Products

Simple mechanical models of combinations of elastic
and viscous elements have been used to represent visco-
elastic behavior of fruits. These mechanical models are
described in more detail in Flugge (1967) and Mohsenin
(1970). They all represent a possible approximation for
describing the viscoelastic behavior with an elastic or
a viscous model as the limits. The material is often
considered to be elastic in dynamic experiments (Fridley
ct a£., 1968; Horsfield ct a£., 1972). Poisson's ratio
has been evaluated both as an elastic constant (Finney,
1963; Morrow, 1965; Hughes and Segerlind, 1972) and a
time dependent value (Chappell and Hamann, 1968).

Finney (1963) calculated an elastic Poisson's ratio
for potatoes from a mean bulk modulus and mean uniaxial
compression modulus. Morrow (1965) used an elastic bulk
modulus and Boussinesq solution for a cylindrical plunger

on a half space to simultaneously determine an elastic

9
uniaxial modulus and elastic Poisson's ratio. Hughes and
Segerlind (1972) derived an elastic Poisson's ratio by
comparing the axial force-deformation relation of two
cylindrical samples, one specimem was free to expand in
the radial direction while the other was not.

Bulk compression and uniaxial compression are most
commonly used to determine the relaxation functions,
while torsion and tension tests are very difficult to
apply because of gripping problems (Morrow and Mohsenin,
1966). Morrow (1965) and Sharma and Mohsenin (1970)
used hydrostatic compression for evaluation of the bulk
creep function. Uniaxial compression has been extensively
used for evaluation of the uniaxial relaxation function
E(t) (Finney, 1963; Chappell and Hamann, 1968; Hammerle
ct a£., 1971; Morrow ct a£., 1971).

Direct methods for measuring Viscoelastic Poisson's
ratio have been investigated. Chappell and Hamann (1968)
measured lateral displacement of an axially compressed
specimen using linear variable differential transformers
(LVDT) on each side of the specimen. They observed that
Poisson's ration of apples decreased with increasing time.
Mohsenin (1970) mentions the measuring microscope as well
as a Nikon Shadowgraph for direct measurement of Poisson's
ratio. No time dependency of this parameter is mentioned.
Hammerle and McClure (1970) used a photomicrometer and
found that Poisson's ratio of sweet potato flesh increased

‘with time.

10

A widely used form of the relaxation function is an

exponential series representation known as the generalized

Maxwell Model

E(t) = (2.9)

"MB
t1:l
(D

c...

Several authors have discussed methods to determine
the parameters Ej and dj from experimental relaxation
curves (Gradowczyk and Moavenzadeh, 1969; Mohsenin, 1970;
Hammerle and Mohsenin, 1970; Chen and Fridley, 1972;
Bashford and Whitney, 1973).

Creep functions are sometimes approximated using
elastic elements and viscous elements in parallel, as
represented by the series (Gradowczyk and Moavenzadeh,
1969)

DCt) = ) (2.10)

"MB
U
A
T
(D

Models of this kind were used to approximate the bulk
compressive behavior of apples (Morrow, 1965; Sharma and
Mohsenin, 1970).

Hamann (1969) attempted to experimentally determine
the dynamic axial compression relaxation function and the
shear relaxation function by measuring complex moduli
and the use of an approximate equation relating relaxation
functions and complex moduli. He found that a simple

Maxwell Model (spring and dashpot in series) is a good

ll
approximation of uniaxial dynamic behavior, While the
shear experiments gave no reliable results.
Peterson and Hall (1973) found no consistent inter—
dependency of the complex modulus and temperature when

studying potato flesh.
2.1.4 Some Other Experiments

Theocaris (1964) has shown that the information
obtained from a simple tension test along the whole
response spectrum of the material together with an
initial value of the lateral contraction ratio are
sufficient for the complete description of the visco—
elastic properties of the material. The lateral contraction
ratio functions are monotonically increasing functions.
Rigbi (1967) also recognized this delayed lateral deform-
ation of viscoelastic materials. Gottenberg and
Christensen (1964) demonstrated that the complex shear
modulus function is a convenient property to experimentally
determine and that the shear relaxation function may be
obtained by direct conversion from the frequency to the
time domain.

Parsons at at. (1969) used a specimen under tensile
stress and a ”shear-sandwich” specimen for the measurement
of the complex uniaxial relaxation function and the complex
shear relaxation function.

Laird and Kingsbury (1973) described an experiment

for determining the longitudinal complex modulus function.

12
Their experimental setup consisted of a single degree of
freedom system with the material specimen acting as a
massless stiffness element, whose spring constant is
represented by a complex number.

Hayes and Morland (1968) proposed different constrained
compression tests for determination of response functions
of anisotropic linear viscoelastic materials. Their
derivations were based on the hereditary integral
representation and no actual experiments were described.

Arridge (1974) determined the bulk or dilatational
relaxation function of solid polymers from measuring the
extensions of a tube of the material under internal

pressure.

2.2 Analysis of Stress in Fruits Under Loading

Limited work has been done to analyze stresses in
fruits under loading. Hamann (1970) solved the contact
problem involving one Viscoelastic spherical body falling
onto another for the approach of the bodies, surface
indentations, surface pressures and internal stresses.

He used a simple Maxwell model for the uniaxial relaxation
function and a constant Poisson's ratio when specifying
the constitutive equations. The finite element method

has been used to determine the stresses in apples sub-
jected to a contact type load. Apaclla (1973) assumed

an elastic material. Rumsey and Fridley (1974) used a

13
material with constant bulk modulus and time dependent

shear relaxation function.

2.3 Criteria for Maximum Allowable Load

 

Considerable work has been done on experimental
determination of loads that cause failure or rupture in
fruits or vegetables. Nelson and Mohsenin (1968) have
determined a relation between bruise volume and load.
They report that bruises caused by dynamic loads are
considerably larger than those caused by equivalent
quasi-static loads. In an effort to understand the
criteria associated with bruising, researchers have
attempted to define parameters associated with damage of
agricultural products. Some of those parameters are
energy to bruising, maximum force during impact, maximum
stresses within the product or maximum deformation.
Energy required for bruising was reported greater under
impact conditions than under quasi-static loading condi—
tions for apples (Mohsenin and Géhlich, 1962), apples and
peaches (Fridley ct a£., 1964). However, others have
found lower energy to damage under impact than under
quasi—static conditions for sweet potatoes (Wright and
Splinter, 1968) and pears (Fridley and Adrian, 1966).
Fletcher ct at. (1965) reported that energy force and
deformation to rupture first decrease with increasing

loading rate, then increase. Maximum force during impact

14
was studied by Davis and Rehkugler (1971) for apple-limb
impact and by Hammerle and Mohsenin (1966) and Simpson
and Rehkugler (1972) for cushioned impact of apples.
Wright and Splinter (1968) reported that rupture forces
of cylindrical samples of sweet potatoes during impact
were about one-third of those under slow loading. Fluck
and Ahmed (1972) studied impact bruising of whole fruits
and concluded from their experiments that it was impossible
to say whether energy or force is the more important
parameter in bruising.

The location of the bruise has suggested that
maximum shear stress can be a possible failure parameter
(Fridley and Adrian, 1966). Horsfield at at. (1972)
predicted damage of peaches caused by impact based upon
the theory of elaStic impact, the radii of curvature, the
impact energy, the elastic modulus and the shear strength.
The shear strength of the material was determined from
calculating the shear stresses that exist in an impacting
elastic sphere and the observation of whether or not a
bruise had occurred.

Miles and Rehkugler (1971) attempted to define a
failure criterion for apple flesh using a uniaxial
compression force superimposed with a hydrostatic
compression. The hydrostatic pressure was of the same
order of magnitude as the mean normal stress at failure.
They confirmed that stress at failure is a function of
strain rate. They concluded that shear stress is the most

significant failure parameter.

15

2.4 Comments and Conclusions

 

Several researchers have established the almost
linear viscoelastic behavior of agricultural products.
Attempts have been made to experimentally determine the
relaxation or creep functions describing this visco-
elastic behavior. It was noted that a uniaxial relaxation
function can easily be found from a relaxation experiment.
However, the determination of a second relaxation function
to completely describe the properties of a linear material
has been less successful. Assumptions have been made
that either Poisson's ratio or the bulk modulus are
constant. Other experiments were thereby based on the
formulae of the theory of elastic contact in which the
boundary conditions are only approximately described.
Christensen (1971) states that the ideal experimental
procedure is the one where the analysis, used to relate
the mechanical property of interest to an observable
quantity, must yield the exact solution of the field
equations of the theory and represent the exact boundary
conditions of the specimen. A torsion test of a
cylindrical sample as used for the determination of the
shear relaxation function satisfies the above conditions.
However, the nature of the agricultural product makes the
torsion test nearly impossible. Direct measurement of
Poisson's ratio during a uniaxial compression test is an

alternative solution, but no standard procedure has

l6
evolved. This is also partially due to the nature of
the material.

Stress analysis in fruits under loading has been
limited either by the insufficient knowledge of the
material properties or by the lack of analytical solutions
to various -— sometimes complicated -— boundary conditions
or by both. The recent introduction of the finite
element technique in the agricultural field offers and
excellent alternative to analytical solutions. However,
some of the applications were restricted by the limited
availability of material properties. Assumptions were
made that one or both of the material property functions
were elastic instead of viscoelastic.

The determination of the maximum allowable load on
fruit under different loading conditions has been the
subject of several experimental investigations while
there have been no analytical approaches to it. The
recent work of Miles and Rehkugler (1971) was a first
attempt to establish a failure criterion for apple flesh.
After analysis of their data, it remains unclear whether
the maximum shear stress theory or the distortion energy
theory or any variation of these can be used as a valid
failure criterion.

In summary, many attempts have been made to describe
the mechanical behavior of fruits. The determination of
the constitutive equations as well as the stress analysis

within the fruits subjected to a load have been based on

l7
assumptions that limit their applicability, and a firm
criterion for material failure is missing. No unified
study which determines the material properties, and then
uses these properties to determine the state of stresses

at failure has been undertaken.

3. CONSTITUTIVE EQUATIONS OF APPLE FLESH

The determination of the material properties is a
basic step in modeling the behavior of an apple subjected
any loading. The review of literature indicated there
are some difficulties in the experimental determination
of the stress—strain relations of apple flesh due to
the nature of the product. A new experimental technique
which allows the determination of two relaxation functions
for an isotropic homogeneous material is presented in

this section.

3.1 Equations of the Linear Theory of Viscoelasticity

 

3.1.1 Laplace Transform of Viscoelastic Equations

The convolution integral form of the viscoelastic
constitutive equations was given in Chapter 2, equations
(2.l)—(2.8). The relationship between the different
relaxation and creep functions can be established by
introducing the Laplace transformation. Let f(t) be a

<

continuous function over 0 < t _ m. The Laplace trans—

form of this function is
' m -st
f(s) = L.[f(t)] = f f(t)e dt (3.1)
0

18

19
Application of the transformation to the convolution

integrals (2.1), (2.2), (2.7) and (2.8) yields (Christensen,

1971)
Eij(s) = sG1(s) 513(8) (3.2)
Ekk(s) = sG2(s) Ekk(s) (3 3)
'Eij(s) = s31(s) Eij(s) (3.4)
'Ekk(s) = s32(s) Ekk(s) (3.5)

In the case of simple uniaxial extension

'5 (s) = s E(s) E' (s) (3.6)
11 11

where E(s) is the Laplace transform of the uniaxial
relaxation function E(t). The equation (3.6) is similar

to the elastic uniaxial stress—strain relationship

011 = E €11 (3.7)
The similarity between (3.6) and (3.7) is expressed more
specifically in the correspondence principle (Flfigge,
1967; Christensen, 1971) which states that a solution to
a viscoelastic problem can be obtained by replacing the

elastic moduli in the elastic solution by the s

20

multiplied transform of the corresponding viscoelastic
relaxation function. The resulting functions are the
Laplace transforms of the solution to the viscoelastic
problem.

The solution in the time domain can be obtained
using the inverse transforms of a function. This is
given by

—l l a+ioo

“f(t) = L [I(8)1= . I f(s) estds (3.8)
. 2N1 a_im

 

The inverse Laplace transform of several common
functions can be obtained from a table of Laplace
transforms and a partial fraction expansion of f(s).
The residue theorem (or integration along a contour in
the complex s—plane) can be used when f(s) is expressed
as a quotient of polynomials in which the polynomial in
the denominator is of a higher order than that in the
numerator (Wylie, 1960; p. 711-716). The above methods
have some disadvantages in numerical analysis. An
approximate numerical method to evaluate the inverse

Laplace transform is discussed later in this chapter.

3.1.2 Complex Modulus Representation

It is sometimes desirable to express the constitutive
equations as a function of frequency rather than time.

Conversion from the time domain to the frequency domain

21

and vice versa is accomplished using the Fourier transform

+00 ,
f(w) = f f(t) e‘1

.00

wt dt (3.9)

and the inverse Fourier transform

+®_ iwt
f(w) e dw (3.9a)

1
f(t) 2E— f_m

According to Christensen (1971), the Fourier transform

of the relaxation function for deviatoric stress is

+m t d ei'(T) ,iwt
_ _ J
sij(w) f_m[[mel(t T) “ET:““ dTJG at
(3.10)
If eij(r) = 0 for t<0, and A: (t-T), then
m_d ei'(T) ' m -iwA
_ j —1wt
.. = -—-————- d G A dA
lJ(w) [f0 d T e TJIIO 1( )e 1
(3.11)
and
w d e..(T) A m -iwt _
f 13 e-led = iw f e (T)e d1 = iw ei.(w)
o d T 0 Li J
r
(3.12)

Decomposing G (t) into two parts

G1(t) = 81 + G1(t) (3.13)

22
where G1(t) + 0 as t + m and substituting into the
second integral of (3.11) yields
m iwk 8 m —iwt

[0 61(1) e- =T—l + foél(1)e dA (3.14)
10)

Equation (3.11) can be rewritten as

oo

_ _. 0 A
.. = .. + A ' +
le(w) elJ(w)[G1 waG1( ) Sln wkdx

oo

iw [061(1) cos midi] (3.15)

01‘

_ * __
sij(w) = G1(im) eij(w) (3.16)

*
after some modification where G (iw) is the complex

1
deviatoric modulus with real and imaginary parts given
by

(D

O A
ul + w [ G1(A) sin midi (3.17)
o

G'(w)
1

and

w [OG1(A) cos wde (3-18)

G”(w)
2

Similar expressions can be derived from the Fourier
transform of the dilatational stress-strain relations

yielding

23

_ * , _
okk(w) = G2 (1w) ekk(w) (3.19)

whereby the complex dilatational modulus has as the real

and the imaginary part

1 0 00"
G (m) = G + m f G (i) sin midi (3.20)
2 2 02
and"
G:(w) = m ] G2(i) cos midi (3.21)
0

Equations (3.17), (3.18) and (3.20), (3.21) convert
the relaxations functions from the time domain to the
frequency domain. Since they have the form of a Fourier
sine or cosine transform they can be inverted to yield

(Christensen, 1971)

 

2 °° G'(w)
G (t) = — [ _9___.sin mt dm (3.22)
CI TT 0 (1)
A 2 ”Ga(w)
Ga(t) = g f cos mt dm (3.23)
0 w

and d = 1, 2

Equations (3.22) and (3.23) provide a means of obtaining

the relaxation functions from dynamic experiments.

24

3.2 Analysis of a Relaxation Experiment

 

The experimental procedure discussed in this section
is based on a method developed by Hughes and Segerlind
(1972), who used the procedure to measure the elastic
modulus and Poisson's ratio of biological materials under
the assumption of time independence. The procedure
involves the compression of two cylindrical cores of
material. One sample is compressed axially while free
to deform in the transverse direction. The second sample
is compressed inside a rigid die which prevents lateral
deformation (Figure 3.1).

This procedure can also be used to obtain the time
dependent material properties of apple flesh. The proof
of this follows.

Equations (3.2) and (3.3) are rewritten here in a

simple manner whereby the Laplace transform is indicated

by a bar
E,, = s G _.. (3 24)
13 1 1J
Gkk = 8 G2 Ekk (3.25)
and from (2.3) and (2.4)
s —3 la 3 (326)
ij ij 2 ij kk '

      

.///////////

........................

 

 

 

////////////////

26

6.. .
1j Ekk (3 27)

toll--J

When a simple uniaxial compression exists in the

direction of o , the boundary conditions are
11

E = E = o (3.28)

22 33

Combining (3.24), (3.26), (3.27) and (3.28) yields

3 = s G [§E11 — %(E + e )] (3.29)

3 = s G (E + e + e ) (3.29a)

Elimination of E32 and Esafrom (3.29) and (3.29a) gives

an expression for the uniaxial relaxation function

3 = s E E (3.30)
11 11
whereby
_ 3 G ‘G
E = 1 2 (3.31)
G1 + 2G2

If a sample is compressed in a rigid die with no

transverse deformation possible then

27

e = e = 0 (3.32)

Combining (3.24), (3.25), (3.27) and (3.32), and (3.25)

and (3.32) yields

and

-E + o + o = s G’ E (3.34)

The last two equations provide an expression for a

”constrained" relaxation function, E(s)

E = S H E (3.35)

with

x = (2‘61 + Gz)/3 (3.36)

The functions E(t) and X(t) can be experimentally
determined and the Laplace transform numerically
calculated.

The Laplace transform of the deviatoric and
dilatational relaxation functions are also needed in
this analysis.

Starting with (3.36)

62 = 3 x — 2 ‘G1 (3.37)

28

and substituting G2 in (3.31) yields the quadratic

equation in G1

3 _ _ _
+ E X) + X E = 0 (3.38)

NIH

(3.39)

 

E 3_ __
X) + [(‘2‘+§X)2— 4X E]
2

Numerical calculations have shown that only the positive
sign in front of the square root gives true values for
the relaxation function G1(t). The root with the
negative sign is deleted.

Hughes and Segerlind (1972) derived an equation for
the elastic Poisson's ratio. Using the correspondence
principle, this equation can be rewritten for the visco-

elastic case as

i
2
— 1)2 — 8(= - 1)} ] (3.40)
3.2.1 Experimental Determination of the Relaxation
Functions E(t) and X(t)

The relaxation functions can be expressed as a sum

of exponential terms of the form

n
E(t) = X E. e (3.42)

29

The shape of experimental constrained relaxation functions
indicated that (3.42) also can be used to obtain an
equation for these functions. Moreover, it can be shown
from (3.36) than if the dilatational and deviatoric
relaxation functions have the form of an exponential
series, then the constrained relaxation function is of
the same form.

Consider a state of simple uniaxial compression
with non-zero stress and strain components olft) and
e (t). Take the strain as specified in terms of a unit

11
step function u(t) and the strain amplitude so

6 (t) = e u(t) (3.43)
11 0

then the convolution integral

 

t dIe: u(T)1
o (t) = j E(t-t) 0 dt (3.44)
1' o d T
becomes
0 (t) = E(t) so (3.45)
11

Equation (3.45) suggests that E(t) can be obtained by
measuring the force on a specimen as a function of time
when the specimen is subjected to an instantaneous strain
80 at time t=0. An 'instantaneous' deformation, however,

is physically impossible. Moreover, it would also

30
require special recording equipment for the force signal,
a feature not commonly available on testing machines.
A ramp-step function as shown in Figure 3.2 is more
easily applied. Chen and Fridley (1972) have derived
the equations of the force on a specimen subjected to

this type of deformation

a A n E- - 't
F(t)=— z_1.(1—e0‘1)ror05t2t
Lo i=1 0‘i 1
(3.46)
and
a A n E1 ‘ait -Gi(t‘t1)
F(t) ='TT— Z .__ (l-e 5e for t > t1
01:101.
1 (3.47)

where a is the slope of the ramp deformation function, A
is the cross-sectional area of the specimen and t1 is the
time after which the deformation is held constant.

Figure 3.3 gives an example of a response curve of
a specimen subjected to the deformation in Figure 3.2.

After the substitution t' = t—t equation (3 47) can

1 7

be rewritten as

v n “Git, '
F(t ) = 2 Ci e t > 0 (3.48)
i=1
where
a A E. —d.t
C, = jg“ a1(1 — e 1 1) (3.49)

31

 

DEFORMATION

 

 

n----—-—--

1 TIME

Figure 3.2. Ramp—Step Deformation Function

 

 

FORCE

 

 

.--.----- --- ------

 

H

1 TIME

Figure 3.3. Force on a Specimen Subjected to a Ramp—Step
Deformation Function

32
Similar equations can be written for the force acting
on a laterally constrained specimen subjected to an axial

ramp-step deformation function.

* n * 'aIt'
F (t') = z c].L e 1 t' > 0 (3.50)
i=1
With *
* a A Xi -0t.t1
c1 = -—— —; (1 — e 1 ) (3.51)

O

The determination of the coefficients C1 and Ci* and the
exponents di and a: from the experimental values of the
force for t > t1 leads directly to the determination of
the uniaxial and constrained relaxation function.
Difficulties arise in trying to fit an exponential
series through experimental data. First, the exact number
of terms is an unknown. Furthermore, Lanczos (1956, p.
272-280) has shown in a numerical example that a
perfectly satisfactory representation of the same data
can be obtained by an exponential series with different
number of terms. He attributes this fact to the extra-
ordinary sensitivity of the exponents and coefficients to
small changes of the data. The only remedy to obtain the
”true” model would be an increase of the accuracy of the
data to limits which are beyond the possibilities of
most measuring devices. Hence, the limitation of the

relaxation method for experimental determination of the

relaxation function is that the exponential series obtained

33
is only adequate for the time range of the data used.
Extrapolations of the function beyond that time range
most likely do not accurately describe the physical
phenomenon. Considering the above problem, it is
possible that dynamic experiments or a combination of
dynamic and relaxation experiments may lead to a more
general relaxation function.

A non—linear least squares method is used here to
determine the number of terms as well as the coefficients
and exponents of the exponential series model. The
final selection of a model is based on the following
criteria:

(i) The model results in the smallest estimated

variance of observation errors;
(ii) All the exponents and coefficients have a
positive value.

A non—linear least squares statistics program is
described by Beck and Arnold (1974). The parameters
(i.e. coefficients and exponents) are estimated in an
iterative manner according to

l
(k+1) (k) T<k> no mo (k)
.9 = 12. + (Z. Y .4 > .2. WEI-:1 >

(k)

where: b vector of estimated parameters

(k) T T
sensitivity matrix : g = (V n )

Z
_ —B_

34
W = weighting matrix. In this application the
diagonal terms are set equal to l, the off-

diagonal terms are equal to zero

Y = observation vector. Contains the measured
values of the force.
(k)
E = model vector. Contains the predicted values

of the force using the estimated parameters

after the kth iteration.
k = number of iterations.

This iterative estimation procedure needs initial or
starting values for the parameters. These starting
values are obtained through a curve fitting procedure
based on the method of successive residuals and described
by Chen and Fridley (1972).

The described technique for obtaining the coefficients
and exponents of the most generalized Maxwell model is

contained in a computer program, GENMAX.

3.2.2 Numerical Interconversions of the Relaxation
Functions: Approximate Laplace Transform
Inversion

The relaxation functions for simple uniaxial
compression and constrained compression were discussed

in the previous section. The deviatoric and dilatational

relaxation functions and the viscoelastic Poisson's ratio

35
can now be calculated using the expressions (3.39),
(3.37) and (3.39a). However, these expressions are the
Laplace transforms of the desired functions and an
inverse Laplace transformation has to be performed. A
numerical inversion procedure must be used to obtain the
inverse Laplace transform.

Numerical inversion of the Laplace transform has
been described by Lanczos (1956), Papoulis (1957), Cost
(1964), Bellman ct at. (1966), Cost and Becker (1970) and
others. The procedure as described here is adopted from
Miller and Guy (1966) and uses and orthogonal polynomial
series expansion. The Laplace transform E(s) of a
function f(t) is defined by

_ ”. _st
f(s) =f0f(t) e dt (3.54)

Assume that E(s) is known at discrete points along

the real s-axis. After the substitution
X = 2 e —1 (3.55)

where 6 is a real positive number, a new function is

defined over the interval (-1 s x g l)

g<x> = f(t) = f(—% ln[(l+X)/2]) (3.56)

36

With this substitution, (3.54) becomes

8
_ 1 +1 g “5'1' (3.57)
f(s) = 7;; {_l[(1+2 1 g(X)dx
(0,8)
Jacobi Polynomials Pn (X)form a Set of orthogonal poly—

nomials in the interval [—1 f x E l]. The expansion of

g(x) over the interval [—1, l] in terms of a series of
Jacobi polynomials can then be written as

(0.8)

Cn Pn (x) (3.58)

”M8

g(X) =

n 0

where B > 0. The major task is to evaluate the

coefficients C which can be done by inserting (3.58)

n)

into (3.57) an substituting

S = (B + 1 + k) 5 (3.59)

After integration and algebraic simplification, Miller

and Guy (1966) obtained the expression

 

_ k k(k-l) ... [k-(m—l)]
6f[(B+l+k)0] = Z Cm
m=0 (k+B+l)(k+B+2)...(k+B+l+m)
(3.60)
This result is true for k e=0, 1, 2 ... and by successively
allowing k = 0, l, ... the following system of equations

iS generated

37

 

 

 

6f[(8+1)61 =
8+1
_ C Cl
6f[(8+2)6] = 0 + (3.61)
(8+2) (8+2)(B+3)
0 2c 20
d¥[(s+3)d] = O + 1 + 2

 

 

 

8+3 (8+3)(B+4) (8+3)(B+4)(B+5)

The coefficient CO is obtained by setting k = 0 and from
the knowledge of E(s) at s = (B+1)5. The coefficient C1
is found from setting k = l and from the knowledge of CO
and the value of E(s) at s = (B+2)6.

When N coefficients are calculated, the function
g(x) may be approximated by this finite number N of terms
in (3.58). The approximation of f(t) is

f(t) =

N (0,8) —6t
2 0 pH [26 —1] (3.62)

The Jacobi polynomials in (3.62) are evaluated using the

recurrence formula (Szegé, 1959)

(0,8)
2n(n+8)(2n+B-2)Pn (X) =
(0,8)

(2n+B-l){(2n+8)(2n+B—2)X-82}Pn_1 (X)
(0.8)

- 2(n—l)(n+B—1)(2n+8) P n = 2,3,4
n-2

(0.8)
P (x) = l

 

38

(0,8)
P1 (x) = 5(B+2)x - 58 (3.63)

Equations (3.61)—(3.63) were programmed for a digital
computer solution. The programs were checked using
functions whose inverse is known. These programs were
combined with a computer program RELAX which calculates
the values of the bulk and shear relaxation functions
(equations (3.37), (3.39) and (2.5), (2.6)) and of the
Viscoelastic Poisson's ratio (equation (3.40)) at
different values of time. The input to the program
consists of the coefficients and exponents of the uni—
axial relaxation function E(t) and the constrained

relaxation function X(t).

3.2.3 Accuracy of the Method

The experimental procedure described above was
simulated on a digital computer to evaluate its accuracy.
Hypothetical bulk and shear relaxation functions were

assumed.

800 e ' + 50 e (3.64)

ll

B(t) = G2(t)/3

-.1t —5t
G(t) = Gl(t)/2 = 450 e + 100 e (3.65)

The uniaxial relaxation functions and the constrained

relaxation function were found using equations (3 31)

 

39

and (3.36)
-.11574 t -l.90753 t
E(t) = 1136.444e + 7.604e +
—4.84951 t
213.211e (3.66)
-.1t —.2t —2t
X(t) = 600 e + 800 e + 50 e +
—5t
133.333 e (3.67)

A relaxation force versus time was then calculated for
both uniaxial and constrained relaxation.

The following parameters were used ((3.49) and (3.51))

a = 1

L0 = .75
A = .5
t1 = .06

An error term ER was introduced with zero mean and

standard deviation SD such that

F'(t) = F(t) + ER(0,SD) (3.68)

Values of F'(t) at discrete time points and for different
values of SD were used as input to the parameter estimation
package GENMAX. It was found that the coefficients and
exponents of the three-term model (3.66) could be obtained

only when there was zero input error. The four—term model

40
posed some problems which were assumed to be due to the
almost equal values of the exponents .l and .2 of the
first two terms in (3.67). The presence of errors no
longer allowed determination of the exact values of the
parameters. A two—term model seemed adequate to
describe the force as a function of time for the time
period considered. A summary of the estimated variances
of observations errors is given in Table 3.1.

'The estimated coefficients and exponents were then
used in the program RELAX for numerical evaluation of the
bulk and shear relaxation functions. These calculated
values were compared with the theoretical values and the
variances and maximum values of the errors are also
summarized in Table 3.1 for the different errors of the
force. The absolute value of the errors in the relaxation
functions generally increased as the variance of the error
in the applied force increased. The maximum errors,_
however, were relatively small. For example, the largest
error variance in the force value produced an error for
the bulk relaxation function of about two percent of the
value of this function.

The two—term approximation of the force curve
resulted in accurate values of the bulk and shear
relaxation functions only for the time period in which
data points for the force were available. Calculation of
values of bulk and shear relaxation functions beyond that

period resulted in meaningless values.

41

 

 

 

 

 

 

 

 

 

 

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42

3.2.4 Summary of the Procedure

The following is a summary of the steps involved in
the proposed procedure for determination of relaxation

functions:

i) Measurement of the force relaxation curve of a
uniaxial compressed specimen and of a laterally

constrained specimen under axial compression.

ii) Estimation of the coefficients and exponents of
a generalized Maxwell model (i.e. exponential
series) that adequately describe the experimental

force curves.

iii) Using the correspondence principle, find the
Laplace transforms of the shear and bulk
relaxation function. Transformation to the time
domain is performed through numerical inversion

of the Laplace transform.

It was shown in an example that errors in the force
data as well as the approximative nature of the inverse
Laplace transform method lead to errors of the bulk and
shear relaxation function. However, the results can be
considered satisfactory for relatively small errors of
the force data. The method does not allow extrapolation
of the bulk and shear relaxation functions beyond the time

period for which experimental force values were available.

43

3.3 Determination of Dynamic Properties

 

The complex modulus representation of the constitutive
equations as given by equations (3.16) and (3.19) can serve
as a basis for the derivation of an experimental technique
for determination of the complex deviatoric modulus G:(iw)
and complex dilatational mOdU1US'G:(iw)r The derivation is
similar to the one for the relaxation functions. A complex
uniaxial modulus E*(iw) and a complex restrained modulus
X*(iw) are introduced. Their relation to deviatoric and

dilatational moduli is expressed by the equations

36:(iw) G:(iw)

 

* .
E (1w) = * . 14. (3.70)
G1(1w) + 262(1w)
and
* . * . * .
X (1w) = [2G (1m) + 2G2(1w)]/3 (3.71)
1
from which the following equations are derived
* E* . 3X* . * ,
G (1w) = %[__Ai92.+ ___Alfll] + %{[E (1w) +
1 2 2 2
3X* ' * %
——~él§l]2 — 4X*(im) E (im)} (3.72)
*. *. *.
G2(1w) = 3X (1w) — 2G1(1w) (3.73)

The complex form of Poisson's ratio is

44

 

 

E* (im) {[Fl:(iw) E* (iw) }A
“1 + *112 -8[— -1]
v*(iw)= X*(im) X*(iw) X* (im) (23.74)
4

The complex moduli E*(iw) and X*(iw) can be
experimentally determined from the amplitude ratio and
phase angle between stress and strain of a specimen
subjected to a sinusoidal displacement. Simple axial
sinusoidal displacement is used for E*(im), while
sinusoidal compression of an axially constrained specimen

*
is used for X (im).

 

* O
E (im) = 1' (im) (3.75)
E:11

and

E*(iw) = E'(w) + i E”(w) (3.76)
with

E'(m) = |E*(im)| cos 9

E"(w) = |E*(im)] sin 9 (3.77)

. * . .
The phase angle is 6 and IE (1w)| denotes the magnitude of

the complex value, i.e.

|E*(im)| = IE'(m)2 + E"(m§ 5 (3.78)

Similar equations can be written for the constrained

complex modulus

45

 

* O I
x (im) = 11(im) (3.79)
e 8 = E = 0
11 22 33
and
X*(im) = X'(m) + i X"(m) (3.80)
with
X'(m) = |X*(im)| cos d
and
* (3.81)
‘X"(w) = [X (im)] sin ¢

Some preliminary experiments suggested that difficulties
may arise in the determination of the phase angle e for

the constrained tests due to friction of the specimen
against the sidewall of the sample holder. It was also
found that the phase angle cannot be determined with
sufficient accuracy from a ”Lissajous" figure on the
oscilloscope screen, and that more accurate instrumentation
is required. If these difficulties can be overcome, the
determination of the complex moduli through the use of
(3.22) and (3.23) might result in a more general expression
for the relaxation functions than the relaxation experi-

ment described in the previous section.

3.4 Closure

 

The constitutive equations of viscoelastic materials

were given and a boundary value problem was formulated

46
such that it served as the basis for the experimental
determination of the stress—strain relations. A constrained
modulus function, based on the stress—strain relationship
of a sample subjected to axial deformation but with no
lateral deformation possible was introduced. Deviatoric
and dilatational stress—strain relations were obtained
from the simple uniaxial modulus function and the con—
strained modulus function. A method for determining
either relaxation functions or complex moduli was

proposed and its limitations and advantages were discussed.

4. EXPERIMENTAL PROCEDURE

4.1 Apple Selection and Storage

 

The apples used in the determination of the
material properties were of the variety Red Delicious,
Miller Spur, grown at the Michigan State University
Horticulture Research Center. The apples were selectively
harvested for size (diameter larger than 80 mm) and
immediately put in controlled atmosphere storage at 00C.
The apples were removed from storage 20 hours before the

experiments were conducted and placed at room temperature.

4.2 Sample Preparation

 

Cylindrical samples with a diameter of 19.05 mm were
cut by driving a corkborer into the apple parallel to the
stem calyx axis. The samples were then placed in a hole
in a plexiglass plate of 19.05 mm thickness and the ends
were cut parallel to the plate using a thin sectioning
machine, as shown in Figure 4.1. The final length of the
specimen was measured to a tenth of a millimeter. The
samples were coated with a silicone spray to prevent
excessive moisture loss during the relaxation tests.

Four cylindrical samples were cut from each apple.
Two of these were used for the determination of the

47

 

 

Figure 4.1. Sample Cutting Machine

 

Figure 4.2. Force and Deformation Measuring and
Recording Equipment

 

 

49
uniaxial and constrained relaxation functions. The
other two were used to determine the failure force of a
sample under uniaxial compression and under constrained
compression at a constant strain rate. A semi-spherical
sample with a diameter of 35.56 mm was also removed from
the cheek of the apple using a utensil normally used to
make small meatballs. This sample was used for the
experimental determination of the force—displacement
relation of a spherical apple flesh sample in contact with

a rigid flat plate.

4.3 Relaxation Tests

 

Relaxation tests on the free and constrained samples
were performed using an Instron testing machine with
some added peripheral equipment as shown in Figure 4.2.

A schematic diagram of the equipment is given in Figure
4.3.

The deformation rate, or the slope of the ramp-step
deformation function in Figure 3.2 was 25.4 mm/min. The
deformation was monitored using a linear variable dis—
placement transducer. The crosshead of the testing machine
was stopped whenever the deformation was approximately
1.65 mm, corresponding to a strain of .0864 mm/mm.

The force was measured using the strain gage bridge
circuit of the Instron load cell and a d.c. bridge

amplifier. The amplifier output was fed into a data

pmoe :oflpmxmaom a now pcoemfisvm Mo Emnmwfla OprEonom .m.v onswfim

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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51
acquisition system. The force signal was sampled every
.4 second and punched on paper tape. The relaxation
experiments were conducted for approximately ten
minutes. The paper tape output was read on the CDC
6500 computer of the Michigan State University Computer
Laboratory. After decoding and correcting occasionally
defective recordings, the maximum force value corresponding
to t' = 0 in (3.48) and (3.50) was determined. A force

vector and time vector were obtained in a form ready for

 

use in the program GENMAX for estimation of the coefficients
and exponents in the generalized Maxwell model.

The output of GENMAX was inspected for the nature of
the signs of the coefficients and exponents, their
variance-covariance matrix and the estimated variance of
the input error. A two—term exponential series gave the
best results for most samples.

The coefficients and exponents of the Maxwell model
for the uniaxial and constrained compression tests were
introduced as input to the program RELAX, together with
control parameters for the polynomial expansion. Discrete
numerical values of bulk and shear relaxation functions

and time dependent Poisson's ratio were the final results.

52

Data on Paper Tape

 

l TRANS J

Decode

 

{ CLEAN I

 

Check for erratic recording

I GENMAX I

 

Uniaxial relaxation function

Constrained relaxation function

 

 

{ 1

 

Bulk relaxation
Shear relaxation

Poisson's ratio

 

Figure 4.4. Analysis of Relaxation Test Data to
Obtain Relaxation Functions and Poisson's
Ratio

5. EXPERIMENTAL RELAXATION FUNCTIONS

The theoretical developments and experimental
techniques described in the previous chapters were used
for the determination of relaxation functions of apple
flesh. The results of these experiments are first
given for a single apple; the average values for a

larger sample of apples are shown thereafter.

5.1 Relaxation Force Curves

 

Measured relaxation force values for a uniaxial
and a constrained compression are given in Figures 5.1
and 5.2. The forces are shown from the moment the
displacement became constant, for for t' > 0 in (3.48)
and (3.50). The test specimen had a diameter of 19.05
mm and an initial length of 19.1 mm. The deformation
was held constant at —l.65 mm and —1.70 mm respectively
for the uniaxial compression and the constrained com—
pression. Only every twentieth data point is shown in
the graphs. The force is in newtons, the time in
minutes. The solid lines represent the exponential
series that were used to model the experimental force
curves. Both curves were represented by a two—term

exponential series.

53

 

 

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8
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cm

 

55

 

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oo.m oo.b oo.m oo

— p —

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56
The uniaxial relaxation force model was

—2. 5 t' —. 29 '
F(t) = 13.0 e 96 + 70.24 e O t (5.1)

which resulted in an estimated variance of the observation
errors of 0.12092. The variance—covariance matrix of

the parameters C1 and d- (equations 3.48 and 5.1) is given

1
in Table 5.1. The model for the constrained relaxation
force was

* —3.10 t' _.027 t'
F (t) = 35.24 9 + 83.44 e (5.2)

and the estimated variance of the observation errors was
1.42. The variance—covariance matrix of the parameters
in this model is also given in Table 5.1.

Observation of the model and the experimental values
in Figures 5.1 and 5.2 indicated that the model showed
large deviations from the measured values for small
values of time, especially for time t' = 0. Therefore,
the estimation of the coefficients in (3.48) and (3.50)

was subjected to the linear restraints

n
F(t'=0) = E C. (5.3)

* n
F (t'=0) = 2 c. (5.4)

 

Table 5.1.

57

Variance-covariance matrices of parameters in

a two-term exponential series.

F = z c. e”°‘it
l

(a)

Uniaxial Relaxation Force

 

C C

[Equations (5.1) and (5.2)]

 

 

 

 

1 2 1 2
C1 .00930 —.00012 —.00202 0
C2 .00052 —.00040 0
al. .00129 0
dz 0 0
Constrained Relaxation Force(a)
* * * *
C1 C2 0L1 0‘2
C1 .11450 —.00248 —.00936 .00001
C2 .00907 -.00259 -.00002

 

 

(a)

Zero value when zero in

—.00256 0

the fifth decimal p1ace.

58
Kmenta (1971) comments that imposing such restraints
increases the value of the estimated variance of the
observation errors, but a better model can be obtained
due to the inclusion of some known values.

The linear restraints were introduced such that

n
C1 = F(t'=0) — E C. (5.3a)
. 1
i=2
and *
* * n *
C1 = F (t'=0) — E C. (5.4a)
i=2 1

The results of forcing the models through the observed
force value at time t'=O are shown in Figures 5.3 and
5.4. The resulting equations were

—4.152 t' —.029 t'
F(t) = 16.06 e + 70.46 e (5.5)

for the uniaxial relaxation, and

* ~4.631 t —.028 t
F (t) = 43.91 e + 84.37 e (5.6)
for the constrained relaxation. In this case the
estimated variance of the observation errors were
respectively 0.1665 and 1.847. These values were higher
than in the previous case as was expected. The variance—
covariance matrices of the three parameters in these

models are given in Table 5.2.

59

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60

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61

Table 5.2. Variance-covariance matrix of parameters in
a two—term exponential series subjected to the

linear restraint C1 = F(t'=0) - 02

[Equations (5.5) and (5.6)]

a
Uniaxial Relaxation Force< )

 

C a d

2 l 2
C2 .0006 -.00042 0
61 .00154 0
d2 0

 

Constrained Relaxation Force(a)

 

 

C2 d1 d2

*

C2 .00927 —.00260 — 00002
*

al .00331 .00001
*

d 0

2

 

 

(a) Zero value assumed when aero in the fifth decimal

place.

62

5.2 Relaxation Functions

 

The uniaxial and constrained relaxation functions,
E(t) and X(t), were determined from the relaxation force
curves using equations (3.49) and (3.51), and (5.5) and

(5.6). The uniaxial relaxation function

 

_ —.029 t
E(t) = .744 e 4'152 t + 2.863 e (5.7)

is shown in Figure 5.5. Figure 5.6 represents the con—

strained relaxation function

—4.630 t —.028 t
X(t) = 2.011 e + 3.325 e (5.8)

The dimensions of the relaxation functions E(t) and X(t)

are in megapascals (MPa) or lOeN/mz, while t is time in

 

 

minutes.

Discrete values of the bulk and shear relaxation
functions and Poisson's ratio were calculated using the
numerical procedure described in Section 3.2.2. The
results are displayed in Figures 5.7, 5.8 and 5.9. It
can be seen that while all these material properties are
time dependent, the bulk relaxation function exhibited the

largest changes with time.

 

63

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5.3 Average Values of Relaxation Functions

 

The uniaxial and constrained relaxation functions and
discrete values of the bulk and shear relaxation functions
and Poisson's ratio for 24 apples are given in Appendix A.
They are summarized in Table 5.3. The tests on these
apples were run within less than three weeks from the
time of harvest. In Figures 5.10 through 5.14, the
average values and standard deviations of the respective
relaxation functions are given for different time values.
It can be noted that the bulk relaxation function
decreased very rapidly during the initial time periods.
The shear relaxation functions and Poisson's ratio also
decreased with time, but their decrease was less rapid
than for the bulk relaxation function. These properties
also varied among apples. The standard deviations of
the values of the relaxation functions at a single time
were as large as ten percent of the average value. They

were generally larger at the small time values.

 

 

 

 

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6. FINITE ELEMENT FORMULATION IN VISCOELASTICITY

A method for determining the viscoelastic constitutive
relations of biological material was presented in the
previous section. These constitutive equations are
essential for the analysis of the behavior of these
materials under different loading conditions.

A finite element method for solving these visco—
elastic boundary value problems is now presented. Its
application will be illustrated in a following section.

Finite element methods have already been used by
several authors in solving viscoelastic boundary value
problems (for example, Taylor at a£., 1968; Herrmann
and Peterson, 1968; Heer and Chen, 1969; Malone and
Connor, 1971; Carpenter, 1972). The following derivations
are similar to those by Taylor at at. (1968) and Heer

and Chen (1969).

6.1 Viscoelastic Boundary Value Problem:
Variational Theorem

 

 

A viscoelastic quasi-static boundary value problem is
governed by the following relations which have to be
satisfied, and which are similar to the elastic boundary

value relations (Christensen, 1971):

75

76

(i) Equilibrium equations
0.. . + F. = 0 i,j = l, 2, 3 (6.1)
where a comma denotes a differentiation and F, is a body
1

force vector.

(ii) Constitutive equations

t 3€k1(T)
.0ij — [OGijk1(t-i) __§¥___ dt (6.2)
where
G - l [G t G t 15 0
ijkl ‘ 3 2( ) ‘ 1( ) ij kl

(6.3)

1:
+ 2G1(t) [51k 6ij+5i1 6jk]

Using convolution notation (6.2) can be written as

,, = *
01J Gijkl Ekl (6.4)

(iii) Strain displacement relations

... —1— .
61.1 “”15 + “3.1) (6 5)

(iv) Prescribed boundary values

Oij nj = Si on B0 (6.6)

77

and

B0 is the part of the boundary on which the tractions Si
are prescribed and Bu is that part on which the dis—
placements Ai are prescribed. nj denotes the components

of the unit normal vector to the boundary.

Let I be a functional defined as (Christensen, 1971)

= 1 * .. * _ _ * _
I IVEZGijkl €13 Ekl F1 ul] dV
- [E(Si * ui) da (6.7)
0

where it is assumed that the displacement boundary
conditions are satisfied. V is the total volume of the
solid. It can be shown that the first variation 61 of 1
due to a displacement variation vanishes when the
equilibrium equations and the boundary conditions are
satisfied. In other words, the solution of the boundary
value problem stated in (6.1), (6.2), (6.5) and (6.6)

can be obtained by finding the stationary value of the
functional I in (6.7). The finite element method can

be used as a numerical technique based on the minimization

of this functional.

78

6.2 Axisymmetric Viscoelastic Solids

In the case of axial symmetry (no displacement or
displacement gradients in the 9 direction), Figure 6.1,

the strain displacement relations are (Sokolnikoff, 1956)

Bur
E:I‘I‘ €11 _ 3r
auz
EZZ _ 822 = DZ
5 e = BE
80 n r
= E = 1—(33111: t .932. )
Erz 12 2 DZ Sr (6.8)

Combining (6.2) and (6.3) the constitutive equations are

t dek1
011 = Orr = IOGllkl(t-T) _CET— dT

ti .2.
IOE3(Gz-Gl> dT (err + 222 +€ee> +

de

G ——££] dr
1 dT
or

1 l
= — — + 3G * + - G —G * +
Orr 3[(G2 G1) 1] Err 3( 2 1) (EZZ 600)

(6.9)

 

79

 

 

 

Figure 6.1. Stresses in Axially Symmetric Problems

80

Equation (6.9) can be rewritten as

0 = A * e + A * e + A * e (6.10)

Similar expressions can be derived for the other stress

components Ozz’ At this moment, it is

0 , 0 .
06 rz
convenient to simplify the notation by setting (Figure 6.1)

1 rr 2 ZZ 3 99 u rZ

(6.11)

The stress—strain relations can then be expressed as

0 = A * e K,M = 1,2,3,4 (6.12)
K KM M
where
AKM = AMK
and
l
A = A = A = “(G +2G ) = B+_G
11 22 33 3 2 1 3
g
A = A = A = (G —G ) = B — G
12 13 23 1 3
A = AG = G
w 1
A = A = A = O (6.13)

101 '42 '43

 

 

81

The axisymmetric expression for the functional (6.7)

becomes

I = A A * * — F * dV
IVEZ KM E:M 8K d ud]

— [B (sa * ua)da (6 l4)
0

where K,M = l,2,3,4

and d = 1,2
6.3 Discretization of a Region:
Finite Element Formulation
6.3.1 Nodal Displacements

The volume and surface integrals in (6.14) can be

expressed as a sum of integrals over a set of subregions

I g 1] ( e * A * e)dv
= 5 8 E:
e=1 Ve K KM M
p e e p
— 2 f e(u * F )dV — z I e(ue * Se)da
e=1 V a “ e=1 B a a

O (6.15)

where the superscripts e indicate the subregions.
The displacements in each subregion or element are
approximated by algebraic polynomials relating them to

displacements of nodal points of that element (Zienkiewicz,

1971)

 

 

82

or {ue} = [Ne]{U} (6.16)

[Nejis the matrix of shape functions relating displacements
in the element Tue} to the nodal displacements {U}.

An example of an axisymmetric element is given in
Figure 6.2, and the nodal displacements are indicated.

The strain vector is obtained by appropriate space

differentiation of (6.16)

{ce} = [Be]{U} (6.17)

Substitution of (6.16) and (6.17) into (6.15) and the

use of matrix notation instead of indicial notation

yields
p T T
I = 2 AI ({U} [B] * [A] * [Bl {U})dV

e=1 V
p T T

— 2 f ({U} [N] * {F})dV
e=1 V

p T T

— z f ({U} [N] * {S})da (6.18)

e=1 BO

T
Where [B] is the transpose of matrix [B] and the element

superscripts are deleted for clarity. The order of

integration can be changed. The integration over space is

performed first, then the convolution. Hence,

 

83

 

 

 

Figure 6.2. Triangular Axisymmetric Element and Nodal
Displacements

 

 

84
P T T
1 = z AIU} * [I [B] IA][B]dV] * {U}
:1 V

p T
— 2 {U} * [IENJT{F}dv]

e 1 V
P T T

— 2 {U} * [f[N] {Slda] (6.19)
e=1 Bo

This functional can be written in a simpler form as
T T
I = A{U} * [K] * {U} — {U} * {R} (6.20)

The stiffness matrix is

.|

GT 0
f eIB ] [AJIB ]dV (6.21)

P
[K] = ZIke] = 2
=1 V

e

The force vector is

p T
{R} = z{re) = 2 (f e[Ne] Fe dV +
e=1 V

I eINeJTISelda) (6.22)
B

It should be noted that the displacement vector {U} now
contains all the nodal displacements of a region and {R}
is the vector of all the nodal forces.

Taking the first variation of (6.20) and setting it

equal to zero yields

 

85

T T
51 = 6{U} * [K] * {U} — 6{U} * {R} = 0

or

[K] * {U} - {R} = 0 (6.23)

Equation (6.23) resembles the finite element equations of
elasticity where the displacements are also the unknowns.
However, the explicit form of (6.23) contains a time

integral

t

f [K(t—T)] d{U(T)} ={R(t)} (6.24)
T=O

These integral equations can be solved numerically by the
use of time increments (Gupta, 1974). Rewriting the
integral in (6.24) as a summation over time steps results

in

n
z [K(tn-tm)]{AU(tm)} = {E(tn)} (6.25)
m=l
where {AU(tm)} is a vector of displacement increments from
time tmto tm+1 ,approximated as a displacement at the
beginning of the time increment.
The last displacement increment can now be found from
the previous displacement increments and a possible
initial step displacement {U0} at time t=0. This is done

by reordering the terms in the summation in (6.25)

 

86

[K02 _t )]{AU(t )} =00: N—
n n n n

n—l

z [K(tn-tm)]{AU(tm)} — [K(tn)]{UO} (6.26)
m=l

Hence the equation for the first few time steps are

[K(o)]{U } = {R(t-)}

o O
[K(o)]{AU(t1)} = {R(t1)} — [K(t1)]{UO}
[K(o)]{AU(t3)} = {R(t2)} - [K(t1)]{AU(t1)}

- [K(t2)]{UO} (6.27)

A disadvantage of this method is that the number of
terms on the right—hand side in (6.27) increases with an
increasing number of time steps. If the stiffness matrix
for each time interval is stored, an enormous amount of
computer storage space is required. An alternate method
is to rebuild these stiffness matrices each time they are
used. This procedure, however, rapidly increases the
required computer time, and, therefore, the cost. The
latter method was used in the computer programs written
for this study. Another approach is the use of logarithmic
time increments which can be organized such that only one
stiffness matrix has to be stored at each moment (Gupta,

1974). More simplifications could be made in the case of

87
exponential series representation of relaxation functions

(Taylor ct a£., 1968; Heer and Chen, 1969).

6.3.2 Element Stresses

The stress in each element can be derived from the

nodal displacements by substituting (6.17) into (6.12)

“(69) [A] * [8911081
01‘
e

{0 }

[AJIBel * {Ue} (6.25)

Writing (6.25) as a convolution integral, and dropping the

superscript notation gives
t
{G(t)} = f [A(t—T)][B] d {U(T)} (6 26)
0

Replacing the integral by a summation over discrete time
steps yields the stress as a function of the displacement

increments
{o(tn)} = [A(tn)l[Bl{UO} +

n
m:1[A(tn—tm)l[B]{AU(tm)} (6.27)

Note that the calculations performed in (6.27) have to be

repeated for each element.

7. NUMERICAL ANALYSIS OF MECHANICAL BEHAVIOR
OF APPLES UNDER LOADS

A finite element method for the solution of visco-
elastic boundary value problems was presented in the
previous section. Existing finite element programs for
two-dimensional elasticity were modified to accommodate
the Viscoelastic problems related to the loading of
apples. The modification included a change to axially
symmetric triangular elements, allocation of computer
storage for the displacement increments calculated
during each time step and the recalculation of the force
vector on the right-hand side of the equation (6.27)
after each time step. The finite element method was
used with the previously discussed constitutive equations
to analyze the behavior of apples under different loading
conditions.

The behavior of a spherical specimen in contact with
a rigid flat plate was studied. It was used as a model
for the behavior of an apple in contact with a rigid wall
or floor.

An analysis of the compression of a simple cylindrical
sample was performed in order to evaluate the accuracy of
the procedure. Comparisons were made between the numerical
results and analytical solutions for simple problems.

88

89

7.1 Division into Elements

 

The finite element grids of a cylindrical and a
spherical sample are shown in Figures 7.1 and 7.2. The
cylindrical sample had a 9.025 mm radius and a length of
19.1 mm. The radius of the spherical sample was 17.8 mm.
Each triangle represents a ring-shaped element as was
shown in Figure 6.2. The region in Figure 7.2 represents
a half sphere, in which the other half was omitted because
of symmetry with respect to the r-G plane. The element
size of the spherical sample was varied such that a very
fine grid was obtained in the vicinity of the applied
contact loads. This fine grid was necessary to obtain
accurate results.

The division of the regions into elements was done
with an automatic grid generating program which also
labeled the nodes for minimum computer memory requirements
relative to the storage of the stiffness matrix [K] in
(6.23). There were 42 elements with 32 nodes in the
cylindrical sample and 370 elements with 215 nodes in the

spherical sample.

7.2 Special Techniques for the Solution
of Contact Problems

 

The solution of the viscoelastic contact problems
using finite elements required Special care in the

determination at each time step of those nodes that make

90

20.00

 

6.00

 

 

 

 

 

 

 

 

 

 

 

C3

C3

c0.00 4100 afoo
R-HXIS MM

Figure 7.1. Finite Element Grid for a Cylindrical
Specimen

 

 

91

MM

HXIS

 

D
9
c0.00

4.00 8.00 12.00 16.00 20.00
R—HXIS MM

Figure 7.2. Finite Element Grid for a Spherical Specimen

92

contact with the flat plate after each time step. The
force between the sphere and the plate was assumed in the
z-direction. A special subroutine was written to monitor
the displacement of boundary nodes and, if necessary, to
impose nodal displacements to make them compatible with
the flat plate displacement. Nodes in contact with a flat
plate are illustrated in Figure 7.3.

The calculation of the resultant contact force between
the flat plate and the specimen also required the knowledge
of the nodes in contact with the flat plate. They
determined over which elements the stress had to be
integrated to obtain the resultant contact force. Integration
was done by multiplying the element stress in the z-
direction with the projection in the r—e plane of the
outside surface area of these boundary elements.

The formulation of the finite element method in terms
of nodal displacements assumes that nodal forces and/or
some nodal displacements can be specified, after which all
the other nodal displacements can be calculated. This
formulation creates some special problems when analyzing
the behavior of a spherical sample under a constant force
(creep) loading. The actual distribution of the force
over the contact area depends on the size of the contact
area, but this in turn depends on the axial displacement.
Hence, no nodal contact forces can be accurately specified
since the axial displacement is the unknown of the problem.

The formulation of a constant force loading was solved by

93

 

 

 

Figure 7.3. Spherical Sample in Contact with Rigid
Flat Plate: Prescribed Displacement of
the Contact Nodes

94
assuming an axial displacement, and calculating the
contact force for this displacement. If this contact
force was not equal to the specified creep load, the
specified displacements were corrected and the contact
force was recalculated. This iterative procedure is
illustrated in Figure 7.4. FC is the constant creep load,

Ul is the initially assumed displacement. The displace—

ment U1 requires a force F which is larger than the creep

1
force. Multiplying U1 by FC/F1 results in a displacement
U2 which requires a force F2. This iteration is continued
until a displacement is obtained which requires a force

very close to the constant creep load.

7.3 Material Properties

 

Most of the numerical analyses in this chapter were
done for apple material having the following experimentally
determined relaxation functions in the zero to two minute
time region

—15.897t _,
.888 e + 2.731 e 106t (7,1)

E(t)

—l9.303t —.119t
1.386 e + 3.549 e (7.2)

X(t)

These relaxation functions are shown in Figures 7.5 and
7.6. The values of bulk and shear relaxation functions,
which were used in the finite element analysis, are given

in Figures 7.7 and 7.8 respectively. In some examples

 

 

 

95

     

 

 

FIL—
/
F3 7' / '
/
.___.L_.-—— ...
F 4;
I
I
(:3. F2 I ',
I I ,
: I I
, l I
It
: ..
I A '
I I i
I ' I
I I I
l l I
I I '
I I I
1 11
U2 U3 [H-
DEFORMATION
Figure 7.4. Iterative Procedure for Determining the

Deformation under a Constant Load

96

4.80

4.00

[MP9]

3420

l

2.40

J

UNIHXng RELRXRTION

0.80

 

.00

 

C0.00 0.40 3.80 17.20 1T.60
TIME IN MINUTES

Figure 7.5. Uniaxial Relaxation Function for
Sample I

1
2.00

4-00
J.

3.00
i

2.00
A

1.00

J

CONSTRRINED RELRXHTIUN FUNCTION

 

97

 

 

c:
9
Ch .00 0.40 0.80 11.20 11.50
TIME IN MINUTES
Figure 7.6. Constrained Relaxation Function for

Sample I

1
2.00

98

4.80

4.00

[MP9]
3.20

l l

l

2.40

+
+++
+

l

BULK RELHXRTIDN
1.80

0.80

 

 

0.00

0.00 0140 0380 1I20 1160 2300
TIME IN MINUTES”

Figure 7.7.‘ Bulk Relaxation Function for Sample I

 

 

99

2.40

I;
r

1.20

1

0.80

SHERR RELRXRTIUN

0.40

l

 

 

0340 0380 1320 1360 2300
TIME IN MINUTES

£000
8
C)

Figure 7.8. Shear Relaxation Function for Sample I

100
specimens were used whose material properties are ex-

pressed by the relaxation functions

-l7.561t3 —.O96t
1.018 e + 3.092 e (7.3)

E(t)

and

—22.127t _ 129
1.964 e + 3.886 e ' (7.4)

X(t)

The latter sample is referred to as Sample II, while the

former will be designated as Sample I.

7.4 Specimen Subjected to a Constant Deformation Rate

 

The finite element models of both the cylindrical
and the spherical specimen were subjected to a deformation
rate of -2.54 mm/min in the z-direction (axial compression).
The required compressive force on the cylindrical sample
to obtain this deformation rate was analytically found
from (3.46) and (7.1)

-15.897t _,106t
F(t) = 978.61 — 2.12 6 -976.61 e
(7.3)

where the force is in newtons.

This force versus time function was plotted in Figure 7.9
for comparison with the force values obtained from the
finite element analysis. Excellent agreement between the
analytical and the numerical technique was observed,

except at time t=0 where a non—zero force value was

 

 

FORCE (N)

 

101

 

Analytical

0 0 Finite Elements

 

Figure 7.9.

l l I l
.2 .3 .4 .5 .6

TIME (Minutes)

Comparison of Analytical and Numerical
Force Versus Time Curves of a Cylindrical
Specimen Under Constant Deformation Rate
of 2.54 mm/min

102
obtained, an error inherent in the summation over time
in equation (6.27).

The calculated force versus time curve was also
compared with experimentally obtained force-time curves
of cylindrical samples subjected to a deformation rate
of 2.54 mm/min. Figure 7.10 shows two experimental
curves and a calculated curve. The variation in response
among samples of the same apple can readily be observed.
The Calculated value does not exactly coincide with either
of the two experimental curves. In most cases the
calculated values are slightly lower than the measured
ones. This can also be seen in the results for a
different apple (Sample 11) represented in Figure 7.11.

Comparison of experimental and calculated force—time
curves for several samples taken from different apples
indicated that the slope of the calculated curves was in
general smaller than the slope of the measured values.
The slope of these force—time curves at this deformation
rate depends on the value of the relaxation functions for
very small values of time. It was already discussed in
Section 3.2.1 that relaxation experiments do not always
result in very good values of the relaxation functions
for small time values, and that dynamic experiments would
be necessary. However, the relaxation functions
determined from the relaxation experiments would be

sufficient for smaller deformation rates.

FORCE (N)

70

60

50

40

30

20

10

 

103

 

 

0
- I
I
I
l
I
/
I
I
d
_ I
I
l
I
l
I
I
I
,d
I
-I I
I
I
I
I
I
I
0'
’I
d l
I
/
I
/
I
I
- ,/ o----o Finite Elements
/
I
I
I,
/ Experimental
I
I T‘ l l l I I
0 l .2 3 4 .5 6 .7

Figure 7.10.

TIME (Minutes)

Experimental and Calculated ForcevTime
Curves of a Cylindrical Specimen,
Sample I. Deformation Rate = 2.54 mm/min

FORCE (N)

70

60

50

4O

30

20

10

104

 

 

/
/
fif
I
/
I
I
I
I
;{
I
I
I
I’
I
;{
/
//
II
[I
V
I
/
/
I
I
I
I
I,“
I
I
I
’I
/
I
/’
p’ °----0 Finite Elements
I
I

x, -— Experimental
l I l l I I
1 2 3 .4 5 6

TIME (Minutes)

Figure 7.11. Experimental and Calculated Force-Time

Curves of a Cylindrical Specimen, Sample II.
Deformation Rate = 2.54 mm/min

 

105

The compressive force on a spherical sample subjected
to a constant axial deformation rate of 2.54 mm/min is
illustrated in Figure 7.12 and Figure 7.13. The
calculated values in both cases are higher than the
measured values. This difference was believed to be
caused by differences in the radius of curvature between
the experimental sample and the model in Figure 7.2.

The calculated force—deformation relation for a 15.1 mm
radius sphere given in Figure 7.12 illustrates this
effect of the radius of curvature. Slight irregularities
in the specimen boundary at the contact point were very
difficult to eliminate during the sample preparation.
Unfortunately, it was seldom possible to obtain more than
one spherical sample from one apple after the cylindrical
samples were removed.

The comparison of the calculated and experimental
force deformation or force—time curves in this section has
shown that the numerical technique can be used to
simulate the mechanical behavior of the fruit. It is also
clear that the experimental relaxation functions have to
be applied with caution and do not always give close
agreement between calculated and measured force values
for rapidly changing loading conditions. This indicates
the need for dynamic experiments if dynamic loadings are

to be studied.

 

 

50
J

FORCE (N)

 

‘00...

 

106

(r-uo Finite Elements (17.78 mm radius)

m“ Finite Elements (15.1 mm radius) 9'
I

Experimental ‘
,6

 

Figure 7.12.

 

TIME (Minutes)

Experimental and Calculated Force-Time
Curves of a Spherical Specimen, Sample I.
Deformation Rate = 2.54 mm/min

FORCE (N)

o--- -o

40

 

 

107

Finite Elements

Experimental I

 

Figure 7.13.

 

TIME (Minutes)

Experimental and Calculated Force—Time
Curves of a Spherical Specimen, Sample II.
Deformation Rate = 2.54 mm/min

 

 

108

7.5 Behavior of Fruit under a Constant Load

The experiments and calculations up to this point
were performed assuming a fixed displacement. In the
following, the displacement of a specimen is analyzed
as it evolves in time under the influence of a constant

load.

7.5.1 Creep Behavior of a Cylindrical Specimen

A creep compliance D(t) can be defined to express
viscoelastic strain of a one-dimensional specimen as a
function of stress (Christensen, 1971)

t d o (T)
e (t) =fD(t-T)-——1-1———— (it (7.5)
11 0 d T

This creep compliance can be related to the uniaxial
relaxation function via their respective Laplace trans—
forms

13(5) = __1__. (7.6)

$2 E(S)

Combining (7.6) with the uniaxial relaxation

function of (7.1) yields the creep compliance

-12.02t
D(t) = .365 + .0387t — .0883 e (7.7)

which is shown in Figure 7.14. The units of D(t) are

(megapascalf1

 

 

109

0.48

l

0.44

0.40.

l

0.38

l

.32

l

§0MPLIRNCE l/MPa

28

 

 

.00 0140 0 2200

Cp.24

0380 1220 115
TIME IN MINUTES

Figure 7.14. Uniaxial Compliance (Sample I)

 

110

Equations (7.7) and (7.5) were used to calculate the
change in length of a cylindrical specimen subjected to a
constant compressive load of 80 N, applied at time t=0.
The deformation of the specimen is shown in Figure 7.15
as a function of time.

The nodal loads could have been specified for this
particular specimen. The iterative procedure described
in Section 7.2, however, was used to demonstrate its
applicability. The calculated force had to be within
plus or minus 2.5 percent of the creep load to terminate
the iteration. The results shown in Figure 7.15

indicate the validity of the technique.

7.5.2 Creep Behavior of a Spherical Specimen

The deformation of and the stress distribution in a
spherical body composed of apple parenchyma and subjected
to a constant flat plate load of 50 N was calculated.

The evolution of the axial deformation, calculated using
the iterative finite element procedure, is depicted in
Figure 7.16. The creep deformation rate became very
small after only 0.5 minute.

Isostress curves were interpolated from nodal stress
values which were obtained from the element stress values
through a ”consistent stress analysis” technique (Oden and
Brauchli, 1971). The isostress lines at time t=0, or the

instant of loading, are shown in Figures 7.17 through 7.20.

 

DISPLACEMENT (mm)

111

 

____ Analytical

° ° Finite Elements

 

 

TIME (Minutes)

Figure 7.15. Creep Response of a Cylindrical Specimen of

19.1 mm Length Subjected to a Constant
Compressive Force of 80 N

 

 

 

112

2
J

.8

-— d" To_ 9-9. .9- a.
-0- "b"

1

1 4 1.6
l
‘o\
\

DISPLACEMENT (mm)
1 2
$1

1.0

 

 

TIME (Minutes)

Figure 7.16. Calculated Creep Response of a Sphere
With a 17.8 mm Radius Under Flat Plate
Compressive Load of 50 N

 

 

113

 

‘.50
“.40

 

 

Figure 7.17. Lines of Constant Stress in the
z—Direction at Time t=O

114

—§5
mw‘
—35
—25
—J5

 

 

 

Figure 7.18. Lines of Constant Maximum Principal
Stress at Time t=O

 

115

—.65
—.60

—.5O
-.4O

 

 

Figure 7.19. Lines of Constant Minimum Principal
Stress at Time t=0

116

 

94

 

.15 ‘8
.20

.20

.15

.10

 

 

Figure 7.20. Lines of Constant Maximum Shear Stress
at Time t=O

 

 

117
In Figures 7.21 through 7.24 those lines were repeated
at time t = 0.76 minutes. The stresses in the z—direction
and the principal stresses appeared largest near the
initial point of contact and decreased with increasing
distance from the contact point. The shear stress,
however, had a maximum near the contact point farthest
from the axis of symmetry. The maximum value of the
shear stress at that point was .476 MPa at t = 0 and
decreased to .2 MPa at t = .76 minute. The decrease
with time of the maximum compressive stress and maxi—
mum shear stress is illustrated in Figure 7.25.

Closer observation of the principal stress values
also showed that there existed a tensile stress at the
circular boundary of the surface of contact with a
maximum value of 1.02 MPa at time t = O and decreasing
to 0.57 MPa at t = .76 min. The tensile stress acted in
the axial direction. Some of the stresses in spherical
bodies in elastic contact were discussed by Timoshenko
and Goodier (1971). They indicated the existence of a
radial tensile stress at the circular boundary of the
surface of contact, but the magnitude of this tensile
stress was smaller in comparison with the compressive
stresses at the center of contact. The location of the
maximum shear stress and the maximum tensile stress is
indicated in Figure 7.26, which also shows the deformed

grid at time t = 0.76 min.

 

118

 

 

Figure 7.21. Lines of Constant Stress in the
z—Direction at Time t = 0.76 min

 

 

119

 

 

Figure 7.22. Lines of Constant Maximum Principal
Stress at Time t = 0.76 min

 

 

120

 

—.20'

 

 

Figure 7.23. Lines of Constant Minimum Principal
Stress at Time t = 0.76 min

121

20
/ .24
.15
.20
.15
.10

 

 

 

Figure 7.24. Lines of Constant Maximum Shear Stress
at Time t = 0.76 min

122

 

 

 

b _
I
o — V ‘V Tr—fir—4L—o
I
a m
n, ...-I
s I,
V )\
m . o———o Maximum Compressive Stress
m
m
E w _ x__€‘ Maximum Shear Stress
(j) .
I
m-
I
N_ WWW—H
I
H _
I
I I I I I l
1 2 3 .4 5 6

Figure 7.25.

TIME (Minutes)

Maximum Compressive and Shear Stress in a
Spherical Sample Under a 50 N Creep Load
(Sample Diameter = 17.8 mm)

 

 

 

20.00

    

16.00

Location of the
Maximum Shear and
Tensile Stress

Z-HXIS
8.00 12.00

4.00

     

cp.00

4.00 8.00 1 .
R—RXIS MM

Figure 7.26. Deformed Finite Element Grid of the
Spherical Specimen at Time t = 0.76
min

00 20.00

 

124‘

7.6 Closure

 

Experimental loadings of spherical samples up to
failure were performed. Inspection of the resulting
bruises showed bruise shapes which passed through the
point of maximum tensile and shear stress as presented
in Figure 7.20. Similar bruise shapes were reported by
Horsfield at afi. (1972). The application of the
numerical methods presented in determining the
occurrence of a bruise depends on the availability of a
criterion to describe the condition of material failure.
It follows from the previous section that if a maximum
stress is a failure criterion, then bruising would
occur at the instant of loading of the apple with the
creep load. A discussion on failure criteria is

presented in section 9.4.

8. SUMMARY AND CONCLUSIONS

A new experimental procedure for the determination
of bulk and shear relaxation functions and Poisson's
ratio of apple flesh was developed. This procedure
utilizes the relaxation properties of free and
constrained cylindrical specimens and numerical tech-
niques related to the inverse Laplace transform to
obtain numerical values for these viscoelastic
properties. The experimentally determined constitutive
relations were used in a viscoelastic finite element
analysis to calculate the stresses in apples under loads.
An iterative method was presented for finding the
deformation of a spherical specimen under a constant
force.

The following conclusions can be drawn from this
study:

1) The bulk and shear relaxation functions and

Poisson's ratio of apple flesh are time

dependent properties.

2) There is a large variation in the magnitude of

the relaxation functions between apples.
3) Apples subjected to constant creep loads

125

4)

126
experience maximum stresses at the initial

application of the force.

Maximum shear and tensile stresses exist at
the circular boundary of the contact surface

during a flat plate loading.

9. FURTHER DEVELOPMENTS AND SUGGESTIONS
FOR FUTURE RESEARCH

The research reported in this dissertation was part
of an ongoing regional research project related to the
harvesting and handling of fruits and vegetables (NE-93).
Michigan's contribution to this project is the development
of failure criteria for apple and potato flesh. This

dissertation was the first step in this study and

 

focused on the definition of the material properties of
apples and the determination of the stress components
within the apple when it is subjected to static loads.
Some of the important areas related to apple bruising
which must still be investigated are discussed in the

following sections.

9.1 Change of Material Properties
During Ripening

 

A method has been presented within this dissertation
for the determination of viscoelastic constitutive
equations of apple parenchyma. These constitutive
relations were used in a numerical technique for the
solution of Viscoelastic boundary value problem and the
analysis of stress in fruit under loading. The under—
lying assumption in these calculations was that the

127

 

 

128
material structure is not altered during the time period
of the experiment. It seems logical that for the study
of long time fruit behavior this assumption has to be
modified.

Water loss during storage and the chemical changes
due to ripening can have a great effect on the material
properties. These factors are very difficult to
incorporate into the described analytical technique
merely because ripening itself is a phenomenon that is
difficult to quantify. The hypothesis is made here that
a ripeness index (T) can be established. This ripeness (
index could be based on either starch content, sugar
content, pectin content (itself a component of the
structural strength of the fruit), or simply on the time
in storage under steady state conditions.

The relaxation functions in Section 5 have to be
determined for different values of the ripeness index
W. Ga(t) denotes the value of the relaxation function
at the reference value of the ripeness index W. If the
ripeness index changes then the relaxation function for
that ripeness index can be designated as ga(t,W).

Thus,
ga(t,wo) = Ga(t) d = 1,2 (9.1)

The equation (9.1) is similar to the one used by

Christensen (1971) to express temperature dependence

129
of mechanical properties. Considerable simplification
can be achieved if a unique shift function X(W) can be

found for which

gums?) = Ga(€) (9.2)

E = t X(W) (9.3)

'This shift function allows the superposition of time
and ripeness in the same way time and temperature can be
superimposed for thermorheological simple materials
(Ferry, 1970; Hammerle and Mohsenin, 1970). The use of
the reduced variable 5 means that the behavior of material
in long—term storage can be analyzed with the same ease
with which the mechanical behavior was studied in the

previous section.

9.2 The Effect of the Skin

 

The presence of the skin was neglected in the models
used in this study. Rumsey and Fridley (1974) found
that the presence of an elastic skin produced no
significant change of the internal stress distribution
of the parenchyma. Gustafson (1974), however, showed
that the restraint created by the skin can cause increased
stresses in the body if the turgor pressure is accounted

for. The effect of the skin properties on the stresses

 

130

in the apple parenchyma demands more investigation.

9.3 Dynamic Mechanical Properties

 

It was mentioned in section 7.4 that the values of
the relaxation functions for small values of time are
important for the study of material behavior under
rapidly changing conditions. A theoretical analysis
for the determination of dynamic material properties was
presented in section 3.3. An improved experimental
technique should be developed for the measurement of the

complex moduli.

9.4 Development of a Failure Criterion

 

Combination of the stress analysis techniques with
a criterion for material failure is required to determine
under which state of external loading apples start to
bruise. Work of Miles and Rehkugler (1971) towards the
development of a failure criterion has been mentioned in
section 2.3. Theirs and some other work on yield
criteria of polymers is analyzed here and some suggestions
for further development are proposed.

Figure 9.1 is a plot of the principal stresses 01
and 02 at failure from data given by Miles and Rehkugler
(1971). This failure locus has more characteristics of
the yield locus of maximum distortional energy theory

than of the locus associated with the maximum shear stress

131

l
—0.6

‘—-~-‘~‘
\
\
\
\
447*
—O 8

(MPa)

 

\
I
Cr2

—1

Strain rate = -0055 SEC

—1
+—-+ 7.0 sec

Failure Locus of Parenchyma of Northern
Spy Apples at Two Strain Rates (Calculated
From Data by Miles and Rehkugler, 1971)

Figure 9.1.

 

132
theory (Mendelson, 1968). Moreover, data on compressive
failure strength at higher hydrostatic pressures are
needed to complete the compressive part of the failure
locus graph.

Yield criteria for polymer materials have been
investigated. Mears at az. (1969) showed that tensile
yield stress of polyethylene and polypropylene increases
significantly with increasing hydrostatic pressures.

The applied hydrostatic stresses were about ten times
higher than the mean normal stress at the yield point.
Raghava at at. (1973) used thin—walled polymer tubes
under different internal pressures and axial tension

or compression to study the yield behavior of poly—
carbonate and polyvinylchloride. They proposed the yield

criterion

(0 — o )2 + (o — 0 )2 + (o — o )2 +
1 2 2 3 3 1
2(C—T)(o1 + 02 + 03) = 2CT (9.4)

where 01, 02 and 03 are the principal stresses of the
applied stress state while C and T are the absolute
values of the compressive and tensile yield strengths
respectively. The influence of the hydrostatic pressure
is indicated by the term (01 + 02 + 03). When C = T,
the criterion reduces to the maximum distortion energy
theory (von Mises criterion). Tensile strength of apple

flesh should be investigated in View of the tensile

 

133
stresses mentioned in section 7.5. They are required to
determine whether equation (9.1) or any variation thereof
is applicable in this case.

The changing of material properties during ripening
can have an effect on: (i) the failure criterion to be
used at different ripeness stages and (ii) the level of
distortional energy 9r shear stress at failure, whichever
may be the criterion. It was mentioned in section 4.2
that some cylindrical specimens were loaded to failure.
The average axial compressive stress in cylindrical
samples at failure was found to be —.49 MPa at the be-
ginning of the storage period and -.34 MPa after four
months of storage. It is conceivable that bruise develop—
ment later than the time of application of the creep load

may result from these changes.

 

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APPENDIX A
EXPERIMENTAL RELAXATION FUNCTIONS FOR APPLE FLESH

(24 apples, Red Delicious)

 

 

142

APPENDIX A
EXPERIMENTAL RELAXATION FUNCTIONS FOR APPLE FLESH

UNIAXIAL RELAXATION FUNCTION
.7780 EXP! -3.33358
2.6532 EXP( -.03357

001
‘T)
’T)

CONSTRAINEO RELAXATION FUNCTION 002

1.2415 EXP‘ -3.62835
3.6281 EXP( -.02657

TIME SHEAR H00

(MIN.) (HEGAPASC0)
.0 .139805001
031006E‘01 .137OQE001
0520005'01 .13k52E001
0125005+00 .13007E601
.250005§00 .12337Et01
o50000€+00 .115§ZE+01
010000E031 .108895f01
0200005901 .10397E+01
.00000E+01 .96900E900
.830005‘01 .90978Ef00

UNIAXIAL RELAXATION FUNCTION
.9B38 EXP( -3.“7148
3.h920 EXP( -.02973

‘T)

’T)

BULK HOD
(MEGAPASC.)

.3005#E+01
.29005E+01
0280785f01
.26512E*01
.2h360E001
.222k5E+01
.21080E+01
.205QGE+01
.19752E+01
.19527E601

003
‘T)
’T)

CONSTRAINEO RELAXATION FUNCTION 005

1.79h9 EXPT ~5.85637
4.0180 EXP( -.03313

TIME SHEAR H00
(MINo) (NEGAPASCo)
03 017190E*01
.31000E'01 .16965E*01
.52000E‘01 .1675“E+01

.125008+00
.250EOE+00
.500COE+00
.10000E+01
.20000E+01
.40000E+01
.00000E+01

.16365E+01
.15735E+01
.14881E+01
.19055E*01
.13078EI01
.1275%E+01
.12536E+01

‘T)

‘T)

BULK H00
(MEGAPASC.)

.35205E+01
.32087E+01
.302h3E+01
.26826E+01
.23021E+01
.206406+01
.201BOE+01
.1963ZE+01
.18255E+01
.17885E+01

P015.

P018. RATIO

.29061E000
.29595E000
.293595+00
.28960E000
.28k405+00
.280005*00
.27980E§00
.28311E+00
.28817E+00
.289265000

RATIO

.269995+00
.27801E+00
.26761E+00
.25063E+00
.22937E000
.21h50E000
.21679E+00
.22078E000
.216888+00
.215628000

UNIAXIAL RELAXATION FUNCTION
.8126 EXP( -3.50315 *T)

2.6765 EXP( -.03152 ’T)

143

 

306

CONSTRAINED RELAXATION FUNCTION 007

1.2592 EXP(
3.7083 EXP(

TIME
(HIh.)

.0

.31000E-01
.62000E-01
.12500E+00
.25000E+00
.50000E+00
.100006‘01
.200006+01
.40000E+01
.80000E+01

UNIAXIAL RELAXATION FUNCTION
.6906 EXP( -2.87651
2.6616 EXP(

SHEAR HOD
(HEGAPASC.)

.10187Ei01
.13862E+01
.13570E+01
.13063E+01
.12326E901
.115176601
.10929E901
.10512E+01
.98797E+00
.970535000

-.03032

-3.28398 ‘T)
-.02986 ‘T)

BULK HOD
(HEGAPASC.)

.30757E+01
.2993QE+01
.29194E+01
.27880E+01
.25911E901
.23616E+01
.21890Et01
.2093QE+01
.19796E+01
.19h77E+01

008
‘T)
'T)

CONSTRAINED RELAXATION FUNCTION 009

1.2021 EXP(

-0.32877 ‘T)

3.5872 EXP( -.02705 l’T)
TIME SHEAR MOD BULK MOD
(MIN.) (MEGAPASC.) (MEGAPASC.)
.0 .12699Et01 .36961E+01
.31000E-01 .12510E+01 .2967QE+01

.62000E-01
.12500E+80
.250COE+00
.50000E+00
.100COE+01
.200COE+01
.40000E+01
.80000E+01

.12331E+01
.12003E+01
.11976E+01
.1077EE+01
.10100E+01
.9620kE+00
.90332E+00
.86533E+00

.28562E001
.267HQE+01
.2b402E+01
.2241OE+01
.21605E+01
.21162E+01
.202“0E+01
.200“1E*01

P015.

P015.

RATIO

.30015Ef00
.29907E+00
.29881E000
.297505+00
.29505Et00
.29112E+00
.286936600
.23532E+00
.286R7E+00
.28722Efi00

RATIO

.31950E+00
.315255+00
.31157E600
.30560E+00
.29835E+00
.290105+00
.29733Et00
.30210E*00
.3003ZE+00
.30458E+00

 

 

 

UNIAXIAL

RELAXATION FUNCTION
.9378 EXP(
2.9911 EXP(

01A

-2.7h615 ‘T)
’003062

‘T)

CONSTRAINED RELAXATION FUNCTION 015
1.7878 EXP( -3.#0880 'T)

3.7663 EXP(
SHEAR HOD
(HEGAPASC.)

TIME
(WIN.)

.9

.31000E-01
QSZUDUE‘UI
.12550E+00
.25000E+00
.50000E+00
.10000E+01
.200006+01
.00000E+01
.80000E+01

UNIAXIAL RELAXATION FUNCTION
.7599 EXP(
2.7509 EXPK

.1u9245.01
.1aeaee+o1
.1uu31£+01
.1uooze+o1
.13323£+o1
.1zuuae+o1
.11635E+01
.110505+o1
.10n225+01
.102505.01

-3.A5898
’002276

-.03088 ‘T)

BULK MOD
(HEGAPASC.)

.3563TE+01
.3a1555+o1
.32822E+01
.3052«E+01
.27236E+01
.237u7e+01
.2159ue+01
.2066~E+01
.19h26E+01
.19033E+01

016
‘T)
‘T)

CONSTRAINED RELAXATION FUNCTION 017
1.2339 EXP( -3.27637 ‘T)

3.1936 EXP(

TIME
(MIN.)

.0

.310005-01
.620COE-01
.12500E+00
.25000E+00
.500005f00
.10000E+01
.20000E+01
.40000E+01
.80000E+01

SHEAR MOD

(MEGAPASC.)

.13836E+01
.13507E901
.13290E+01
.12895E+01
.12204Et01
.11509E+01
.11036E+01
.10733E+01
.10266E+01
.10163E+01

-.01969

*T)
BULK M00
(HEGAPASC.)

.25321E+01
.24097E+01
.237“5E+01
.22“21E*01
.20998E+01
.1818JE+01
.1657AE+01
.15928E+01
.15397E001
.15206E+01

POIS. RATIO

.31626E+00
.3122“E#00
.30855E000
.30202E500
.29220E+00
.28098E000
.2739BE+00
.27321E+00
.27320E+00
.272915t00

POIS. RATIO

.26896E+00
.266585*00
.26“30E+00
.25997EO00
.25250E+00
.24165E+00
.23056Ef00
.22583E+00
.22825E*00
.22909E+00

UNIAXIAL RELAXATION FUNCTION
.7696 EXP( -3.08525 ‘T)

2.8499 EXP( -.02071 ‘T’

145

018

CONSTRAINED RELAXATION FUNCTION 019
1.3622 EXP( -3.50002 ‘T)

3.4560 EXP( -.01954 ’T)
TIME SHEAR MOD BULK MOD
(MIN.) (MEGAPASC.) (MEGAPASC.)
.0 .13963E?01 .29562E001
.31080E-01 .13726E+01 .28459E+01
.62000E-01 .13507E+01 .27473E601
.12560E+00 .13119E+01 .25779E+01

.25000E+30
.50000E+00
.10000E+01
.20000E+01
.43000E+01
.80000E+J1

UNIAXIAL RELAXATION FUNCTION
2.7105 EXP(
.7799 EXP( -4.66051

.12529E+01
.11819E+01
.11246E+01
.10885E001
.10432Et01
.10296E401

-.01756

.2336SE+01
.20833Et01
.19307E+01
.18734E+01
.18094E+01
.17931E+01

020
'T)
‘T)

CONSTRAINEO RELAXATION FUNCTION 021
2.4472 EXP( -6.23685 ‘T)

3.7800 EXF(

TIME
(MIN.)

.0

.310COE-01
.620ECE-01
.12500E+00
.25000E+00
.53000E+00
.10000E+01
.20000E+01
.40000E+01
.800006+01

-002519 ‘T)
SHEAR MOD BULK MOD
(MEGAPASC.) (MEGAPASC.)
0127266201 O“5296E*01
.12405E*01 .414COE*01

.12127E+01
.11668E+01
.11060E+01
.1049TE+01
.10193E+01
.10011E+01
.9729GE000
.96723E600

.38196E+01
.33347Eé01
.27964E+01
.24415E+01
.23317E+01
.22589E*01
.21225E+01
.20774E+01

P015.

P013.

RATIO

.29597E000
.29232Et00
.28902Et00
.28319E*00
.27445E400
.26443E+00
.25786E+00
.25688E900
.25824E600
.25901E§00

RATIO

o37159€*00
0364105000
035753E§00
0346575000
033189E500
o31799£§00
031093E+00
o30790€*00
.30328E000
.30193E*00

UNIAXIAL RELAXATION FUNCTION
.6658 EXP( -3.03202 ‘T)

2.4942 EXP( -.02260 'T)

 

022

CONSTRAINEO RELAXATION FUNCTION 023

1.2927 EXP(
3.4987 EXP(

TIME
(“10.)

.0

.31000E-01
.62000E-01
.125006+00
.25000E+00
.50000E+00
.10000E+01
.200005+01
.40000E+01
.80000E+31

UNIAXIAL RELAXATION FUNCTION
.5034 EXP(
2.2716 EXP(

SHEAR M00

(MEGAPASC.)

.11828E+01
.11622Ei01
.11435E001
.11101E+01
.10584E001
.99469E+00
.94135E+00
.90850E900
.87181E000
.86536E‘00

-3.39397
‘0028H1

-3.86304 ’T)
'002277 ‘T)

BULK HOD
(MEGAPASC.)

.32147E+01
.30934E901
.29864E+01
.28062E*01
.25596EI01
.23200E601
.21917E+01
.21329E+01
.20416E+01
.20168E+01

032
.‘T)
’T)

CONSTRAINED RELAXATION FUNCTION 033
1.7824 EXP( -3.76473 'T)

2.5951 EXP(

TIME

(MIN.)
.0
.31000E-01
.62000E-01
.12500E+30
.25000E+00
.50000E+00
.10000E+01
.200006+01
.40000E+01
.80000E+01

SHEAR MOO

(MEGAPASC.)

.10306E+01
.10164Ef01
.10036E+01
.98148E600
.95020E+00
.91791E*00
.89947E+00
.88931E+00
.86523E+00
.85618E+00

-.05160 'T)

BULK MOD
(MEGAPASC.)

.30029E+01
.28217E+01
.26601E+01
.23831E+01
.19905E+01
.15766E+01
.13065E+01
.11555E+01
.96252E+00
.91072E+00

POIS. RATIO

.33612E900
.33311E+00
.33041E+00
.32584E000
.31963E000
.31402E’00
.31254E000
.31352E*00
.31334E¢00
.31281E+00

POIS. RATIO

0305925000
03393BE§00
0333145900
032148E§00
030185E+00
02729“E*00
0239215900
0208245500
0179305900
0165135500

147

UNIAXIAL RELAXATION FUNCTION 034
.7781 EXF( ~2.95406 'T)
2.4524 EXP( -.02814 ‘T)

CONSTRAINEO RELAXATION FUNCTION 035
1.5068 EXP( -4.04959 ‘T)
3.2468 EXP( -.02794 'T)

TIME SHEAR M00 BULK MOD POIS. RATIO
(MIN.) (MEGAPASC.) (MEGAPASC.)
.0 .12163E+01 .31316Ei01 .32808E000
.31000E-01 .11940Et01 .2981OE+01 .32343E+00
.62000E-31 .11735EO01 .28488E+01 .31927E§00
.12500E+00 .11365E+01 .26284E+01 .31220E§00
.25000E+00 .10786E+01 .2333BE+01 .30246E+00
050000E+00 .10048E*01 .20612E+01 .293525t00
.10000E+01 .93946E+00 .19309Ek01 .29134E600
.20000E+01 .89702E+00 .18746E+01 .29362Efi00
.40000E+01 .84973E*00 .17748Ef01 .29409E+00
.80000E+01 .83855E+00 .17432E+01 .29391E+00
UNIAXIAL RELAXATION FUNCTION 036
.7335 EXP( -3.26040 ‘T)
2.5434 EXP( -.02737 ‘'T)
CONSTRAINED RELAXATION FUNCTION 037
1.7364 EXP( -4.45328 ’T)
3.3146 EXP( -.02786 ’T)
TIME SHEAR MOD BULK M00 POIS. RATIO
(MIA.) (MEGAPASC.) (MEGAPASC.)
.0 .12224E+01 .34212E+01 .34039E'00
.31000E-01 .12006E+01 .32234E+01 .33450E+00
.62050E-01 .11807E+01 .30521E+01 .32924Ef00
.12500E+00 .11453E+01 .27711E+01 .32017E+00
.250005+00 .10916E+01 .24065E+01 .30724Et00
.50000E+00 .10267E+01 .20871E+01 .29402E000
.100COE+01 .97285E+00 .19464E#01 .28763E+00
.20000E+01 .93581E+00 .188785+01 .28761E000
.400£GE+01 .88760E+00 .17899E+01 .28772Et00
.80000E+01 .87487E+00 .17650E+01 .28797E+00

 

UNIAXIAL RELAXATION FUNCTION
.5859 EXP(
2.3484 EXP(

148

038

-2.77435 ‘T)
-.02773 ‘T)

CONSTRAINEO RELAXATION FUNCTION 039
1.0738 EXP( -2.99313 ‘T)

3.2166 EXP(
SHEAR M00
(MEGAPASC.)

TIME
(MIN.)

.0

.31000E-01
.62000E-01
.12500E+00
.25OCOE+80
.50000E+00
.10000E+01
.20000E+31
.40000E+01
.800COE+01

UNIAXIAL RELAXATION FUNCTION
.6060 EXP(
2.2361 EXP(

.11058E+01
.10885E*01
.10723E+01
.10430E+01

-.02344 ‘T)

BULK MOD
(MEGAPASC.)

.28156E+01
.27415E+01
.26741E+01
.25551E+01

.99735E+00 .23762E+01
.93922E600 .21674E+01
.88711E+00 .20130E+01
.849205+00 .19395E+01
.79972Et00 .18654E+01
.78282E+00 .18477E+01
040
-2.37079 ‘T)
-.03184 ‘T)

CONSTRAINED RELAXATION FUNCTION 041

1.6123 EXP(
3.1743 EXP(

TIME
(MIN.)

.0

.310COE-01
.620005-01
.12500E+00
.25000E+00
.50000E+00
.10000E+01
020000E+31
.40000E+01
.80000E901

SHEAR MOD

(MEGAPASC.)

.10447E+01
.1C309Eé01
.101826’01
.99453E*00
.95574E+00
.90177E+00
.84572E*00
.80228E+00
.75655E+00
.74591E+00

-3.37739 ‘T)
-.04169 ‘T)

BULK MOD
(MEGAPASC.)

.33926Ef01
.32474E+01
.31162E+01
.28887E+01
.25603E§01
.22046Et01
.19720E+01
.18515E+01
.16789E+01
.16218E+01

POIS. RATIO

.32632E000
.32469E000
.32321E900
.32058E900
.31656E*00
.31182E+00
.30868E900
.30890E+00
.31154E*00
.31248E000

POIS. RATIO

.36044E+00
.356555'00
.35297E+00
.34656E+00
.33666Et00
.32466E600
.31576E+00
.31222E+00
.30745E+00
.30613E000

149

UNIAXIAL RELAXATION FUNCTION 042
.4973 EXP( -2.48030 ‘T)
2.3227 EXP( -.03064 ‘T)

CONSTEAINED RELAXATION FUNCTION 043
1.9459 EXP( -3.47533 'T)
3.6249 EXP( -.03449 'T)

TIME SHEAR HOD BULK M00

(MIN.) (MEGAPASC.) (MEGAPASC.)
.0 .10154E+01 .42160E+01
.310COEe01 .10039Et01 .40295E+01
.62000E-01 .99317E+00 .38616E+01
.12500E+00 .97345Et00 .35716E+01
.25000E900 .94147E+00 .31549E+01
.50000E000 .89782E+00 .27084E+01

.10000E+]1
.20000E+01
.40000E+01
. 30000E+01

.85310E+00
.81554Er00
.76888E+00
.75692E+00

.24247E*01
.22970E+01
.21349E+01
.20835Et01

UNIAXIAL aELAXATION FUNCTION 044
2.8628 EXP( -.02892 ‘T)

CONSTRAINED RELAXATION FUNCTION 045

2.0112 EXP(

-4.63030 ‘T)

3.3256 EXP( -.02840 ‘T)

TIME SHEAR MOD BULK M00

(MIN.) (MEGAPASC.) (MEGAPASC.)
.0 .13563E+01 .35285E+01
.310COE-01 .13288E+01 .32932E+01
.620COE~01 .13347E+01 .30895E+31
.12500Ef00 .12644E+01 .27553E+01
.25000E+00 .12100E+01 .23206E+01
.530COE+00 .11577E401 .19336E+01
.100EOE+01 .11259E+01 .17507E+01
.200005+31 .109606+01 .16813E+01
.40000Ef01 .10383E001 .15929E+01
.8]0€0E+31 .10239E+01 .15698E+01

POIS.

'0'.

P015. RATIO

.38854E+00
.38506Ef00
.38183E400
.37600E400
.36685E+00
.35533E000
.34603E000
.34243Ef00
.34066E+00
.34012E400

RATIO

.32966E+00
.32241E‘00
.31574E+00
.30377E+00
.28514E000
.26178E000
.24225E+00
.23420E+00
.23363Ef00
.23385E*00

 

 

150

UNIAXIAL RELAXATION FUNCTION 046
.5797 EXP( -4.85286 ‘T)
2.5519 EXP( 0.02972 ‘T)

CONSTRAINED RELAXATION FUNCTION 047
3.1240 EXP( -5.25084 ‘T)
3.0730 EXP( -.05344 ‘T)

TIME SHEAR MOD BULK MOD

(MIN0) (MEGAPASC0) (MEGAPASCo)
00 0112736901 0“693SE§01
031000E‘01 011038E901 0425095901
0620(05-01 010837E+01 038733E+01

.12500E+00
.25000E§00
.50000E+00
.10000E+01
.20000E+01
.40060E001
.800C05+01

.10518Et01
.10132Ef01
.98497E+00
.97778E600
.96825Ef00
.93513Et00
.92497E+00

032707E+01
0252205+01
.19050E+01
.16256E+01
.14704E+01
.12428Ef01
011812E+01

UNIAXIAL RELAXATION FUNCTION 050
.8353 EXP( -2.57384 ‘T)
3.1536 EXP( -.03147 ‘T)

CONSTRAINED RELAXATION FUNCTION 051
1.3105 EXP( -3.42813 'T)
3.8451 EXP( -.02748 ‘T)

TIME SAEAR MOD BULK MOD

(MIN0) (MEGAPASC0) (MEGAPA300)
00 0155265201 0308505001
031000E‘01 .15312E+01 029785E+01
062000E'01 015111E§01 .238325001

.12500E+30
.25000E+00
.500EOE§00
.100CCE+01
oZUUCUEfOl
.40000E+01
.80050E+31

.14741E+01
.14133E001
.13292E001
.12426E+01
.11756E+01
.11006E+01
.10808Et01

.27202Ef01
.24906E+01
.22565E+01
.21264E+01
.20732E+01
.19823E+01
.19545E+01

POIS. RATIO

030073E+00
038087E§00
037345E+00
035972E§00
033709E‘00
030509E*00
027056E+00
0242925+00
0216025+00
020863E+00

POIS. RATIO

.28452E000
.28062E000
.27714E400
.27120E000
.26302E+00
.25580E+00
.25564E*00
.26103E000
.26533E+00
.26625E+00

151

UNIAXIAL RELAXATION FUNCTIONI 052

3.0238 EXP(

.6996 EXP( -3.83343 ‘T)
-.03406 ‘T)

CONSTRAINEO RELAXATION FUNCTION 053

1.2360 EXP(
3.8003 EXP(

TIME
(MIN.)

.0

.310COE-01
.620COE-01
.125COE000
.250COE+30
.50000E+00
.10000E+01
.20000E+31
.40000E+01
.80000E+01

UNIAXIAL RELAXATION FUNCTION
.7501 EXP( -3.17312
2.9222 EXP(

-3.06739 ‘T)

-002267 ‘T)
SHEAR MOD BULK M00
(MEGAPASC.) (MEGAPASC.)
.14301E+01 .31292E+01
.13992E+01 .30558E+01
.13718Et01 .29879E+01

.132515+01
.12599E+01
.11923£+01
.110505+01
.11013£+01
010233E+01
.1ecoss+01

-.03818

.28651E+01
.26732E+01
.243455+o1
.22458E+01
.216586+01
.21097E+01
.209395+o1

054
’T)
‘T)

CONSTRAINED RELAXATION FUNCTION 055

2.3671 EXP(
4.2212 EXP(

TIME
(MIN.)

.0

.310COE-01
.620COE-01
.12500E+00
.25000E+00
.53000E+00
.100LOE+31
.200COE+01
.400EOE+31
.80000E+11

*3.66107 ‘T)

-.04113 ‘T)

SHEAR M00 BULK M00

(MEGAPASC.) (MEGAPASC.)
.13376E+01 .48044E+01
.13143E+01 .45766E+01
.1293LE+01 .43730Et01
.12553E+01 .40237E+01
.11984E+01 .35282E+01

.11302E+01
.1C725E+01
.10247E+01
.95291E+00
.93321E+00

0300815*01
0263158201
0252295001
0231895+01
022615E+01

POIS.

POIS.

RATIO

.30174E§00
.30138E000
.30093E+00
.29974Er00
.29676E+00
.29074E*00
.28367E+00
.28274E000
.290665000
.29348E+00

RATIO

.37271EG00
.36905E+00
.36573E000
.35973Ef00
.35025E+00
.33811Et00
.32763E*00
.32285E+00
.32157E+00
.32228E000

 

152

UNIAXIAL RELAXATION FUNCTION 056
.6686 EXP( -2.55635 'T)
2.7517 EXP( -.03327 ’T)

CONSTRAINEO RELAXATION FUNCTION 057
1.7637 EXP( -3.08777 ‘T)
3.0195 EXP( -.04004 ‘T)
TIME SHEAR MOD BULK MOD POIS. RATIO

(MIN.) (MEGAPASC.) (MEGAPASC.)
.0 .13026E+01 .30463E+01 .31288E000
.310CCE-31 .12882E+01 .290096+01 .30673E§00
.62000E-01 .12749Et01 .27687E+01 .30092E000
.125COE+00 .12508E+01 .25357E+01 .29003E*00
.25000E+00 .12129E+01 .21873E+01 .27167E000
.50000E+00 .11644E001 .17839Ef01 .24479E000
.10000E+01 .11201E001 .14878E+01 .21461E000
.21000E401 .10847Ef01 .13444E+01 .19139E600
.40000E+01 .10301E+01 .12058E+01 .175805+00
.30000E+11 .1£148E+01 .11685E+01 .17260E+00

UNIAXIAL RELAXATION FUNCTION 060
.6872 EXP( -3.48049 ‘T)
2.5574 EXP( -.04137 ‘T)

CONSTRAINED RELAXATION FUNCTION 061
1.4710 EXP( -3.57430 ‘T)

3.6786 EXP( -.03413 ’T)

TIME SHEAR MOO BULK M00 POIS. RATIO

(MIN.) (MEGAPASC.) (MEGAPASC.)
.0 .12041E501 .35438E+01 .34746E+00
.31000E-01 .11784Ef01 .34201E+01 .34551Et00
.62000E-01 .11553E+01 .33093E+01 .34375E*00
.12500E*30 .1115CE+01 .31172E+01 .34059Et00
.25000E+30 .10557E+01 .28418E+01 .33573E000
.500005+00 .98793E+00 .254555+01 .32986E400
.130COE+01 .93390Et00 .23510Ef01 .32573E000
.200COE+31 .88846E+00 .22523E+01 .32587E+00
.40000E+01 .81866E+00 .21243E+01 .32960E000

.800005+01

 

.80109F+00

.20862E+01

.33105Et00

 

153

UNIAXIAL RELAXATION FUNCTION 064
.7654 EXP( -3.54544 ¥T)
2.8442 EXP( -.04543 ‘T)

CONSTRAINED RELAXATION FUNCTION 065
1.7495 EXP( -3.42369 ‘T’
3.4250 EXP( -.03241 ‘T)

TIME SHEAR M00 BULK MOD POIS. RATIO

(MIN.) (MEGAPASC.) (MEGAPASC.)
.0 .13667E+01 .33521Et01 .32055E+00
.31000E-01 .13386E+01 .32099E+01 .31701E+00
.620i0E-01 .13135E+01 .30817E+01 .31367Ei00
.12500E930 .12699E+01 .28583E+01 .30745Ef00
.25000E+30 .12067E+01 .25317E+01 .297095000
.500CGE+00 .11365E+01 .21704E+01 .28260E+00
.13000E+01 .1[813E§01 .19308E+01 .26899E+00
.23000E+01 .10272E+01 .18425E+01 .26562E000
.40000E+01 .93304E*00 .17728E+01 .27487E+00
.80000E+01 .90762E+00 .17551E+01 .27791E+00

UNIAXIAL RELAXATION FUNCTION 066
.8096 EXP( -2.55665 ‘T)
2.6134 EXP( -.04454 ’T)

CONSTRAINED RELAXATION FUNCTION 067
3. 013 EXP( -4.24651 ‘T)
3.4463 EXP( -.09141 ‘T)

TIME
(MIN.)

.0

.3100CE-01
.62000E‘01
.12500E+00
.25000E+00
.50000E+30
.13000E+01
.20000E+01
.400EOE+]1
0300005+01

SHEAR MOO
(MEGAPASC.)

.12328E+01
.12153E901
.1199CE+01
.11692E+01
.11214E+01
.1C575E*01
.99486E+00
.94665E+00
.891586t00
.87669E*00

BULK MOD
(MEGAFASC.)

.51023E+01
.47100E+01
.43652E+01
.37897E+01
.30156E901
.22779E+01
.18658E+01
.16074E+01
.12067E+01
.10893E+01

POIS. RATIO

038808E+00
.38144E000
.375145500
.36354E+00
03““73E000
.31894E000
029249E*00
.26931E+00
.23567E000
.22523E900

 

 

 

"11007MM