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International Journal of Agriculture and Crop Sciences.
Available online at www.ijagcs.com
IJACS/2013/5-16/1739-1744
ISSN 2227-670X ©2013 IJACS Journal
Simulation of a Tomato Sample Using Finite Element
Method
Mahmood Mahmoodi-Eshkaftaki *1, Ali Maleki1
1.Department of Agricultural Machinery Engineering, Faculty of Agriculture, University of Shahrekord, Shahrekord,
Iran.
*Corresponding author email: [email protected]
ABSTRACT: The purpose of this study was to investigate deformation behavior of the tomato fruit
(hereafter tomato) upon crashing. To achieve this, a 3D simulation was used to generate a model of
tomato using finite element method and then compared with experimental results. For finite element
modeling with software, some mechanical properties of tomato justly punching from limited sides (head,
bottom, two sides, and four sides) must be determined with compression test. To this goal, an apparatus
according to Bussinesk method was manufactured. A comparison between experimental data and finite
element simulation showed that the two sets of results agree well.
Keywords: Elasticity Modulus, Finite element method, Simulation, Tomato.
INTRODUCTION
Tomato (Lycopersicon esculentum) is classified under subfamily of the Solanaceae. It is a self–fertile
vegetable and has 4–6 percent crossbreeding. Tomato is one of the most consuming vegetables used as fresh,
dried or processed in human nutrition and is secondary important in its family (Delina Felix and Mahendran, 2009).
At present, tomato harvesting and processing are mostly manual while as increased use of tomato in large
industries, it is required sufficient knowledge of physical and mechanical properties (MP) of the tomato to design
some proper machines for harvesting, grading, cleaning, packaging, and transporting (Perez et al., 2007).
Furthermore in these processing, tomatoes are exposed to mechanical loading with impact from different sides.
This can lead to puncture injury such as makes a hole in tomato with stem, bruises or cuts, all causing qualitative
and quantitative losses (Tianxia et al., 2002; Mazaheri Tehrani, 2007). Researchers have spent a lot of their time
and efforts to reduce horticultural losses (Tabatabaeefar and Rajabipour, 2005; Papadopoulou and Manolopoulou,
1990). Thus, by performing studies on MP of tomatoes and effective parameters on them and with aid of obtained
information, it can be possible to design mechanical systems for processing tomatoes, therefore by decreasing
losses in different process sectors, increase the production output.
In order to minimize mechanical damage, the handling stresses must be kept under a certain value. The
internal stresses caused by an external force are very difficult to measure. An alternative approach to estimate
these stresses is finite element analysis (FEA) (Cardenas and Stroshine, 1991). Presently, the most studies on
macroscopic and microcosmic mechanical characteristics and mechanical damages of fruit and vegetables are
concentrated on apple and potato (Abbott and Lu, 1996) and several theories of bruising and failure of fleshy fruits
have been proposed. Many reports are available in the scientific literature on the FEA and resonant frequencies of
various kinds of near-spherical agricultural objects such as apples (Chen and De Baerdemaeker, 1993; Lu and
Abbott, 1997), peaches (Verstreken and De Baerdemaeker, 1994), melons (Chen et al., 1996) and tomatoes
(Langenakens et al., 1997; Kabas et al., 2008). The researches show that the FEA is a suitable method for
calculating the resonance properties of near-spherical solid materials.
Tomato is much more susceptible to mechanical damage because of its soft texture. While there are a lot
of studies on tomato damages, the number of literatures about the tomato’s compression test studies punching
from limited sides by finite element method (FEM) is little; therefore some MP of tomato punching from four sides
were measured and then used in FEA to determine the stress and deformation distributions for a tomato model,
finally compared the software results with experimentally measured methods.
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MATERIALS AND METHODS
Five tomato varieties – ‘Kariz’, ‘Darbigo’, ‘Falkato’, ‘Newton’, and ‘Shaqayeq’ – in three ripeness – green,
pink, and red – used in this study were obtained from greenhouse of Kesht Nesha Jonoub located at Shahreza,
grown at similar conditions. The samples were around full maturity size cleaned manually to remove all foreign
matter, dust, dirt, injured samples. Through a completely random factorial design test,
test, effects of independent
independent factor
of tomato varieties and ripeness levels of product at the time of harvesting on MP of tomato were considered. In
this experiment, for each test both 48 replications were used and the
he tests were carried out at (21±1
21±1) °C and
relative humidity of (30±1) % of environment (Papadopoulou
Papadopoulou and Manolopoulou, 1990).
1990)
Mechanical properties (MP)
Fruit stiffness
To determine the external stiffness in this research, pressures applied from four concurrent zones – head,
bottom, two symmetric sides, and four sides–
sides on the fruit skin for each sample. According to the references of fruit
stiffness determination, the suitable probe diameter was 8 mm and the penetrating duration for depth of 4–5
4 mm
was 2–3
2 s (Mahmoodi,
Mahmoodi, 2008).
2008 . To determine the feature, an apparatus was manufactured as shown in Fig. 1 and a
500 N loadcell with 0.001 N precision was fixed on the apparatus by a pawl, furthermore a micrometer was timely
used to measure the tomato deformation (Fig.
(
1).
). The loadcell was connected to a PC with a special cable and the
data were recorded with its software (Data collection speed of this device was 200 Hz). The equivalent force for
penetration of 4–5
4 mm (measured with micrometer) was considered as stiffness index (White
White et al., 2005)
2005).
Elasticity modulus (E)
To determine this feature, the Bussinesk method was used.
used. In this method, by pressing the probe cylinder
on top of the sample, given the maximum applied force (Fmax) and probe penetration inside the sample (Dp), the
fruit E was calculated from Eq. 1 (Abbott
Abbott and Lu, 1996).
1996
E=
Fmax (1 − µ 2 )
×
Dp
2r
(1)
To determine the
the Poisson
oisson ratio (µ),
(µ after
fter measuring the initial length and width of the tomato, it was
compressed until deformation occurred on the fruit body,
body the µ was calculated by using the following formula after
measuring the final diameter and length after deformation (Kabas
Kabas et al., 2008):
2008
µ=
∆D Di − D f
=
∆L L f − Li
(2)
Where Di is the initial diameter, Df is final diameter, Li is initial length, and Lf is final length of the tomato, in
mm (Fig. 2).
Fig
Figure
1. Manufactured apparatus to determine the tomato mechanical
properties
Figure 2. Definition figure for the Poisson ratio
calculation
Finite element simulation
To investigate deformation behavior of the tomato samples upon crashing, a 3D simulation was generated
for each sample using FEM. The FEM is a numerical technique for solving field problems described by a set of
partial differential equations. These types of problem are commonly found in many engineering disciplines, such as
machine design, acoustics,
acoustics, electromagnetism, soil mechanics, and fluid dynamics (Tianxia
Tianxia et al., 2002).
2002 . The FEA
Intl J Agri Crop Sci.
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744, 2013
not only is a powerful tool for engineering analysis but also is used to solve problems ranging from the very simple
to the very complex (Chen
Chen et al., 1996).
1996 Time constraints and limited availability of product data call for many
simplifications in the analysis models. On the other hand, specialized analysts implement FEA to solve very
advanced problems, such as crash dynamics, material forming or analysis of bio-structures
bio structures (Cardenas
Cardenas and
Stroshine, 1991; Chen and De Baerdemaeker, 1993; Lu and Abbott, 1997; Kabas et al., 2008).
2008 . Today, with the
help of advancing technologies, computers and FEA softwares
software allow us to use FEM in nearly all disciplines of
engineering
engineering.
The consequent vector which usually applied on a material is FI(t)+FD(t)+FE(t)-R(t)=0,
R(t)=0, where FI(t) are the
inertia forces, FD(t) are the damping forces, FE(t) are the elastic forces, and R(t) are externally impact forces (The
unit of force is N). All of these
these forces are time dependent. In static analysis, this equation reduces to FE(t)=R(t)
since the inertia and damping forces are neglected due to no velocities and accelerations (Kabas
Kabas et al., 2008).
2008
In this research, the effect of the impact on a part or an assembly with a rigid or flexible planar surface was
evaluated with a new method. At first a tomato was modeled as a 3D solid by CATIA V5R21 software, and then the
simulated tomato was fixed on the rigid plate (No rotations were considered until the initial
initial impact occurs). The
tomato was assumed to be nearly spherical, solid and with a skin that had the same properties over the entire
body. All dimensions, rigid target plane and tomato’s solid model are shown in Fig. 3(a). In the simulation, octree
tetrahedral elements were used for the mesh construction of the 3D model of the tomato. After the meshing
operation, 9613 total nodes and 5502 total elements obtained (Figure
ure 3(b)).
(a)
(b)
Figure 3. (a) Solid model of tomato and its dimensions in scale of 2, in this case the force have been applied from head; (b)
Mesh construction of a 3D model of a tomato
Table 1. Comparison of the elasticity modulus and stiffness of four researched side averages
averages for the variety traits by Duncan
test
Head
Bottom
Both sides
Four sides
E (Mpa)
0.1765
c
0.1785
c
0.1085
a
0.1026
a
0.1423
b
0.1798
b
0.1806
b
0.1007
a
0.1059
a
0.1553
b
8.0750
a,b
8.8637
b
7.0100
a
8.4258
b
8.2014
a,b
0.1559
b
0.1507
b
0.0740
a
0.0769
a
0.1353
b
8.0114
Variety
0.1670
c
‘Kariz’
0.1652
c
‘Darbigo’
0.0893
a
‘Falcato’
0.0906
a
‘Newton’
0.1420
b
‘Shaqayeq’
b
7.8743
b
‘Kariz’
8.1162
b
8.1978
b
‘Darbigo’
6.3112
a
6.5709
a
‘Falcato’
7.3990
a,b
7.7797
b
‘Newton’
7.7171
b
7.8134
b
‘Shaqayeq’
Stiffness (N/mm)
7.3994
a
7.6954
a
6.6511
a
7.8951
a
7.6182
a
Averages with similar letters have no significant differences at level of 5%
RESULTS AND DISCUSSION
Results of variance analysis showed that the relationship between the tomato E of head, bottom, both
sides, and four sides, and variety was statistically significant (P<0.01). While this parameter in different ripeness
Intl J Agri Crop Sci. Vol., 5 (16), 1739-1744, 2013
and the mutual effect of variety–ripeness was not significant. Multi-domain Duncan test (Table 1) showed that the
maximum and minimum E values of head tomatoes were related to ‘Darbigo’ and ‘Newton’ varieties, respectively.
These values of bottom fruit were measured for ‘Darbigo’ and ‘Falcato’ varieties, respectively, also that of both
sides and four sides were related to ‘Kariz’ and ‘Falcato’ varieties, respectively. According to researches, the E
indicates substance elastic states against static and dynamic loads (Abbott and Lu, 1996; White et al., 2005). After
comparing the results, it was concluded that the variety of ‘Kariz’ had the most averages of E for all four researched
sides and therefore this variety more acts as a spring against impact loads and have a damping condition; thus this
variety can be more simulated with a solid model.
Variance analysis results showed that the variety had significant effect on stiffness of samples at four side
and both side measurements (P<0.01).The multi-domain Duncan test showed that the minimum of four side
stiffness averages was measured for ‘Falcato’ variety and there was no significant difference with the other
varieties (Table 1). The stiffness of ‘Kariz’ was ranged from 7.3994 N/mm for tomato head to 8.0750 N/mm for
tomato bottom. After considering the FEA software both MP and physical properties of tomato needed modeling it.
As it is shown in Table 2, the material properties of head and bottom of tomato were measured. Furthermore using
the peck of stress–strain curves of tomato under compression test, the yield strength ( b) for these sides were
determined (Fig. 4). According to references, the thermal conductivity (KT) of fruit ranges around 0.55 w/m c
(Mahmoodi and Kianmehr, 2009).
Table 2. The material properties of tomato solid model used finite element software
Tomato side
E (Mpa)
µ
3
(kg/m )
KT (w/m c)
Fmax (N)
b
(Mpa)
Head
0.1765
0.33
948.48
0.55
8.0750
0.4710
Bottom
0.1798
0.33
948.48
0.55
8.0750
0.4900
=Density (kg/m3)
Figure4. Stress–strain curve of tomato using puncture test
After completing the software settings for the analyses, a solving process was run and the results of the
Von Mises stress and deformation values have been shown in Figs. 5–6. As it is illustrated the minimum and
maximum of stress in tomato with compression of head were 268.461 and 50400 N/m2, respectively and those of
deformation under the force were 0 and 8.8 mm, respectively. These values of stress for tomato with punching from
bottom (Fig. 6) were 1176.89 and 46840.5 N/m2, respectively and those of deformations were 0 and 10.5 mm,
respectively. The result showed that the mean of stresses and deformations of tomato compression of bottom was
more than punching from tomato head, these results was documented in Table 1. When comparing the simulation
results with physical test results good agreement was observed and it was in agreement with the arguments
presented by Kabas et al. (2008).
Intl J Agri Crop Sci. Vol., 5 (16), 1739-1744, 2013
Figure 5. Von Mises stress in tomato and deformation in scale of 2 of tomato under compress force (8.0750 N) of head with
stress principal tensor and estimated local error illustrated as wireframe model. The maximum and minimum of stresses are
shown on the tomato model
Figure 6. Von Mises stress in tomato and deformation in scale of 2 of tomato under compress force (8.0750 N) of bottom with
stress principal tensor and estimated local error illustrated as wireframe model. The maximum and minimum of stresses are
shown on the tomato model
The FEM not only modeled a tomato more reliable but also when comparing the simulation results with
physical test results good agreement was observed. Which documented the FEM approach can be a useful tool to
find the mechanical behaviors of tomato for its processing machine to reach the optimal efficiency that is
considered as a tedious and time-consuming task.
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