Effects of Time after Harvest and Rate of Loading on Force Relaxation Behaviour of Local Variety of Grapefruit ( LemunTaba )

Force relaxation properties of a local variety of grapefruit (Lemuntaba) was determined under quasi-static compression loading using Instron UTM at three levels of time after harvest (freshly harvested, 7 days and 14 days after harvest) and three rates of loading (10, 5, 1 mm/s) for freshly harvested and 10 mm/s for one and two weeks after harvest. Fitting the obtained data for freshly harvested, loaded at 10 mm/s to a three-term Maxwell model; the resulted model equation was of F(t) = 743.521e -t/1.843 +592.817e -t/0.007 +474.254e -t/0.008 , R 2 = 0.97. For freshly harvested loaded at 10 mm/s, the force relaxed (decayed) from an initial value of 2435.647 N to 743.521 N; about 69.473% in 1.834 s; Similar phenomenon was observed for other treatments. From the results, it can be deduced that when this cultivar is loaded with about 65% of the total force at rupture, about 69% of the imposed load will be dissipated upon removal of that in about 1.8 s; an evidence of high elasticity.


INTRODUCTION
Stress relaxation of food and biological material is a measure of the rate at which the same material dissipates stress after being subjected to a sudden load.The principle of stress relaxation is widely used in fruits and vegetable industries as well as food industries generally, the knowledge of stress relaxation parameters is used in the design of containers, i.e., the number of stack of fruits and the maximum depth of container to minimize mechanical damage due to dynamic or vibration loading while in transit and in store (Mohsenin, 1986).
Grapefruits (citrus paradisi) are a subtropical citrus fruits known for its bitter taste, it could be yelloworange in colour when ripe; the flesh is segmented and is generally acidic.It ranges from 10-15 cm in diameter depending on the cultivar.The primary varieties include: Ruby Red, Pink, Thompson, Marsh and Duncan.
Grapefruit is an excellent source of many nutrients and phytochemicals that contribute to a healthy diet.It forms an essential part of a balanced diet as it is an important source of digestible carbohydrates, minerals and vitamins; particularly vitamins A and C. In addition, it provides roughage (indigestible carbohydrates) which is needed for normal healthy digestion.The juice helps lower cholesterol level in Plate 1: Hand picking of grapefruits humans as well as assisting the body's metabolism to burn fats and is an antioxidant; the seed extract has strong anti-microbial properties against fungi and bacteria (www/Grapefruit-Wikipedia, the free encyclopedia).
Grapefruit, just as other fruits is essentially a perishable commodity, it begins to deteriorate as soon as it is harvested and is particularly prone to handling damage at all times.In general, the level of susceptibility of grapefruits to handling damage is greatly underestimated because the effects of mishandling do not appear until sometimes after the damage had occurred.
The physical and mechanical characteristics of grapefruit (citrus paradisi) for temperate regions are well documented (William, 1986) but the viscoelastic properties such as force relaxation and creep are rare if not completely absent for local varieties such as Lemuntaba (Plate 1) hence the near absence of handling and processing equipment and the huge losses of the variety that is often encountered.Given the growing economic and nutritional importance of this tropical local variety, it is imperative that viscoelastic properties of the variety be determined accurately so that handling, packaging and transportation systems are designed with utmost efficiency to minimize losses.
The objective of this study is to determine the relaxation parameters (force relaxation, decay modulus and relaxation time) of Lemuntaba at three levels of loading (10, 5 and 1 mm/s, respectively) and at three levels of time after harvest (freshly harvested, 7 and 14 days).
In stress relaxation, the test specimen is suddenly brought to a given deformation (strain) and the stress required to hold the deformation constant is measured as a function of time (Mohsenin, 1986;Anazodo, 1982;Golacki et al., 2007;Marco et al., 2007;Burubai et al., 2009aBurubai et al., , 2009b)).
It is worthy to note that the initial deformation of the material must be less than the deformation at failure of the test specimen; however, it should be high enough to impose considerable strain on the specimen.It should be at least 65% of the total deformation at failure (Anazodo, 1982).
The most important viscoelastic parameters which can be obtained from a stress relaxation test are decay stress (σ d ) or decay modulus (Ε d ), equilibrium stress (σ e ) or equilibrium modulus (Ε e ) and time of relaxation (T rel ) (Khazaei and Mann, 2004;Pallottino et al., 2010).Relaxation time is the time at which the stress in a body resembling a simple Maxwell model decays to 1/e of the initial stress (Mohsenin, 1986).It is a measure of the rate at which a material dissipates stress after receiving a sudden force.

Mathematical models:
In modeling stress relaxation of biological materials, a generalized Maxwell model with two or three elements is often used.Although when using generalized Maxwell model to characterize food and biological materials, most researchers used 'stress' (Pallottino et al., 2010;Anazodo, 1982); however, because the actual contacting surface area of food material continually changes under applied load making it difficult to calculate exactly the 'true stress' values from the beginning of compression to rupture point, 'stress' can be replaced by any other decaying parameter such as force, modulus of elasticity, (Khazaei and Mann, 2004;Gorji et al., 2010); so generalized Maxwell's model for force relaxation can be represented by Eq. ( 1): where, T 1 , T 2 , … T n are the relaxation time constants corresponding to various Maxwell model elements, F 1 , F 2 .F n are the decay forces and F (t) is the instantaneous force.

Materials:
Grapefruits used for this study were obtained from Kaura CitrusFarm in Toto Local Government Area of Nasarawa State, North Central Nigeria.Four trees in plots of trees typical of the variety were selected from which fruits were harvested for the tests.Some fruits were carefully handpicked from the trees while others were chipped off the tree with a knife leaving a stalk 10-12 cm long and leaves removed (Coppock et al., 1969); this is to maintain some level of physiological freshness for tests concerning freshly harvested.The fruits were kept cooled in a fruit shed by water spray while harvesting was going on; at the end of harvest (between 1.00-2.00pm), they were packed in cardboard boxes at ambient temperature of 27°C and 78% relative humidity as shown in Plate 2. The bottoms of these boxes were lined with foam to minimize mechanical injuries and sides perforated to reduce temperature and ethylene build up (Tabatabaekoloor, 2012).In addition, the heat of respiration is removed through these perforations.
The fruits were transported the same day to Advanced Materials Laboratory of the Engineering Materials Development Institute (EMDI), KM 4, Ondo Road, Akure, Ondo State, Southwest Nigeria and stored in a cool room maintained at about 5°C and 87% relative humidity immediately upon arrival at about 8.30 pm.Tests for freshly harvested was conducted at 7.30 am the following day (about 11 h after harvest) while other tests were conducted after 7 and 14 days respectively.

Methods:
Dimensions: Dimensions of 100 freshly harvested fruits were determined on three mutually perpendicular axes using a digital vernier caliper reading to 0.01 mm and the results presented in Table 1.
Preliminary tests: Because of variations in sizes, the fruits were grouped into two based on geometric mean diameter (nearly the same physical characteristics): For freshly harvested for example, the range of the  Five fruits from each group were randomly selected, cleaned of any surface moisture, placed centrally (axially) in the Instron Universal Testing Machine (Model 3369, No. K334; 50 kN capacity) under parallel steel flat plate (Plate 3); however, to avoid spillage of citrus juice (which is acidic) on the platform of the machine, it was covered with plastic sheet and loaded to rupture point (Plate 4).For freshly harvested, the samples were loaded at 10, 5 and 1 mm/s respectively while for 7 and 14 days after harvest, the samples were loaded at the rate of 10 mm/s.The mean load at rupture for each group was determined and 65% of this load used.
Relaxation test: For each test, 65% of the value of force obtained in the preliminary test was imputed in to the machine and fruits in each group deformed at the set loading rates (10, 5 and 1 mm/s for freshly harvested and 10 mm/s for 7 and 14 days after harvest) (Khazaei and Mann, 2004).The machine automatically stopped when the set value of force is reached, then force decay with time was recorded at time intervals as in Table 2.The duration for each test was 300 sec (Pallottino et al., 2010;Jatuphong et al., 2008), though the duration of the test could be as long as possible since the Freshly harvested -----------------------------------------------------------------------------------1 Wk after harv. 10 mm/s 2 Wks after harv. 10 mm/s 10 mm/s 5 mm/s 1 mm/s 0 2435.647Fig. 1: A typical force relaxation curve of fresh grapeloaded at 10 mm/s theoretical time is infinity (Mohsenin, 1986).Each test was replicated ten times and the mean value of the decay forces for each rate of loading are as presented in Table 2.The decay force versus time was then plotted.Figure 1 shows a plot of Force versus time of freshly harvested, loaded at 10 mm/s; Fig. 2 are plots of the experimental and predicted values, while Fig. 3 and 4 are Force-Relaxation curves at different rates of loading and times after harvest respectively.
As noted by Khazaei and Mann (2004), a threeterm Maxwell model involving six constants are sufficient for many biological materials, thus a threeterm Maxwell model expressed by Eq. ( 2) was used: (2) where, F 1 , F 2 , F 3 are decaying forces; T 1 , T 2 , T 3 are the relaxation time constants and F(t) is the instantaneous force at any time, t.
The Coefficients F 1 , F 2 , F 3 ; and T 1 , T 2 , T 3 were obtained by non-linear regression analysis by iteration method using IBM © SPSS ® Statistics, Version 20.0.The following procedures were followed: • A scatter plot diagram of Force versus Time for each condition was plotted as in Fig. 1 • The curve was segmented into three based on change in shape • Linear regression analysis of force-time data of each segment was run to obtain intercepts and slopes which act as the starting values for iteration • Using the Model equation: F 1 *е -t/XI +F 2 *е -t/X2 + F 3 *е -t/X3 and fixing constraints; F 1 , F 2 , F 3 >0; and X 1 , X 2 , X 3 < 0; the data was then iterated • The values of the intercepts, F 1, F 2 , F 3 give the 1 st , 2 nd and 3 rd exponential coefficients of the threeterm Maxwell model while the slopes; X 1 , X 2 , X 3 equal to
Coefficients of maxwell model for grapefruit: From Table 3, considering freshly harvested, loaded at 10 mm/s; inserting the coefficients in Eq. (2) yields: F(t) = 743.521e-t/1.843+ 592.817e -t/0.007+ 474.254e -t/0.008, R 2 = 0.97 Table 3, the first terms of the three-term Maxwell mode (F 1 ) made major contributions to the total decay forces.For freshly harvested at 10 mm/s for instance, the force relaxes from an initial value (F o ) of 2435.647N (Table 1) to 743.521 N; about 69.473% in 1.834 s after which it slows down.Khazaei and Mann (2004) observed a similar trend with Sea buckthorn berries (Hippophaerhamnoides L.) using three-term Maxwell model where about 80% of the induced force was dissipated at the initial stage though at relatively long period of 370 s.
Relaxation time may be considered as a measure of the rate at which the material dissipates internally imposed stress, thus the shorter the relaxation time the quicker the imposed stress is being dissipated.

CONCLUSION
It has been asserted that high relaxation times are associated with viscous materials while low relaxation times are associated with elastic materials (Mohsenin, 1986).Thus from the initial relaxation times of 1.854, 1.834, 1.727 s for freshly harvested loaded at 10, 5, 1 mm/s; 1.605 and 1.589 s for one-and two weeks after harvest respectively, it can be deduced that grapefruit can dissipate internally imposed stress rapidly resulting in less deterioration as a result of imposed load.

Plate 2 :
Grapefruits to be loaded in to a cushioned perforated Carton Plate 3: Grapefruit placed axially between parallel compression tools of the universal testing machine Plate 4: Grapefruit loaded to rupture

Fig. 2 :
Fig. 2: Experimental and predicted force relaxation curve for fresh fruit loaded at 10 mm/s values obtained are presented in Table3 ) where, A o = {H{ { { , F(t) = force at time (t), F(o) = initial time.For Tarrocco orange, Pallottino et al. (2010) fitted the mean values to a three-term Maxwell model and obtained initial dimensionless decay parameter of 0.59±0.04 with initial relaxation time (T rel ) of 5.1 sec.

Table 1 :
Physical properties of grapefruit geometric mean diameter of the first and second group were 8.240-9.799and 9.899-11.119 cm, respectively.

Table 2 :
Effect of time after harvest and rate of loading on force relaxation (N)

Table 3 :
Maxwell three-term coefficients