Comparison of Axial and Radial Compression Tests for Determining Elasticity Modulus of Potatoes

Abstract

The effect of sample size, loading rate, and compression level on the determination of tangent modulus of elasticity of potatoes in axial (E_a) and radial (E_r) compression tests was studied. Cylindrical potato samples of 10, 15, and 20 mm diameter with length-to-diameter (L/D) ratio 1, 1.5, and 2 and loading rates of 50, 100, 200, 300, and 400 mm/min were used. A third-degree polynomial best described the force-deformation (F-D) plot of the test specimens (R² = 0.98 to 0.99) in both tests. In view of the non-linearity of the F-D plot, E_a and E_r were calculated corresponding to 10, 20, and 30% compression. The values of E_a and E_r obtained for various combinations of sample size and compression level had coefficient of variation (CV) of about 24 and 8%, respectively, indicating relatively greater influence of experimental conditions in axial than radial compression testing. Apparently, E_a increased with an increase in compression level and loading rate whereas E_r exhibited inconsistent trend. The combined effects of sample size and compression level accounted for about 93 and 20% of the changes in E_a and E_r, respectively signifying the minimal effects of these parameters in radial compression testing. Similarly, the loading rate and compression level accounted for about 96 and 28% of the changes in E_a and E_r, respectively. The results of this article revealed that determination of modulus of elasticity of potatoes in axial compression testing was significantly influenced by testing conditions and sample size, whereas radial compression testing appeared to be independent of testing conditions and sample size.

Keywords:

• Potatoes
• Elasticity modulus
• Axial and radial compression

INTRODUCTION

The mechanical properties of solid and semi-solid food products have been frequently measured by axial compression of test specimens between parallel plates.[1,2] The resulting force-deformation (F-D) curves from compression tests are used for estimating several rheological parameters, which may signify the textural attributes of food products.[2] However, an arbitrary selection of sample size and test conditions often yields considerable variations in the values of derived parameters that are difficult to compare, interpret and reproduce.[3] Such problems highlight the importance of standard test techniques, which are scarce for food materials for various reasons. Cylindrical test specimens have been routinely subjected to an axial compression loading for determining the mechanical properties of vegetable tissue in general and potatoes in particular.[3–7] Some studies have been conducted to investigate the effect of experimental test conditions on the determination of mechanical properties of potatoes. The influence of loading rate on the elasticity modulus, fracture strength, and stress developed in potatoes have been studied and reported.[3,4,5,7] The results indicated that elasticity modulus increased with increase in loading rate whereas fracture strength and stress decreased with increase in loading rate. Modulus of elasticity of cooked potatoes has been also reported to increase with increase in loading rate between 20 to 250 mm/min.[6] In a related study, the secant modulus of pear tissue was reported to increase with increase in loading rate.[8] The effect of sample size on stress developed in cylindrical potato samples has been studied, and the result showed that stress slightly increased with increase in sample diameter while a decrease in sample height resulted in increased stress.[2]

It has been reported that modulus of elasticity can also be determined from radial compression of cylindrical test specimens.[9] Both axial and radial compression tests have been shown to give comparable values of the tangent modulus of elasticity of potatoes.[10,11] However, the comparison of the two test techniques in relation to sample size and other experimental conditions has not been studied. Therefore, the objective of this study was to investigate and compare the effect of experimental test conditions on the determination of tangent modulus of elasticity of potatoes in axial and radial compression tests.

MATERIALS AND METHOD

Test Sample Preparation

Potatoes (cv spunta) grown in north Thailand were purchased two days after harvest and stored at about 14°C for a week before conducting the tests. Prior to testing, the potatoes were reconditioned overnight at room temperature (about 24°C). After removing the stem and bud ends, two cylindrical samples were removed from each tuber using cork borers of 10, 15, or 20 mm diameter. The cylindrical specimens were subsequently trimmed to the required lengths corresponding to length-to-diameter (L/D) ratio of 1, 1.5, or 2 using a slicer. The two cylindrical samples taken from each tuber were aligned approximately 3 to 4 mm apart along the length of the tubers to make sure the samples are taken from corresponding locations in different tubers.[5,12]

Axial Compression Test

Cylindrical test specimens of potatoes were subjected to axial compression between two horizontal flat plates through downward movement of the crosshead of a food texture analyzer (Model LRX 5K, Lloyd Instruments, UK) fitted with a 500 N load cell using the geometry as shown in Fig. 1(a). Loading rates of 50, 100, 200, 300, and 400 mm/min were used to compress the samples. The force-deformation (F-D) data for each test specimen was recorded using Nexygen software 4.0 supplied with the Lloyd Instrument and later fitted with a third-degree polynomial using Eq. (1)[10] for determining the instantaneous slope required in the computation of the tangent modulus of elasticity by Eq. (2). The tangent modulus of elasticity corresponding to 10, 20, and 30% compression of original length of test specimens was calculated.

Figure 1 Schematic presentation of test techniques and model equations used for determining the tangent modulus of elasticity of cylindrical samples of potatoes: (a) axial compression (b) radial compression.

Radial Compression Test

In radial compression test, the samples were compressed radially between two parallel plates as shown in Fig. 1(b) using loading rates of 50, 100, 200, 300, and 400 mm/min. The F-D data was fitted to Eq. (1) for estimating the force at a given deformation. The tangent modulus of elasticity was determined using Eq. (4) by first estimating the parameter Z in Eq. 3 for a given deformation by Newton-Raphson method.[9] The value of Poisson's ratio for potatoes was taken to be 0.48.[10] The tangent modulus of elasticity corresponding to 10, 20, and 30% compression was determined.

DATA ANALYSIS

One-way ANOVA was used to examine the effect of sample size and experimental test conditions on the determination of the tangent modulus of elasticity. Stepwise multiple regression was employed to examine the combined effects of sample size, compression level, and loading rate using SPSS version 10 (SPSS Inc., USA).

RESULTS AND DISCUSSION

Effect of Sample Size and Compression Level on Tangent Modulus of Elasticity

Typical F-D plots obtained from axial and radial compression tests of cylindrical test specimens (15 mm diameter and 30 mm long) at a loading rate of 50 mm/min are presented in Fig. 2. A third-degree polynomial (Eq. 1) represented the F-D data of test specimens adequately in both tests (R² = 0.98–0.99). The tangent modulus of elasticity in axial (E_a) and radial (E_r) compression tests calculated for various combinations of sample sizes and compression levels at a loading rate of 50 mm/min are presented in Table 1. The tangent modulus of elasticity ranged from 2.17 to 5.18 MPa and 2.59 to 3.64 MPa in axial and radial compression tests, respectively. The mean and standard deviations of E_a and E_r were 3.546 ± 0.856 and 2.989 ± 0.236 MPa, respectively. The relatively wider range of E_a, coupled with larger standard deviation, indicated considerable influence of compression level and sample size in axial compression testing. This was also apparent from the coefficients of variation (CV) of about 24.1% and 7.9% obtained for E_a and E_r, respectively. The modulus of elasticity of potato has been reported to range from 2.7 to 2.9 MPa corresponding to about 7.8 to 9.4% axial compression of cylindrical test samples.[5] The tangent modulus of elasticity of potatoes in axial and radial compression of test specimens were determined to be 2.9 and 2.1 MPa, respectively, for 3% compression.[10] The differences in reported values of elasticity modulus could be due to the method of calculation and test conditions used. Previous studies have indicated that stiffness or moduli of potato tissue depend on turgor pressure and the cell wall strength.[13,14] Changes in the mechanical properties of potatoes due to storage has been associated to loss of turgor pressure and other biochemical reactions, which affect the cell wall and middle lamella of the tissue.[13,14,15] The degradation of pectic substances, which make up 52% of potato cell wall and are responsible for the binding of the cells together, has been reported to cause reduction in mechanical or rheological properties of potatoes.[14,15]

Table 1 Tangent modulus of elasticity (MPa) of potatoes based on axial and radial compression of cylindrical test samples (Loading rate = 50 mm/min).

		Axial compression			Radial compression
Diam. (mm)	L-D ratio	10%	20%	30%	10%	20%	30%
10	1.0	2.171^a	2.868^a	3.945^a	2.858^a	3.210^a	3.643^a
	1.5	2.613^b	3.195^a	4.080^a	2.690^a	2.741^a	2.942^b
	2.0	2.554^b	3.187^a	4.121^a	2.989^a	3.081^a	3.337^ab
15	1.0	2.792^b	3.334^b	4.385^b	3.104^b	2.949^c	3.132^b
	1.5	2.818^b	3.347^b	4.653^bc	2.610^b	2.585^b	2.727^c
	2.0	2.774^b	3.321^b	5.177^c	2.903^b	3.051^c	3.313^b
20	1.0	2.696^a	3.525^c	4.890^c	3.060^d	2.810^d	3.010^d
	1.5	2.997^b	3.941^c	4.898^c	3.004^d	2.851^d	2.919^d
	2.0	2.909^b	3.522^c	4.916^c	2.883^d	3.044^d	3.259^d
Note: Values in a column for a given diameter followed by different letters are significantly different at 0.05 level.

Figure 2 Typical force deformation plots in axial and radial compression testing of cylindrical samples of potatoes (D = 15 mm; L = 30 mm; LR = 50 mm/min).

The results in Table 1 indicate that the L/D had no significant effect (p > 0.05) on the elasticity modulus in most of the cases for any given sample diameter and compression level. Also no systematic tends were observed for both E_a and E_r with respect to L/D. However, E_a apparently increased with an increase in the percent compression in all cases whereas E_r did not show any such distinct trend as shown in Fig. 3. Hence, it is important to indicate the strain or compression level used while reporting the modulus of elasticity. Cylindrical samples taken in transverse direction of potato tubers were reported to be stiffer than those taken in longitudinal direction under axial compression.[5,14] Such anisotropicity was attributed to the orientation of the pith rays in the transverse direction. However, direct comparison between the results of this and the previous studies is not possible due to differences in the test configuration.

Figure 3 Tangent modulus of elasticity as function of compression level in axial and radial compression testing (D = 15 mm; L = 30 mm; LR = 50 mm/min).

The dependence of tangent modulus of elasticity (E) on compression level (∊), sample diameter (D) and L/D was analyzed statistically by stepwise multiple regression assuming the following relationship:

(5)

Table 2 shows the results of stepwise multiple regression in both axial and radial compression of cylindrical test specimens based on Eq. (5). The result indicated that E_a was significantly influenced by ∊, D and L/D. Among the experimental variables, ∊ was the most important accounting for 83% of the changes in E_a as shown by the R² value. The combined effect of ∊, D and L/D accounted for 93% of the changes in E_a, indicating the strong dependence of E_a on these parameters. In radial compression test, the combined effect of ∊, D and L/D accounted for only 20% of the changes in E_r indicating that these experimental variables had no significant effect on the determination of E_r as indicated by F-test statistics and small R² value (Table 2). This was also supported by the relatively higher magnitude of the constant term (a₀) in Eq. (5) in radial compression in comparison with the axial compression test. These results clearly indicated that the determination of E_r was least influenced by the experimental test conditions in their respective ranges used in this study.

Table 2 Results of stepwise multiple regression for compression testing of cylindrical test samples of potatoes based on Eq. (5).

Tests	Predictors	a0	a1	a2	a3	R^2		SEE	F
Axial	∊	1.682	9.30E-02			0.831	0.824	0.36	122.52∗
	∊, D	0.755	9.30E-02	6.18E-02		0.922	0.916	0.25	142.05∗
	∊, D, L/D	0.443	9.30E-02	6.18E-02	0.208	0.933	0.924	0.23	105.93∗
Radial	∊	2.747	1.21E-02			0.182	0.149	0.22	5.56∗
	∊, D	2.855	1.21E-02	−7.24E-03		0.198	0.131	0.22	2.96
	∊, D, L/D	2.842	1.21E-02	−7.24E-03	9.02E-03	0.198	0.094	0.23	1.90
Note: ∗Significant (p < 0.05).

Effect of Loading Rate and Compression Level on Tangent Modulus of Elasticity

Table 3 shows E_a and E_r values for various combinations of loading rate (LR) and ∊ of cylindrical test specimens of 15 mm diameter and L/D of 2. The mean and standard deviations of E_a and E_r were estimated to be 3.869 ± 0.975 and 3.151 ± 0.245 MPa, respectively, based on the data presented in Table 3. The result indicated that E_a in general increased with an increase in the loading rate except in the case of 30% compression level. An increase in the LR from 50 to 400 mm/min did not apparently influence the determination of E_r at all compression levels and no systematic trend was also observed. An increase in modulus of elasticity up to LR of 250 mm/min in axial compression and then a decrease has been reported for cooked potatoes.[6] The secant modulus and failure stress in axial compression test were also observed to increase with an increase in loading rate for pear tissue.[8] The compression level had pronounced effect on E_a as compared to E_r at all loading rates (Table 3) as evidenced by a progressive increase in E_a with an increase in compression level. Such distinct trends, however, were not observed for E_r.

Table 3 Tangent modulus of elasticity (MPa) of potatoes based on axial and radial compression of cylindrical test samples (D = 15 mm; L = 30 mm).

	Axial compression			Radial compression
Loading rate (mm/min)	10%	20%	30%	10%	20%	30%
50	2.774^a	3.321^a	5.177^a	2.903^a	3.051^a	3.313^b
100	2.553^a	3.534^ab	4.821^a	3.006^a	2.892^a	2.938^a
200	2.796^a	3.690^abc	4.997^a	3.193^a	3.110^a	3.204^ab
300	2.867^ab	4.094^c	5.092^a	3.545^a	3.153^a	3.219^ab
400	3.159^b	3.925^bc	5.228^a	3.512^a	3.093^a	3.104^ab
Note: Values in a column followed by different letters are significantly different at 0.05 level.

The combined effect of LR and ∊ on the determination of modulus of elasticity in axial and radial compression testing was analyzed by a stepwise multiple regression based on the following relationship:

(6)

The results of stepwise multiple regression based on Eq. 6 for both tests are presented in Table 4. In axial compression testing ∊ alone accounted for about 94% of the changes in E_a. The combined effects of ∊ and LR accounted for about 96% of the changes in E_a demonstrating the strong dependence of E_a on these test conditions. On the contrary, these parameters together accounted only about 28% of the changes in E_r. The result apparently showed that the determination of E_r was not significantly influenced by ∊ and LR as indicated by the F-test statistics and very low R² value. In view of the results, it was concluded that radial compression testing is less influenced by experimental test conditions as compared to axial compression testing.

Table 4 Results of stepwise multiple regression for compression testing of cylindrical test samples of potatoes based on Eq. (6).

Tests	Predictors	a0	a1	a2	R^2		SEE	F
Axial	∊	1.635	0.112		0.936	0.931	0.26	189.98∗
	∊, LR	1.374	0.112	1.24E-02	0.964	0.959	0.20	162.97∗
Radial	LR	2.957	9.22E-04		0.248	0.190	0.22	4.29
	LR, ∊	3.065	9.22E-04	−5.38E-03	0.282	0.163	0.22	2.36
Note: ∗Significant (p < 0.05).

CONCLUSION

The determination of tangent modulus of elasticity of cylindrical potato test specimens was significantly influenced by compression level, sample size, and loading rate in axial compression testing. On the contrary, the effect of experimental test conditions on the determination of elasticity modulus in radial compression testing was not evident. However, comparable results for tangent modulus of elasticity could be obtained from both axial and radial compression testing of homologous products like potatoes with careful selection of experimental test conditions.

NOMENCLATURE

A	=	Cross-sectional area [m2]
b	=	Half contact width in radial compression [m]
D	=	Diameter [m]
E	=	Tangent modulus of elasticity [Pa]
Ea	=	Tangent modulus of elasticity in axial compression [Pa]
Er	=	Tangent modulus of elasticity in radial compression [Pa]
F	=	Force [N]
L	=	Length [m]
LR	=	Loading rate [mm/min]
L/D	=	Length-to-diameter ratio [-]
R	=	Radius of the cylindrical sample [m]
Z	=	R/b
ΔL	=	Deformation [mm]
∊	=	Compression level [%]
μ	=	Poisson's ratio [-]
a0, a1, a2, a3	=	Constants