An Experimental Study of Axial Poisson’s Ratio and Axial Young’s Modulus Determination of Potato Stems Using Image Processing

Abstract

Potato stems removal is an important part of mechanized potato harvesting. However, there is still limited research on the physical properties of potato stems, especially the determination of Poisson’s ratio and Young’s modulus. This study determined the Poisson’s ratio and Young’s modulus of the potato main stems at different heights above the ground. Since the Potato stems are viscoelastic cylinders with non-standard circular cross-sections and complex textures, the existing determination methods are difficult to apply. We propose a new method to determine Poisson’s ratio and Young’s modulus by combining image processing in the mechanical compression process. The feasibility of this method was verified by determining the hardness value of 65 Shore ‘A’ nitrile rubber specimens, and the measured Poisson’s ratio and Young’s modulus were close to the relevant literature. This method can be used for the determination of potato stems. Atlantic potatoes are widely grown for their high solids content, resistance to pests and diseases, and good processing quality. Ten Atlantic potato main stems were randomly selected at harvest time. Specimens with a length of 11 ± 1 mm were taken at 0 cm, 10 cm, 20 cm, and 30 cm above the ground from each stem. The average values of the axial Poisson’s ratio were determined as: 0.21, 0.28, 0.30, 0.32, and the axial Young’s modulus as: 15.90 MPa, 12.38 MPa, 11.68 MPa, 11.28 MPa. This study has provided critical basic data for the discrete element model construction of potato stems and numerical simulation of potato haulm killers and potato harvesters, which is beneficial for improving the harvest quality of potato. It also provides new ideas for Poisson’s ratio and Young’s modulus measurement of non-regular cross-sectional cylindrical viscoelastic materials.

1. Introduction

Potato is the largest non-cereal food crop worldwide and is ranked as the world’s fourth most important food crop after rice, wheat, and maize [1]. According to FAOSTAT, the total world production of potatoes amounts to 359,071,403 tons and the total world harvested area amounts to 16,494,810 ha in 2020 [2]. The development of potato mechanization production has a pivotal role in ensuring world food security. The performance of potato harvesting equipment greatly affects the quality of the potato harvest. At present, there is still room for improving the existing potato haulm toppers and potato harvesters in terms of potato stem cutting, potato stem removal and potato tubers damage reduction [3,4,5,6].

With the development of the Discrete Element Method (DEM) and Multi-Body Dynamics (MBD), simulation has become one of the important tools for the development of modern agricultural equipment [7,8,9]. Physical parameters of the material such as density, Poisson’s ratio, Young’s modulus, and coefficient of restitution have a great influence on the simulation results. The physical properties and the damage characteristics of potato tubers have been relatively comprehensively researched in the available literature. Canet et al. [10] studied the fracture behavior of potato samples under uniaxial compression. Bentini et al. [11] researched the changes in the Poisson’s ratio and Young’s modulus of two varieties of potato tubers during cold storage. Feng et al. [12,13] determined the restitution coefficient and dropping impact characteristics of potato tubers. Deng et al. [14] designed a friction-collision test rig to investigate the mechanism of potato peel damage. Compared to the enthusiasm for studying the physical properties of potato tubers, far too little attention has been paid to the potato stems. Gong et al. [15] investigated the compression characteristic of the potato stalk buds and measured their compressive strength and compressive strain. However, there are no studies related to the Poisson’s ratio and Young’s modulus of potato stems. Therefore, this paper aims to determine the Poisson’s ratio and Young’s modulus of the main stems of potatoes at different heights above the ground and to provide basic material parameters for the simulation design and optimization of the potato haulm killers and potato harvesters.

Poisson’s ratio is defined as the negative ratio of transverse strain to the corresponding axial strain resulting from axial stress below the proportional limit of the material [16]. Young’s modulus is the ratio of tensile or compressive stress to corresponding strain below the proportional limit [17]. Common measurement methods of the Poisson’s ratio and Young’s modulus can be divided into mechanical, acoustic, and optical methods [18]. The principle of the machine measurement method is to apply a uniaxial force to the test specimen; the axial force is measured by the universal testing machine, and the transverse strain is measured by the extensometers or strain gauges. According to the ASTM E132-17 standard [16], at least two pairs of extensometers should be used—one pair for measuring axial strain and the other for transverse strain. The strain gauges are used in the GB/T3354-2014 standard for testing the Poisson’s ratio of the orientation fiber reinforced polymer matrix [19]. The acoustic measurement methods include surface Brillouin scattering (SBS), laser-generated surface acoustic waves (LSAW), and acoustic microscopy (AM) [20]. The optical measurement methods include digital holography and digital speckle correlation [21,22].

As most of the agricultural materials are complex in texture and irregular in shape, it is generally difficult to meet the application requirements of acoustic and optical measurement methods. In addition, agricultural materials have a certain moisture content and are easy to break, which makes stable clamping very difficult in tensile experiments. Therefore, the mechanical measurement method of applying uniaxial compression is a common method for measuring the Poisson’s ratio and Young’s modulus of agricultural materials. Chappell et al. [23] measured the Poisson’s ratio and Young’s modulus of apple flesh by making standard cylindrical specimens and obtaining transverse strain and a cross-sectional area with contact sensors under uniaxial compression. Bentini et al. [11] also used this method to measure the changes in the Poisson’s ratio and Young’s modulus of potato tubers during cold storage. Kiani et al. [24] investigated the effect of loading rate on Poisson’s ratio and the effect of moisture content on Young’s modulus by applying uniaxial load to red beans.

The potato main stem is complex in texture and consists of epidermis, xylem and pith. The surface is not smooth, which makes it difficult to meet the application requirements for Poisson’s ratio and Young’s modulus measurement by acoustic and optical methods. For this reason, the mechanical method was chosen for the measurement. Potato stems are anisotropic materials. Since specimens suitable for measuring the radial Poisson’s ratio and radial Young’s modulus could not be made while keeping the potato stalk tissue relatively intact, this paper only focused on the determination of the axial Poisson’s ratio and axial Young’s modulus of potato stems. Drying experiments showed that the relative moisture content of the main stems of Atlantic potatoes was high at harvest time, with an average value of 91.88%. With this moisture content, the stems are easily broken during clamping in tensile experiments, and it is very difficult to find a suitable clamping force. Therefore, we also decided to apply uniaxial compression for the measurement of the Poisson’s ratio and Young’s modulus of potato stems. The pressure and axial strain can be calculated from the load-displacement curve output from the universal testing machine. The measurement of lateral strain and the cross-sectional area is the critical part. The main stems of potatoes are irregularly rounded. The extensometer, strain gauge, and contact displacement transducer can only record the lateral strain at a certain measurement angle, which is not sufficient to describe the overall lateral strain of the specimen and cannot accurately measure the cross-sectional area. To solve this problem, we used image processing techniques to identify the contour of the cross-section, and then measured the area and expressed the average lateral strain of the specimen by the radius of the equal-area fitted circle. The feasibility of this method was tested using buna rubber. The axial Poisson’s ratio and axial Young’s modulus of the main stem of potatoes at 0 cm, 10 cm, 20 cm, and 30 cm above the ground were determined by applying this method.

2. Materials and Methods

2.1. Preparation of Potato Main Stem Specimens

Atlantic potatoes are widely grown for their high solids content, resistance to pests and diseases, and good processing quality. We collected ‘Atlantic’ potato stems at harvest time from Zhanjiang, Guangdong Province, China (21°22′48″ N, 110°14′53.1″ E). The position on the potato main stems were marked where it breaks out of the soil. The leaves and some lateral branches of the potato plant were cut off. The remaining main stems were rapidly transported to the laboratory for experiments under low temperature (2–5 °C) conditions. The remaining lateral branches were pruned in the laboratory and the main stems were retained for experiments. The cutting position of the potato main stems is an important design parameter for potato haulm killers. The stubble height of the main potato stems is an important design basis for the potato harvester’s stem removal mechanism. Therefore, the study of the axial Poisson’s ratio and axial Young’s modulus of the potato main stem at different heights above the ground can provide important basic parameters for the design and simulation of potato haulm killers and potato harvesters. Ten main stems of potatoes at harvest time were randomly selected. Specimens with a length of 11 ± 1 mm were taken at 0 cm, 10 cm, 20 cm, and 30 cm above the ground from each stem. The cut surfaces were required to be perpendicular to the axis of the stems. The diameter of the potato main stem is usually 13–15 mm. To prevent the specimen from tipping during compression, we took a length slightly smaller than the diameter (11 ± 1 mm) as the height of the specimen. The shape of the upper and lower cross-section of each specimen is closer at this length. The stem samples and sampling methods are given in Figure 1.

2.2. Introduction of the Poisson’s Ratio and Young’s Modulus Measurement Test Bench

In this experiment, an electronic universal testing machine UTM6503 (SUNS, Shenzhen, China) was used to apply uniaxial compression to the specimens. We improved the traditional steel plate in order to keep the specimen in the axis of the upper and lower circular steel plate contactor during the compression process and to install the camera to obtain the cross-section image of the specimen during uniaxial compression. The new type of contactor consists of a steel plate, three steel posts, a steel ring, and a glass. The steel plate is connected to the steel ring by three steel posts and transfers the force. The 10 mm thick glass plate is fastened to the steel ring by clamps. The Sony IMX415 camera module and upper led light were placed inside the upper contactor for capturing a clear image of the specimen cross-section. A translucent white background paper is sandwiched between the lower glass plate and the steel ring of the lower contactor to reduce the interference of the background to the image processing. The lower LED light illumination reduces the interference of the specimen projection on the contour identification. A black 10 mm × 10 mm square area block is attached to the lower surface of the upper glass plate to obtain the cross-sectional area of the specimen. To reduce the influence of uncontrollable environmental light, the experiment needs to be conducted in a darker environment to ensure that the light source of the specimen is mainly provided by the upper and lower LED lights. The Poisson’s ratio and Young’s modulus measurement test bench are shown in Figure 2.

2.3. Determination of the Elastic Limit of Potato Stems during Compression

Five potato main stem specimens at 10 cm above the ground were taken for the axial compression pre-test. The universal testing machine’s downward speed was set to 3 mm/min. The load-displacement curves output from the universal testing machine are shown in Figure 3a. Taking specimen 3 as an example (Figure 3b), the load-displacement curves of the stem specimen during compression could be roughly divided into three stages. In the first stage, the slope of the (o~a) segment was increasing. This stage is the process from slight contact to full contact between the glass plate of the contactor and the specimen section. In the second stage, the slope of the (a~b) segment was almost stable. At this stage, the contact between the contactor and the specimen cross-section entered a steady state. This stage was the elastic stage of the specimen, which is used for Poisson’s ratio and Young’s modulus measurements. In the third stage, the (b~d) segment of the curve showed significant fluctuations, and the force did not increase linearly with the increase of displacement. This stage was the yielding stage, which produces plastic deformation. Experiments show that potato stems did not possess a significant bio-yield point, which is consistent with the mechanical properties of rape stalks studied by Liao et al. [25].

Figure 3a shows that the elastic stage was different for each specimen due to individual differences in the main stem of each potato plant. In order to accurately capture the elastic stage of each specimen, a full video recording with a resolution of 1280 × 720 was performed at the same time as the compression started. The elastic stage was found according to the load-displacement curve and the image of the corresponding moment was output. Taking specimen 3 as an example, the image at point (a) corresponded to the video moment of 00:00:13:00; the image at point (b) corresponded to the video moment of 00:00:25:00.

2.4. Measurement of the Cross-Sectional Area and the Fitted Circle Diameter of the Potato Stem

Images of point (a) and point (b) were captured from the video by corresponding moments. The images need to contain the potato stem cross-section and black standard area square. Image processing and operations were performed in Python 3.8 with the Open CV2 command set. Figure 4a shows the picture of the stem specimen at point (a) and Figure 4b shows the picture of the stem specimen at point (b).

Take Figure 4a as an example to illustrate the image processing. Firstly, we converted the RGB coordinates to the HSV space of the captured images. HSV (Hue, Saturation, and Value) is widely used in computer graphics. RGB coordinates can be easily converted to HSV space using the following equations [26]:

$$V=\frac{1}{3(R+G+B)}$$

(1)

$$S=1-\frac{3}{R+G+B}[\min (R,G,B)]$$

(2)

$$H=\arccos \left\{\frac{[(R-G)+(R-B)]/2}{{(R-G)}^2+(R-B){(G-B)}^{\frac{1}{2}}}\right\},$$

(3)

where R, G and B represent red, green, and blue components. H stands for hue, V stands for value and S stands for saturation.

Secondly, the mask is constructed based on the threshold value and the extraction effect is checked using bitwise operations. Thirdly, the images are made grayscale and are binarized. Fourth, the removal of internal holes was carried out by closing operations. The image processing process is shown in Figure 5.

Fifth, the external contour was found, and the result is shown in Figure 6a. Last, since the cross-section of the main potato stem was an irregular circular cross-section, a circle with an area equal to the cross-section of the potato stem specimen was fitted, and the diameter of the fitted circle was used to express the diameter of the specimen. The center of the fitted circle is located at the shape center of the potato stem specimen’s cross-section and the result is shown in Figure 6b.

The pixels of the potato stem cross-section and the pixels of the black area squares can be output from the program. The side length of the square block was 10mm and the area of the square blocks was 100 mm². The potato stem specimen cross-sectional area and the fitted circle diameter can be calculated from Equations (4) and (5).

$$S_1=\frac{P_1}{P_2}\cdot S_2$$

(4)

$$d_1=\sqrt[2]{\frac{S_1}{\pi }},$$

(5)

where S₁ represents the area of the potato stem specimen cross-section in m², P₁ represents the pixels of the potato stem specimen cross-section, P₂ represents the pixels of the area of the square block, and S₂ represents the area of the square block in m².

2.5. Equations of Calculating the Poisson’s Ratio and the Young’s Modulus

Poisson’s ratio is defined as the negative ratio of transverse strain to the corresponding axial strain resulting from axial stress below the proportional limit of the material. Poisson’s ratio, usually denoted by ν, can be expressed by Equation (6) [27]. Young’s modulus is the ratio of tensile or compressive stress to corresponding strain below the proportional limit. Young’s modulus, usually denoted by E, can be expressed by Equation (7) [28]. The schematic diagram of the specimen being compressed is shown in Figure 7. The specific equation for this experiment is shown in Equations (8) and (9).

$$v=-\frac{transverses strain}{axia strain}=-\frac{\epsilon \prime }{\epsilon }$$

(6)

$$E=\frac{\sigma }{\epsilon }=\frac{F/A}{\Delta l/l}$$

(7)

$$v=-\frac{\epsilon \prime }{\epsilon }=-\frac{\Delta d/d_a}{\Delta l/l_a}=-\frac{(d_a-d_b)/d_a}{(l_a-l_b)/l_a}$$

(8)

$$E=\frac{\sigma }{\epsilon }=\frac{F/A_a}{\Delta l/l_a}=\frac{F/A_a}{(l_a-l_b)/l_a}=\frac{(F_a-F_b)/A_a}{(l_a-l_b)/l_a}.$$

(9)

ν is the Poisson’s ratio, ε′ is the transverse strain, ε is the axial strain, E is the Young’s modulus in MPa, σ is the axial stress in N/mm², F is the axial force in N, is the cross-sectional area of the object in mm², Δl is the is the amount by which the length of the object changes in mm, l is the origin length of the object in mm, d_a is the fitted diameter of the object at point (a) in mm, d_b is the fitted diameter of the object at point (b) in mm, l_a is the length of the object at point (a) in mm, l_b is the length of the object at point (b) in mm, F_a is the axial force at point(a) in N, F_b is the axial force at point (b) in N, and A_a is the cross-sectional area of the object at point (a) in mm²_.

We used a vernier caliper (Pro’sKit PD1-151) with an accuracy of 0.001 inches to measure the distance between the two glass plates before the compression started. The l_a, l_b, F_a, and F_b can be obtained combined with the load-displacement curves from the universal testing machine. The A_a and A_b were obtained by contour recognition. The d_a and d_b were obtained by fitted circle.

3. Results

3.1. The Feasibility Verification of Poisson’s Ratio and Young’s Modulus Measurement Methods

To verify the feasibility of this measurement method, we measured the Poisson’s ratio and Young’s modulus of nitrile rubber (NBR). Five 12 mm diameter, 11 ± 1 mm height, 65 Shore ‘A’ nitrile rubber cylindrical specimens were taken for testing. Figure 8a showed the compression load-displacement curves of the NBR specimens. Figure 8b showed the contour finding of the cross-section of the NBR specimen and the square block.

The calculated parameters are shown in

Table 1. The calculation parameters of the Poisson’s ratio (ν) and Young’s modulus (𝐸).

Specimen No.	ν	E/MPa	la/mm	lb/mm	Aa/mm2	Aa/mm2	da/mm	db/mm	Fa/N	Fb/N
1	0.42	7.16	11.17	10.25	116.08	124.26	12.16	12.58	7.23	75.38
2	0.41	7.00	11.04	10.22	118.19	125.43	12.27	12.64	17.64	79.09
3	0.40	7.29	10.60	9.98	119.19	124.88	12.32	12.61	27.81	78.60
4	0.42	6.54	10.72	10.13	118.30	123.84	12.27	12.56	24.17	66.78
5	0.42	6.63	10.37	9.69	121.03	127.71	12.41	12.75	20.07	72.66

. The mean value of the Poisson’s ratio of nitrile rubber measured in this experiment was 0.41, with a coefficient of variation of 2.05%. The mean value of Young’s modulus was 6.92 MPa, with a coefficient of variation of 4.70%.

Sotomayor-del-Mora et al. [29] measured the Possion’s ratio of nitrile rubber via DIC creep testing. The Poisson’s ratio of nitrile rubber was finally stabilized at 0.43 under a sustained loading of 200 kPa at 25 °C. Due to the lack of creep test bench, this experiment could not be carried out for a long time with constant stress loading, and the Poisson’s ratio of nitrile rubber measured in this experiment was 0.41, which was slightly less than 0.43.

D. Murali Manohar et al. [30] measured the Young’s modulus of nitrile rubber with different hardnesses and fitted the following relationship based on experimental data, as in Equation (10). The Young’s modulus of nitrile rubber with a hardness value of 65 Shore ‘A’ is 6.87 MPa, calculated from Equation (10). The Young’s modulus of 65 Shore ‘A’ nitrile rubber is calculated to be 6.87 MPa. The mean Young’s modulus of nitrile rubber (NBR) measured in this paper was 6.92 MPa, which is closer to the results of D. Murali Manohar et al.

$$\log (E)=0.0308S-1.1653,$$

(10)

where E is the Young’s modulus of the nitrile rubber and S is the Shore ‘A’ hardness.

The Poisson’s ratio and Young’s modulus of nitrile rubber determined by this method are close to those in the existing literature, and the errors are within acceptable limits. Therefore, this method can be used to measure the axial Poisson’s ratio and axial Young’s modulus of potato stems, and provides a new idea for the determination of the Poisson’s ratio and Young’s modulus of viscoelastic materials.

3.2. Measurement Results and Analysis of the Axial Poisson’s Ratio and the Axial Young’s Modulus of Potato Main Stems

The mean values of the axial Poisson’s ratio of potato main stems at 0 cm, 10 cm, 20 cm and 30 cm above the ground were measured as 0.21, 0.28, 0.30 and 0.32; the mean values of axial Young’s modulus were 15.90 MPa, 12.38 MPa, 11.86 MPa and 11.28 MPa. The measurement results are shown in Figure 9. We can see that the Poisson’s ratio of the potato main stems showed an increasing trend with increasing height above the ground. The Young’s modulus of potato main stems showed a decreasing trend with increasing height above the ground. The letters ‘a’ and ‘b’ in Figure 9 are the results of the Tukey multiple comparisons test.

The standard deviation may originate from the following. Each potato plant varied in soil environment, light environment, moisture and other growing conditions, which results in inconsistent tissue strength from plant to plant. Potato stems were irregular in cross-section, not standard circular cross-section, and different cross-sectional structures may bring about changes in mechanical properties. The location of the lateral growth of the main potato stalk differed from plant to plant, and the tissues near the lateral branches may have changed and brought about different mechanical behavior. The total plant height of the stems varied from plant to plant, so the development status at the same height above ground may not be the same. In addition, measurement errors need to be taken into account. Therefore, the measurement results at the same height also show some differences.

For this reason, it is necessary to examine whether different heights above the ground had a significant effect on the axial Poisson’s ratio and axial Young’s modulus of the potato main stems. The testing homogeneity of variances and the analysis of variances performed on the data are as follows.

According to

Table 2. The testing homogeneity of variances.

	Levin Statistics	df 1	df2	Sig.
axial Poisson’s ratio	0.720	3	36	0.547
axial Young’s modulus	1.408	3	36	0.256

, the P value of the axial Poisson’s ratio is 0.547 and the p value of the axial Young’s modulus is 0.256. In general, the testing of homogeneity of variances can be considered valid when the p value is larger than 0.05, and the analysis of variance can be continued.

From

Table 3. The analysis of variances.

		Sum of Squares	df	Mean Square	F	Sig.
axial Poisson’s ratio	Between Groups	0.067	3	0.022	8.547	0.000
	Within Groups	0.093	36	0.003
	Total	0.160	39
axial Young’s modulus	Between Groups	129.457	3	43.152	5.367	0.004
	Within Groups	289.428	36	8.040
	Total	418.885	39

, the F value of axial Poisson’s ratio is 8.547 and the p value is 0.000. The F value of axial Young’s modulus is 5.367 and the p value is 0.004. In general, a p value less than 0.05 means that the factor has a significant effect on the results. Therefore, different heights above the ground had a significant effect on the axial Poisson’s ratio and the axial Young’s modulus of the potato main stems.

To further explore the difference between the two groups, a Tukey multiple comparisons test was performed on the data. The Tukey multiple comparisons test is often used for experiments with equal sample numbers in each group. The groups with small differences are indicated by the same letters, and the groups with large differences are indicated by different letters; the results are shown in

Table 4. The Turkey multiple comparison test.

The Height above the Ground	Number of Samples	Axial Poisson’s Ratio	Axial Young’s Modulus
0 cm	10	0.21 ± 0.04 b	15.90 ± 3.63 a
10 cm	10	0.28 ± 0.06 a	12.38 ± 1.99 b
20 cm	10	0.30 ± 0.04 a	11.86 ± 2.83 b
30 cm	10	0.32 ± 0.06 a	11.28 ± 2.64 b

The letters a and b are the results of the Turkey multiple comparison test.

.

As seen in Table 4, the difference between 0 cm above the ground and other heights above the ground was extremely significant. The differences were smaller for heights of 10 cm, 20 cm and 30 cm above the ground, and the mean values of axial Poisson’s ratio and axial Young’s modulus varied almost linearly with increasing height. The reason for this phenomenon may be related to the tissue structure of the potato stems. Figure 10 showed the cross section of the potato main stems at different heights above the ground.

As seen in Figure 10, the potato stem consists of the epidermis, the xylem and the pith. The chlorophyll content in the stems was low at 0 cm above the ground, and the cross section showed a yellow color. The chlorophyll content of the stems was higher at the other heights above the ground, and the cross-section of the stems showed a green color. In addition, compared with the stalks at other heights above the ground, the area of the epidermis and xylem at 0 cm above the ground accounted for a significantly larger proportion of the stem, while the area of the pith occupied a smaller proportion. This may be due to different degrees of lignification at different heights above the ground. Lignin deposition in plant cell walls and provides rigidity to the plant tissues, which is one of the mechanisms that allowed the development of upright plants adapted to a terrestrial habitat [31]. The lignification phenomenon is also reflected in the sharp changes of the Young’s modulus. As seen in Figure 9, the Young’s modulus of the potato main stem at 0 cm above ground height was significantly greater than that at 10 cm. This is consistent with the study by Gibson [32], which indicated that lignin can have a significant reinforcing effect on any cell wall and provide additional Young’s modulus. That means the stems are stiffer at 0 cm above the ground and the Poisson’s ratio is significantly smaller than other heights above the ground. This phenomenon is consistent with the results of Malekabadi et al. [33] who measured a smaller Poisson’s ratio for the harder onion varieties.

The area share of the epidermis, xylem and pith of the stem cross-section at 10 cm, 20 cm and 30 cm above the ground were not significantly different, so their axial Poisson’s ratio and axial Young’s modulus were not significantly different. However, the structure of the pith gradually loosens, and a through-hole even appeared in the center of the pith, reducing the ability to resist deformation when subjected to axial load. Therefore, the mean value of axial Poisson’s ratio showed a linear increase with height above ground, and the mean value of axial Young’s modulus showed a linear decrease with height above ground.

4. Discussion

The measurement of Poisson’s ratio and Young’s modulus has been a popular research direction in materials science, and the commonly used methods for measuring Poisson’s ratio and Young’s modulus are mainly divided into three categories: mechanical, acoustic and optical methods. Currently, the Poisson’s ratio and the Young’s modulus measurements of commonly used industrial materials such as metals, glass, and rubber have been studied extensively. However, it is difficult to measure Poisson’s ratio and Young’s modulus because of the complex texture, irregular shape and individual differences of agricultural materials. Therefore, there are still few relevant measurement standards, especially for the Poisson’s ratio and Young’s modulus of stems, and there is no common measurement standard yet.

For this reason, this paper proposes a Poisson’s ratio and Young’s modulus measurement method for cylindrical viscoelastic materials with irregular cross-sections. By improving the traditional contactor of the universal testing machine and combining image processing technology, we are the first to obtain the real-time change of the cross-sectional area of the specimen during the axial compression. This method has a good measurement effect and operability for the measurement of the axial Poisson’s ratio and axial Young’s modulus of stem-like agricultural materials. However, this method also has some limitations. In agricultural materials, in addition to stems of cylindrical viscoelastic materials, spherical viscoelastic materials are also extremely common, such as apples, sorbets and potatoes. This method is not suitable for the Poisson’s ratio and the Young’s modulus measurement of spherical agricultural materials.

The equations for calculating Young’s modulus of spherical viscoelastic materials is defined by the ASAE S368.4 standard [34]. This equation is adapted from the theory of contact between convex surfaces made of elastic materials. Voicu et al. [35] measured the Young’s modulus of wheat seeds based on this standard, which was calculated as in Equation (11).

$$E=\frac{0.338F(1-{\mu }^2)k_g^{3/2}}{D^{3/2}}{\left[\frac{1}{R^{\prime }}+\frac{1}{R_1^{\prime }}\right]}^{1/2},$$

(11)

where E is the Young’s modulus in Mpa, k_g is the coefficient which depends on the geometrical characteristics of wheat seeds, F is the compression force in N, D is the seed deformation in m, ν is the Poisson’s ratio, and R′ and R′₁ are the small and large radius of the curvature of convex surface seed in contact with the flat surface in m.

In Equation (10), the compression force ($F$) and the seed deformation ($D$) can be directly put out from the universal testing machine. The measurement of R′ and R′₁ is the critical part, which greatly influenced the result. According to the standard (ASAE S368.4), also presented by Shelef et al. [36], the R′ and R′₁ in Figure 11 can be calculated using Equations (12) and (13).

$$R\prime \approx \frac{H}{2}$$

(12)

$$R\prime \approx \frac{H^2+\frac{L^2}{4}}{2H},$$

(13)

where $H$ is the seed thickness in m and $L$ is seed length in undistorted in m.

Estimation of R′ and R′₁ based on the overall dimensions of the material may not accurately express the true radius of curvature of each specimen. When applying this criterion to measure the Young’s modulus of spherical agricultural materials, we can also reference the image processing and real-time contour recognition applied in this experiment to obtain the contour line function. Measurement errors of the R′ and R′₁, introduced by non-standard spherical surfaces, can be reduced by solving for the curvature of the contour line function at the contact point.

5. Conclusions

In this paper, we sampled the main stem of the Atlantic potato at 0 cm, 10 cm, 20 cm, and 30 cm above the ground and measured the axial Poisson’s ratio of 0.21, 0.28, 0.30, and 0.31; and the axial Young’s modulus of 15.90 MPa, 12.38 MPa, 10.82 MPa, and 9.90 MPa. It was found that the height above the ground of the stems had a significant effect on both the axial Poisson’s ratio and the axial Young’s modulus. The axial Poisson’s ratio and axial Young’s modulus of the stems at 0 cm above the ground were significantly different from those at other heights above the ground. The differences between the axial Poisson’s ratio and axial Young’s modulus of the stems at 10 cm, 20 cm and 30 cm were small, and their mean values basically varied linearly with the height above the ground. This study provides an important basic parameter reference for the design and simulation of the potato haulm killers and potato harvesters. The differences in axial Poisson’s ratio and axial Young’s modulus measured in this paper at different heights above the ground may be related to the lignification phenomenon. This also provides the macro-mechanical reference data for research on potato stems in botany and biomechanics.

We propose a new method for Poisson’s ratio and Young’s modulus measurement by improving the compression contactor of the universal testing machine and combining with image processing technology. The cross-sectional area and the diameter of the fitted circle can be calculated using image processing methods. Combining the load-displacement curves’ output with the universal testing machine, the axial Poisson’s ratio and axial Young’s modulus measurement can be calculated. The feasibility test was performed with 65 Shore ‘A’ nitrile rubber. The Poisson’s ratio was measured to be 0.41 and the Young’s modulus was 6.92 MPa, which is very close to those reported in the available literature. Therefore, the present method has good accuracy and provides new ideas for the Poisson’s ratio and the Young’s modulus measurement of columnar viscoelastic materials with irregular cross-sections. It also provides an accurate and operable measurement method for the axial Poisson’s ratio and the axial Young’s modulus of stem-like agricultural materials.