“BROCADE” METHOD FOR DETERMINATION OF THE STRAIN TENSOR ELEMENTS OF THIN BIOLOGICAL MATERIALS

Bożena Gładyszewska¹, Izabela Szczurowska¹, Dariusz Chocyk^2
1 Department of Physics, Agriculture University of Lublin, Poland
² Institute of Physics, Lublin University of Technology, Poland

ABSTRACT

In this paper a video-extensometer method for studying the mechanical properties of wet biological samples is presented. The method is based on the relationships between a unit elongation of linear element on the surface and strain tensor elements. The linear elements are represented by randomly deposited markers. The analysis of the changes of markers positions during uniaxial tension directly allows to determine the strain in thin coat layers. As a test case, the method was applied to analyze the strain of the dwarf bean (Phaseolus coccineus) seed coats. Young’s modulus and Poisson’s ratio were determined for different moisture content. Young’s modulus clearly decreases with increasing moisture content of the sample.

Key words: Strain tensor, Young’s modulus, Poisson’s ratio.

INTRODUCTION

The mechanical properties of seed coats, including strain tensor elements, Young’s modulus and Poisson’s ratio are of great importance to seed production and to storage. The dependence of seed mechanical properties on moisture content is one of the most important problems, particularly during grain drying [7, 8, 11]. Too big stresses arising during drying process may cause the seeds damage and lead to decline quality of seeds. Therefore, the knowledge of the mechanical properties is essential.

The extensometer methods have been used for measuring the strain for different kinds of materials [1, 5, 12]. Applying the CCD cameras causes the reviving of these methods [9]. The change of pattern deposited on the surface studied is observed under applied different load. The change of deposited pattern is directly related with the sample deformation. A system of lines is often used as a pattern deposited on sample surface. For some biological materials a discrete Fourier transform was applied to determine the strain tensor elements [2, 3, 4, 6]. Then a two-dimensional Fourier transform was used to analyze changes of thin metal nets during stretch of sample. The metal nets were deposited by vacuum sputtering on sample surface. In many cases however, due to the fact that samples are not dry, the deposition by vacuum method is impossible. Therefore, the modification of the method is necessary that will allow applying the video-extensometer method for wet samples.

In this paper a “brocade” technique for determination of strain tensor elements, Young’s modulus and Poisson’s ratio is presented. As a test case the method has been used to analyze the strain of the dwarf bean “KONTRA” (Phaseolus coccineus) seed coats. The Young's modulus and Poisson’s ratio were determined for different moisture content.

METHOD

In the presented method the change of the randomly deposited markers position on a sample surface is analyzed during uniaxial tension (fig. 1). Due to the fact that the sample is under uniaxial tension it is enough to determine the deformation state in tension plane. Microscopic images of some markers are transferred through a camera to a computer memory. Then the image analysis is performed to collect the changes of markers positions. Each pair of markers produces the linear elements. For each analyzed linear element the length and direction cosines are calculated. The direction cosines are calculated as relative to the system axes and related with the edge of the image. The relative change in the length of the linear element before and after deformation and direction cosines directly allow to determine the strain in thin coat layers.

The method bases on the relationships between a unit elongation of linear element λ and strain tensor elements [10]:

(1)

where ν_i and ν_j are direction cosines of linear elements before deformation. The ε_ij are the strain tensor elements. If we consider three linear elements, we get a system of linear equations that allows to obtain all deformation tensor elements:

(2)

The solution of this system of linear equations gives a tensor elements corresponding with the image edge coordinate system. However, the tensor elements in coordinate system rotated at angle φ relative to the coordinate system corresponding with the image edge can be expressed by [13]:

(3)

In the case when the rotated coordinate system axes overlap the principal stress axes, deformations in the principal directions have extreme values, but the shear deformation is equal zero. Under uniaxial tension the principal stress axes are parallel to the principal strain axes. That allows us to determine the angle between the coordinate system connected with the image edge and the coordinate system related to tension direction. The angle φ can be calculated from the expression:

(4)

Young’s modulus and Poisson’s ratio are expressed as follows:

(5)

(6)

EXPERIMENTAL

The dwarf bean (Phaseolus coccineus) seeds were used as a test case of the presented method. Young’s modulus and Poisson’s ratio were determined for samples with different moisture content.

The moisture content of wet solid body is defined as [11]:

(7)

where W_d  is dry mass of the sample mass (W) and W_w is mass of water. As W = W_d + W_w the moisture content may be always expressed as a percentage moisture:

(8)

Procedure of a sample preparation was as follows. First, seeds of the dwarf bean were placed in the water for 1 day until they wetted to minimum 60% (1.5 kg/kg) moisture. After taking seeds out of the water the coat samples were cut out of the seed covers. Next, the coats were weighted and measured. Sample dimensions were determined using a micrometer screw. Then depending on required moisture content the sample was dried. The time of this process depended on the assumed moisture content. In this work we studied samples with the moisture content from 25% (0.33 kg/kg) up to 55% (1.22 kg/kg).

Before the sample was installed in stress tester the graphite powder was sprayed on the sample to deposit the markers. The prepared sample thus was installed in a stress tester. During the sample tension the microscopic image of deposited markers was captured by a computer program via video-camera. The operation resolution of a captured image was 320×240 pixels. Values of ε’_x and ε’_y were determined using the method given above. The program also acquired the value of the tensile force. Knowing the cross-sectional area of sample and value of tensile force, a stress was calculated. After strain measurements the sample was weighted again to control the change of the sample mass. In order to determine the moisture content the sample was placed on moisture balances (RADWAG WPE 30S). The sample was dried up to dry mass and weighted (W_d).The exact weight (±0.1 mg) was obtained by the use of WPS72 balance.

RESULTS

Figure 2 shows results of the “brocade” method used to define Young’s modulus E and Poisson’s ratio ν of the exemplary sample with moisture content equal to 26.7% (0.36 kg/kg). As it is shown, the ε’_x increases (and ε’_y decreases) linearly with the tension σ. The Young’s modulus was derived from the slope of the ε’_x plot as a function of σ. For this sample we obtained E = 18.9 MPa and ν = 0.31. At the same time, we observe very important scatter of the Poisson’s ratio for studied samples. One should remember however that biological samples (as seed covers) are very inhomogeneous materials. This may be connected with occurrence of crosswise changes in sample under influence of tensile force. The changes can be attributed to progressive lengthwise bedding of cover fibers. Explanation of this effect will be possible after more complex research in different directions have been made. Similar results were also obtained for all samples.

In spite of the fact that no relation of the Poisson’s ratio versus tension σ was observed, it was still possible to calculate Young’s modulus E and Poisson’s ratio ν for samples having different moisture content M (see fig. 3 and 4). We observe important decrease of Young’s modulus with the increase of the moisture content. No clear dependence of the Poisson’s ratio versus moisture content has been observed. Such results may be considered as predictable. We would like however to stress that the presented “brocade” method allows us to obtain exact values of the Young’s modulus for any studied material. This is very important data for drying process and we believe that our method will be helpful in that case.

CONCLUSIONS

To conclude, the method presented here bases on the relationships between a unit elongation of linear element and strain tensor elements and allows to define tensor elements of deformations of studied material. The method has been applied for wet dwarf bean covers (Phaseolus coccineus) to determine important mechanical parameters as Young’s modulus and Poisson’s ratio.

One should note that the proposed method allows to avoid influence of tester clips on results. This is particularly important for parameters determination. Together with the method based on Fourier numerical analysis [6], both methods create complementary tools for studying mechanical properties of thin biological materials. The method allows direct connection of tensile force value with sample deformation. Results obtained should be considered as example of broader applications of the method. Research with the use of the applied method is currently being made also for other materials and their characteristic parameters and will be published soon.

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Accepted for print: 16.01.2007

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.