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Three-point bending: An alternative method to measure tensile properties in fruit and vegetables
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Postharvest Biology and Technology 48 (2008) 63–69
Three-point bending: An alternative method to measure tensile properties in fruit and vegetables Marvin J. Pitts ∗ , Denny C. Davis, Ralph P. Cavalieri 209 L. J. Smith Hall, Washington State University, Pullman, WA 99164-612, USA Received 6 June 2007; accepted 19 September 2007
Abstract Tensile mechanical properties of fruit and vegetable tissue are likely to have a significant effect on fruit and vegetable quality evaluations. Very few studies have been made of tensile material properties because sample preparation for uniaxial tensile testing of fruit and vegetable tissue is difficult. Three-point bending is an alternative experimental method to measure tensile elastic modulus. In this study a derivation of bending theory was developed and used in conjunction with a three-point bending procedure using digital image-based analysis to locate the neutral axis of the material. Mechanics of materials theory and concepts were used to determine a relationship between the location of the neutral axis and the ratio of compressive elastic modulus to tensile elastic modulus. The procedure to locate the neutral axis and the derivation to determine the tensile elastic modulus were verified using a homogeneous cork-based material which exhibited distinctly different compressive and tensile properties. Tensile elastic modulus measured using the bending apparatus agreed closely (within 1%) to tensile elastic modulus measured using a uniaxial tension apparatus. This experimental method is well suited to measure tensile properties in many fruit and vegetables. © 2007 Elsevier B.V. All rights reserved. Keywords: Fruit; Tissue; Tensile; Material properties; Elastic modulus
1. Introduction Firmness is a key factor in consumer acceptance of fruit, is measured as an indicator of maturity, and in many governmental jurisdictions, is a part of legislation governing shipment of fresh fruit. With firmness playing such an important role in fruit and vegetable quality, it is ironic that tissue mechanics, which govern firmness, is not well understood. Without this knowledge, development of destructive or nondestructive instruments to measure fruit firmness is hampered. Poorly understood tissue mechanics also leaves a potential for disagreement over the definition of firmness and subsequent disagreement about the firmness of a shipment of fruit. More knowledge about tissue mechanics is essential to a better understanding of firmness (Abbott, 1999). Identifying tissue material properties that predict apple firmness has been a long quest. Magness and Taylor first established their procedure and device in 1925. For most research and proposed instrumentation to measure tissue firmness, elastic modulus is the material property that researchers consider best
∗
Corresponding author. Tel.: +1 509 335 3243; fax: +1 509 335 2722. E-mail address: [email protected] (M.J. Pitts).
0925-5214/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.postharvbio.2007.09.025
correlates with firmness. While the elastic modulus can be measured using an excised sample of tissue, most research has focused on finding a nondestructive method of measuring the elastic modulus in intact fruit. Since the elastic modulus is related to the response of the fruit to many external types of mechanical excitation, a number of methods have been proposed to nondestructively measure elastic modulus, and by extension, fruit firmness (Abbott et al., 1997; Hung et al., 2001; Lu and Abbott, 2004). The tensile and compressive elastic moduli are assumed to be equal in the methods described below. This assumption is not valid, and could lead to lower correlations between the test method and the Magness and Taylor test. One method for measuring tissue elastic firmness deforms the fruit enough to measure the elastic modulus, but not enough to bruise the tissue. Delwiche and Sarig (1991) demonstrated some success in relating apple firmness to the failure stress of apple tissue, but their instrument bruised the apple tissue. Other researchers have tried to find the appropriate probe and distance to deflect the tissue for this type of measurement (Harker et al., 1996; Lu et al., 2006). Because the speed of sound in a material is a function of the material’s elastic modulus, numerous researchers tried to relate the speed of sound in the apple tissue to firmness without
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success. Abbott and her group studied relationships between apple firmness and sonic transmission in apple tissue (Abbott et al., 1992; Abbott, 1994; Abbott and Liljedahl, 1994), as did Garrett and Furry (1972). Ultrasound has been a particular frequency range of interest, and many researchers have tried to determine the elastic modulus of fruit tissue by either the speed of an ultrasonic pulse (Sarkar and Wolfe, 1983; Mizrach et al., 1989), or the attenuation of the signal strength of an ultrasound signal (Mizrach et al., 1997). The natural (resonant) frequency and modes of vibration of a fruit are governed in part by the size and shape of the fruit, and by the elastic modulus of the fruit tissue. Finney tried to relate the vibrational characteristics of fruit to the elastic modulus of the tissue and to the firmness of the fruit (Finney, 1970, 1971, 1972). Armstrong tried to measure apple firmness using the resonant frequency of the apple (Armstrong et al., 1990; Armstrong and Brown, 1991). De Baerdemaeker and his group conducted numerous analytical studies trying to relate apple firmness to the modal response of an apple (Cooke and Rand, 1973; Chen and de Baerdemaeker, 1993; Belie et al., 2000). Abbott (2004) summarized these results and others. The various measures of ‘goodness of fit’ between the various devices and sensory panels were about 80–85% effective. Almost 80 years after Magness and Taylor developed their penetrometer, Harker et al. (2002) stated that there is still not a satisfactory firmness sensor that can replace a human sensory panel. The majority of past research on apple tissue mechanics has been based on engineering mechanics-of-materials concepts commonly used with engineering materials such as steel. A key assumption for engineering materials is that the material responds similarly to both compressive and tensile forces; thus, material properties of steel can be measured using either compressive or tensile forces. The assumption of equal elastic properties in tension and compression is not true for most biological materials, and apple tissue in particular. The data of Harker and Hallett (1992, 1997) indicate a lower apple tissue failure stress in tension than in compression, and a higher elastic modulus in tension than in compression. Analysis of apple tissue failure during biting or cutting with a knife suggests that a dominant failure mode is a combination of tensile and shear failure caused by the wedge action of the knife blade or tooth. So why aren’t tensile material properties used to characterize firmness? Because tensile properties are very difficult to measure in apple tissue. Conducting compression tests only requires cutting a block or cylinder of tissue with parallel ends. Comparatively, conducting tensile tests requires grasping the tissue so that (1) there is no slipping at the grip and (2) the tissue sample does not fail due to bending forces generated by the grip holding the tissue sample (Harker and Hallett, 1992). Compounding the gripping problem are (1) the moistness of the tissue and (2) the short time available to mount and test the sample before the tissue begins to change after removal from the fruit. There is a need for a quick and simple procedure to determine tensile material properties in biological materials. In this study, we attempted to develop a test for measuring both tensile and compressive elastic moduli from a single beamshaped sample with a three-point load. Our objectives are to
(1) develop an analytical relationship between compressive and tensile elastic moduli and the location of the neutral axis in bending, and (2) verify the analytical relationship experimentally using a material with unequal compressive and tensile elastic moduli. 2. Mathematical derivation The theoretical development below follows a well established derivation relating normal stress to bending loads such as is done in Hibbeler (2003). Two assumptions are required for this analysis: (1) the beam material is elastic in compression and in tension, and (2) cross-sectional areas of the beam remain planar and perpendicular to the neutral axis. 2.1. Strain and geometry Consider the beam in Fig. 1. Under the moment induced by force P, cross sections of the beam, such as the one pictured in Fig. 2, will be compressed on the top and elongated on the bottom. A graph of the strain along the height of the cross-section (Fig. 3) is a straight line with the maximum compressive strain at the top, and maximum tensile strain at the bottom. Note that the neutral axis is located at the point where the strain is zero. If the compressive elastic modulus is not equal to the tensile elastic modulus, the neutral axis will not pass through the geometric center of the cross-section. Because the cross-section remains planar, the strain will vary linearly as indicated in Eq. (1): εyc = −
yc εc max Cc
and
εyt = −
yt εt max Ct
(1)
where yc , distance from the neutral axis toward the compressive side of the beam, yt , distance from the neutral axis toward the tensile side of the beam, Cc , distance from the neutral axis to the compression edge of the beam, Ct , distance from the neutral axis to the tensile edge of the beam, εcmax , strain at the compressive edge of the beam, εtmax , strain at the tensile edge of the beam εyc strain in the beam at a distance yc from the neutral axis, εyt strain in the beam at a distance yt from the neutral axis.
Fig. 1. Beam in bending.
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2.2. Elastic modulus and geometry If we consider the stress distribution along the cross section in Fig. 3, it becomes apparent that for the cross-section to remain stationary in space, the internal compressive and tensile forces resulting from the respective stresses must offset each other. Substituting stress for the forces, and introducing the cross-sectional area yields a relationship between stress and geometry. First the internal tensile and compressive forces must be equal: dFc = dFt (4) Ac
At
where Fc , differential compressive force; Ft , differential tensile force; At , cross-sectional area under tension; Ac , cross-sectional area under compression. Substituting F = σA into Eq. (4); σyt dAt = σyc dAc (5) At
We can introduce the maximum compressive and tensile stress by substituting Eq. (2) into Eq. (5): −yc −yt σc max dAc = σt max dAt (6) cc ct Ac At
Fig. 2. Deformation in a beam element under bending.
Noting that the elastic moduli Ec and Et are constant, and σ = Eε, we can substitute stress (σ) for strain (ε): σyc = −
yc σc max Cc
and σyt = −
yt σt max Ct
(2)
where σyc , the compressive stress at a distance y from the neutral axis, σyt , the tensile stress at a distance y from the neutral axis. In Fig. 3 the graph of the strains in the cross-section and the dimensions of the cross-section form similar triangles that define a relationship between the distance from the neutral axis to the surface (cc and ct ) and the amount of compressive and tensile strain (εc and εt ): εc cc = ct εt
Ac
(3)
Because σ cmax , σ tmax , cc , and ct are constant across their respective areas, we can move them outside the integration: σcmax σt max yc dAc = yt dAt (7) cc ct Ac At What remains within the integration is cross-section geometry, and is the first moment of area, Q. Eq. (7) can be simplified to: σc max σt max Qc = Qt (8) cc ct Eq. (8) is significant because it demonstrates that the ratio of compressive to tensile stress can be determined if we can determine the location of the neutral axis. Eq. (8) also is a starting point for relating the elastic modulus to the cross-section geometry. Substituting the basic definition between normal stress and strain (σ = Eε) into Eq. (8): Ec εc Qc Et εt Qt = cc ct
(9)
Gathering the strain terms to one side of the equation, substituting Eq. (3) into Eq. (9), and simplifying the equation yields a relationship between E and Q: Ec Qc εc Et Qt = c c εt ct cc εc subst for ct εt Ec Qc = Et Qt Ec Qt = Et Qc Fig. 3. Strain variation along the beam cross section.
(10)
This equations states that the ratio of the compressive to tensile elastic modulus is equal to the ratio of the first moment of
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area for the tensile to compressive areas. This tells us that if we can identify the areas under compression and under tension, and measure Ec using simple compression testing, we can derive Et . 3. Materials and methods Experiments were conducted to test the validity of the derived equation for materials with different elastic moduli in tension and compression. The test compared the value for the tensile elastic modulus measured by a bending test to the value of tensile elastic modulus measured by a traditional tensile test. Manufactured cork wine closures (ALTOP closures manufactured by Oeneo Closures USA, Napa CA) were selected as test material because the product has substantially different compressive and tensile elastic moduli (tensile E is higher than compressive E), because of the consistency between material used in different closures, and desirable small pore size in these closures. Samples were tested in compression, bending and tension to compute the compressive elastic modulus (required in the bending technique), the location of the neutral axis of samples placed in bending, and the tensile elastic modulus.
the sample’s stress versus strain curve. The mean compressive elastic modulus for the eight replicates was used in later computations.
3.1.1. Compressive test
3.1.2. Tensile test
Eight replicates (closures) were deformed in a parallel plate testing device (Instron Model 1350 with a 400 N capacity load cell). The closures were shaped into cylinders 20 mm in diameter and 40 mm in length. The ends of the samples were sanded to ensure that ends were parallel to each other and perpendicular to the cylinder sides (Fig. 4). The diameter and length of each sample was measured prior to testing. The cylinders were deformed 24 mm (nominally 60% strain) at a rate of 0.485 mm/s. One thousand data points (deformation and force) were collected for each sample. From the deformation and force data, compressive engineering stress and strain was computed. Because cork’s Poisson’s Ratio is very small (about 0.01), engineering stress and strain are very close to true stress and strain values. The compressive elastic modulus for each sample was computed from the linear region of
Nine replicates (straight-sided closures 20 mm in diameter and 40 mm in length) were placed in a lathe and the middle 30 mm were turned down to a nominal diameter of 10 mm. A 20 mm fillet was turned into the transition areas between the middle 30 mm section and the ends of the cork to minimize stress concentration in the transitional region. The diameter and length of the turned section of each sample was measured and recorded for use in computing tensile stress and strain. The ends of the sample were glued to the top and bottom plates of the testing device with a cyanoacrylate-based glue (Fig. 5). The Instron 1350 device used a hydraulic piston to move the lower plate, ensuring that only an axial force (no rotational or bending loading) was applied to the sample. The cork sample was pulled to failure at a rate of 0.485 mm/s. Deformation and applied force was recorded at 0.0125 mm increments of deformation. From the deformation and force data, engineering stress and strain were computed using the diameter and length of the turned down section of each sample. Tensile elastic modulus for each sample was computed from the linear region of the sample’s stress vs. strain curve. The mean tensile elastic modulus of the nine replicates was used in later computations.
Fig. 5. Tensile testing with hour-glass shaped cork.
3.1.3. Bending test
Fig. 4. Compression testing with cylinder of cork.
Ten replicates of cork samples (blocks 10 mm × 10 mm × 50 mm) were tested in bending. Each sample was laid on a bending jig (Fig. 6) on the lower plate of the Instron device. A 10 mm diameter horizontal rod was attached to the load cell on the fixed upper crossbeam. The bending jig was located so that the rod was perpendicular to the long axis of the cork sample and came in contact with the center of the sample (Fig. 7). A digital camera (Olympus 2100C, resolution 320 × 240 pixels) was mounted on
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Fig. 6. A block laid on the bending jig.
the lower plate, and moved with the sample. The center of the sample was deformed 5 mm by the horizontal rod at a rate of 0.485 mm/s. During the deformation, the camera recorded the deformation in a 15 frame/s movie. The amount of deformation and force was recorded at deformation intervals of 0.005 mm. Following each test, two frames were selected from the movie; one prior to deformation, the second at about 3.5 mm of deflection. Using image processing software (Adobe Photoshop) the images were inverted and then the deformed image was subtracted from the undeformed image (Fig. 7). The vertical edges of the sample in the subtracted image formed a white and a black triangle, joined at a point. This point is the pivot point of the vertical side, and thus the point is on the sample’s neutral axis. The ratio of the number of pixels from the bottom of the sample to the point on the neutral axis to the number of pixels from the neutral axis point to the top of the sample is the same as the ratio of ct to cc . 4. Results 4.1.1. Compression test A typical stress versus strain curve is shown in Fig. 8. The compressive elastic modulus for each sample was computed as the average tangent elastic modulus between the strain values of 5–20% strain. The average compressive elastic modulus of the 8 replicates was 2.40 MPa, with a standard deviation of 0.268 MPa. A high compressive strain rate was chosen because the material underwent more compressive strain than tensile strain during the bending. This conclusion falls out directly from the observation that the neutral axis is closer to the tensile side of the test specimen. If the cross-sections of the specimen are to remain planar (one of the assumptions for the derivation), and the neutral axis is displaced toward the tensile side of the specimen, then the compressive strain at the compression side of the specimen has to be greater than the tensile strain at the tension side. In this situation, it was appropriate to use a larger range of compressive strain to compute the average compressive elastic modulus.
Fig. 7. Typical image sequence used to determine the neutral axis. The right image is the difference between the left and middle images.
4.1.2. Tensile test A typical stress versus strain curve for tension is shown in Fig. 9. The tensile elastic modulus for each sample was computed as the average tangent elastic modulus between the strain values of 0–10% strain. The average tensile elastic modulus of the nine replicates was 14.36 MPa, with a standard deviation of 3.846 MPa. The ratio of the first moment of area for the tensile to compressive areas (Qc /Qt ) for each slab in the bending test was computed from the difference images (such as the one in Fig. 7(c), by first counting the number of pixels from the bottom of the slab to the point of the white triangle (pt ) and counting the pixels from the point of the white triangle to the top of the beam (pc ). A pixel-
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Fig. 8. Typical stress vs. strain curve for a cork sample in compression.
to-mm conversion factor was computed from the sum of pt and pc and the measured height (in mm) of the slab. The values of pt and pc were multiplied by this pixel-to-mm conversion factor to determine ct and cc , respectively. The first moments of area for the tensile and compressive regions were computed using Eq. (11): Q = c2
(width of the slab) 2
(11)
Finally, the ratio of the first moment of area for the tensile to compressive areas (Qc /Qt ) was computed by dividing the respective first moments of area. The average value among the nine slabs tested for this ratio was 0.166, with a standard deviation of 0.00313. The tensile elastic modulus computed from the bending test was determined by dividing the average compressive elastic modulus (2.40 MPa) by the average ratio of the first moments of compressive and tensile areas (0.166). The tensile elastic modulus computed from the bending test was 14.47 MPa. The average tensile elastic modulus measured from tensile loading of the cork was 14.36 MPa. The experimental error in the two methods of computing tensile elastic modulus was 0.67%. 5. Discussion The variation in the elastic modulus results was quite low for a biological material. The wine closure material used in this test was unusually homogenous for a biological material because the
Fig. 9. Typical stress vs. strain curve for a cork sample in tension.
Fig. 10. Bending procedure applied to apple tissue (left) and pear tissue (right).
stopper was manufactured by gluing ground cork into the stopper shape. The manufactured closure mimicked the compressive properties of biological materials well in that it had a low compressive elastic modulus, a large compressive failure strain, and exhibited plastic failure in compression. The material also simulated the characteristics of the tensile properties of biological materials in that the material had a high tensile elastic modulus, low failure strain and exhibited brittle failure in tension. The cork material worked well to verify the derivation described in this paper, but it is not a suitable surrogate for fruit or vegetable tissue. The cork material has a very low Poisson’s Ratio, and a very high failure strain in both tension and compression. These characteristics are not typical of fruit or vegetable tissue. Successful use of this technique requires a clear indication of the neutral axis position as seen in the difference of the two images. The more the material is deformed, the easier it will be to identify the point of the triangles (pivot point, or location of the neutral axis). But the deformation must not be so large to violate the assumptions made at the beginning of the derivation: (1) the beam material is elastic in compression and in tension, and (2) cross-sectional areas of the beam remain planar and perpendicular to the neutral axis). Even if the material remains elastic (not bruised) at excessive deformations, the cross-sectional areas will no longer remain planar and perpendicular to the neutral axis. For tissue tested to date (apple, pear), adjusting the length and/or height of the tissue sample was sufficient to clearly identify the point of the triangles (Fig. 10). For materials with higher elastic modulus (tension or compression), or a lower compressive or tensile yield strength, locating the point of the triangles
M.J. Pitts et al. / Postharvest Biology and Technology 48 (2008) 63–69
will be more difficult. As a point of reference, the wine closure material was less stiff and weaker than most reported elastic modulus and compressive yield values for apples (Mohsenin, 1996). The precision of this method is dependent on the overall resolution of the imaging hardware. The bending samples used in this study were about 87 pixels in height. The effective resolution of the imaging system was 1/87 or about 1%. Newer digital cameras are likely to contain more memory, and as a result more vertical pixel resolution, but the overall resolution is a combination of the number of pixels in the vertical direction and the camera optics. The number of vertical pixels in the edge of a sample is also dependent on the telephoto, zoom and macro capabilities of the camera. A digital video camera rather than a digital camera will make a better video record of the bending (frames every 30 s during the deflection of the sample), but most commercial digital video cameras have only 240 pixels in the vertical direction. If a video camera is used to record the sample deflection, it is very important that the sample contain as many vertical pixels as possible. 6. Conclusion Tensile mechanical properties of fruit and vegetable tissue are likely to differ from their compressive mechanical properties, and properties in both load directions are likely to affect fruit and vegetable quality evaluations. Sample preparation for uniaxial tensile testing of fruit and vegetable tissue is difficult due to fixture limitations. This study has shown that bending loading in conjunction with a uniaxial compression loading can be used to determine tensile mechanical properties. The experimental errors in measuring tensile elastic modulus using this procedure derive from camera resolution, and are much lower percentages than random variations in biological material properties typically reported. These results indicate that the methods used in this paper merit wide adoption by researchers seeking to measure elastic properties of biological materials in both tension and compression. References Abbott, J.A., 1994. Firmness measurement of freshly harvested ‘Delicious’ apples by sensory methods, sonic transmission, Magness-Taylor, and compression. J. Amer. Soc. Hort. Sci. 119, 510–515. Abbott, J.A., Liljedahl, L.A., 1994. Relationship of sonic resonant frequency to compression tests and Magness-Taylor firmness of apples during refrigerated storage. Trans. ASAE 37, 1211–1215. Abbott, J.A., 1999. Quality measurement of fruits and vegetables. Postharvest Biol. Technol. 15, 207–225. Abbott, J.A., 2004. Textural quality assessment for fresh fruits and vegetables. Qual. Fresh Process. Foods 542, 265–279. Abbott, J.A., Affeldt, H.A., Liljedahl, L.A., 1992. Firmness measurement of stored ‘Delicious’ apples by sensory methods, Magness-Taylor, and sonic transmission. J. Amer. Soc. Hort. Sci. 117, 590–595 .
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Abbott, J.A., Lu, R., Upchurch, B.L., Stroshine, R.L., 1997. Technologies for nondestructive quality evaluation of fruits and vegetables. Hort. Rev. 20, 1–120. Armstrong, P., Zapp, H.R., Brown, G.K., 1990. Impulsive excitation of acoustic vibrations in apples for firmness determination. Trans. ASAE 33, 1353–1359. Armstrong, P.R., Brown, G.K., 1991. Apple firmness sorting using a nondestructive acoustic technique. ASAE Paper No. 916044. Belie, N., Schotte, S., Coucke, P., de Baerdemaeker, J., 2000. Development of an automated monitoring device to quantify changes in firmness of apples during storage. Postharvest Biol. Technol. 18, 1–8. Chen, H., de Baerdemaeker, J., 1993. Finite-element-based modal analysis of fruit firmness. Trans. ASAE 36, 1827–1833. Cooke, J.R., Rand, R.H., 1973. A mathematical study of resonance in intact fruits and vegetables using a 3-media elastic sphere model. J. Agric. Eng. Res. 18, 141–157. Delwiche, M., Sarig, Y., 1991. A probe impact sensor for fruit firmness measurement. Trans. ASAE 34, 187–192. Finney, E.E., 1970. Mechanical resonance within Red Delicious apples and its relation to fruit texture. Trans. ASAE 13, 177–180. Finney, E.E., 1971. Dynamic elastic properties and sensory quality of apple fruit. J. Texture Stud. 2, 62–74. Finney, E.E., 1972. Vibration techniques for testing fruit firmness. J. Texture Stud. 3, 263–283. Garrett, R.E., Furry, R.B., 1972. Velocity of sonic pulses in apples. Trans. ASAE 15, 770–774. Harker, F.R., Hallett, I.C., 1992. Physiological changes associated with development of mealiness of apple fruit during storage. HortScience 27, 1291– 1294. Harker, F.R., Hallett, I.C., 1997. Texture of parenchymatous plant tissue: a comparison between tensile and other instrumental and sensory measurements of tissue strength and juiciness. Postharvest Biol. Technol. 11, 63–72. Harker, F.R., Maindonald, J.H., Jackson, P.J., 1996. Penetrometer measurement of apple and kiwifruit firmness: Operator and instrument differences. J. Amer. Soc. Hort. Sci. 121, 927–936. Harker, F.R., Maindonald, J., Murray, S.H., Gunson, F.A., Hallett, I.C., Walker, S.B., 2002. Sensory interpretation of instrumental measurements. 1. Texture of apple fruit. Postharvest Biol. Technol. 24, 225–239. Hibbeler, R.C., 2003. Mechanics of Materials, fifth ed. Prentice Hill, Pearson Education Inc., Upper Saddle River, NJ, ISBN: 0-13-008181-7. Hung, Y., Prussia, S., Ezeike, G.O.I., 2001. Chapter 7 Firmness-measurement methods. In: Nondestructive Food Evaluation: Techniques to Analyze Properties and Quality. Marcel Dekker, Inc, New York, pp. 243–285. Lu, R., Abbott, J.A., 2004. Chapter 5 Force/deformation techniques for measuring texture. In: Texture in Food: vol. 2: Solid Foods. Woodhead Publishing Limited, England, 109–145. Lu, R., Srivastava, A.K., Ababneh, H.A.A., 2006. Finite element analysis and experimental evaluation of bioyield probes for measuring apple fruit firmness. Trans. ASAE 49, 123–131. Mizrach, A., Galili, N., Rosenhouse, G., 1989. Determination of fruit and vegetable properties by ultrasonic excitation. Trans. ASAE 32, 2053–2058. Mizrach, A., Flitsanov, U., Fuchs, Y., 1997. An ultrasonic non-destructive method for measuring maturity of mango fruit. Trans. ASAE 40, 1107– 1111. Mohsenin, N.N., 1996. Physical Properties of Plant and Animal Materials. Gordon and Breach Publishers, Amsterdam, The Netherlands. ISBN: 0-67721370-0. Sarkar, N., Wolfe, R.R., 1983. Potential of ultrasonic measurements in food quality evaluation. Trans. ASAE 26, 624–629.
Postharvest Biology and Technology 48 (2008) 63–69
Three-point bending: An alternative method to measure tensile properties in fruit and vegetables Marvin J. Pitts ∗ , Denny C. Davis, Ralph P. Cavalieri 209 L. J. Smith Hall, Washington State University, Pullman, WA 99164-612, USA Received 6 June 2007; accepted 19 September 2007
Abstract Tensile mechanical properties of fruit and vegetable tissue are likely to have a significant effect on fruit and vegetable quality evaluations. Very few studies have been made of tensile material properties because sample preparation for uniaxial tensile testing of fruit and vegetable tissue is difficult. Three-point bending is an alternative experimental method to measure tensile elastic modulus. In this study a derivation of bending theory was developed and used in conjunction with a three-point bending procedure using digital image-based analysis to locate the neutral axis of the material. Mechanics of materials theory and concepts were used to determine a relationship between the location of the neutral axis and the ratio of compressive elastic modulus to tensile elastic modulus. The procedure to locate the neutral axis and the derivation to determine the tensile elastic modulus were verified using a homogeneous cork-based material which exhibited distinctly different compressive and tensile properties. Tensile elastic modulus measured using the bending apparatus agreed closely (within 1%) to tensile elastic modulus measured using a uniaxial tension apparatus. This experimental method is well suited to measure tensile properties in many fruit and vegetables. © 2007 Elsevier B.V. All rights reserved. Keywords: Fruit; Tissue; Tensile; Material properties; Elastic modulus
1. Introduction Firmness is a key factor in consumer acceptance of fruit, is measured as an indicator of maturity, and in many governmental jurisdictions, is a part of legislation governing shipment of fresh fruit. With firmness playing such an important role in fruit and vegetable quality, it is ironic that tissue mechanics, which govern firmness, is not well understood. Without this knowledge, development of destructive or nondestructive instruments to measure fruit firmness is hampered. Poorly understood tissue mechanics also leaves a potential for disagreement over the definition of firmness and subsequent disagreement about the firmness of a shipment of fruit. More knowledge about tissue mechanics is essential to a better understanding of firmness (Abbott, 1999). Identifying tissue material properties that predict apple firmness has been a long quest. Magness and Taylor first established their procedure and device in 1925. For most research and proposed instrumentation to measure tissue firmness, elastic modulus is the material property that researchers consider best
∗
Corresponding author. Tel.: +1 509 335 3243; fax: +1 509 335 2722. E-mail address: [email protected] (M.J. Pitts).
0925-5214/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.postharvbio.2007.09.025
correlates with firmness. While the elastic modulus can be measured using an excised sample of tissue, most research has focused on finding a nondestructive method of measuring the elastic modulus in intact fruit. Since the elastic modulus is related to the response of the fruit to many external types of mechanical excitation, a number of methods have been proposed to nondestructively measure elastic modulus, and by extension, fruit firmness (Abbott et al., 1997; Hung et al., 2001; Lu and Abbott, 2004). The tensile and compressive elastic moduli are assumed to be equal in the methods described below. This assumption is not valid, and could lead to lower correlations between the test method and the Magness and Taylor test. One method for measuring tissue elastic firmness deforms the fruit enough to measure the elastic modulus, but not enough to bruise the tissue. Delwiche and Sarig (1991) demonstrated some success in relating apple firmness to the failure stress of apple tissue, but their instrument bruised the apple tissue. Other researchers have tried to find the appropriate probe and distance to deflect the tissue for this type of measurement (Harker et al., 1996; Lu et al., 2006). Because the speed of sound in a material is a function of the material’s elastic modulus, numerous researchers tried to relate the speed of sound in the apple tissue to firmness without
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success. Abbott and her group studied relationships between apple firmness and sonic transmission in apple tissue (Abbott et al., 1992; Abbott, 1994; Abbott and Liljedahl, 1994), as did Garrett and Furry (1972). Ultrasound has been a particular frequency range of interest, and many researchers have tried to determine the elastic modulus of fruit tissue by either the speed of an ultrasonic pulse (Sarkar and Wolfe, 1983; Mizrach et al., 1989), or the attenuation of the signal strength of an ultrasound signal (Mizrach et al., 1997). The natural (resonant) frequency and modes of vibration of a fruit are governed in part by the size and shape of the fruit, and by the elastic modulus of the fruit tissue. Finney tried to relate the vibrational characteristics of fruit to the elastic modulus of the tissue and to the firmness of the fruit (Finney, 1970, 1971, 1972). Armstrong tried to measure apple firmness using the resonant frequency of the apple (Armstrong et al., 1990; Armstrong and Brown, 1991). De Baerdemaeker and his group conducted numerous analytical studies trying to relate apple firmness to the modal response of an apple (Cooke and Rand, 1973; Chen and de Baerdemaeker, 1993; Belie et al., 2000). Abbott (2004) summarized these results and others. The various measures of ‘goodness of fit’ between the various devices and sensory panels were about 80–85% effective. Almost 80 years after Magness and Taylor developed their penetrometer, Harker et al. (2002) stated that there is still not a satisfactory firmness sensor that can replace a human sensory panel. The majority of past research on apple tissue mechanics has been based on engineering mechanics-of-materials concepts commonly used with engineering materials such as steel. A key assumption for engineering materials is that the material responds similarly to both compressive and tensile forces; thus, material properties of steel can be measured using either compressive or tensile forces. The assumption of equal elastic properties in tension and compression is not true for most biological materials, and apple tissue in particular. The data of Harker and Hallett (1992, 1997) indicate a lower apple tissue failure stress in tension than in compression, and a higher elastic modulus in tension than in compression. Analysis of apple tissue failure during biting or cutting with a knife suggests that a dominant failure mode is a combination of tensile and shear failure caused by the wedge action of the knife blade or tooth. So why aren’t tensile material properties used to characterize firmness? Because tensile properties are very difficult to measure in apple tissue. Conducting compression tests only requires cutting a block or cylinder of tissue with parallel ends. Comparatively, conducting tensile tests requires grasping the tissue so that (1) there is no slipping at the grip and (2) the tissue sample does not fail due to bending forces generated by the grip holding the tissue sample (Harker and Hallett, 1992). Compounding the gripping problem are (1) the moistness of the tissue and (2) the short time available to mount and test the sample before the tissue begins to change after removal from the fruit. There is a need for a quick and simple procedure to determine tensile material properties in biological materials. In this study, we attempted to develop a test for measuring both tensile and compressive elastic moduli from a single beamshaped sample with a three-point load. Our objectives are to
(1) develop an analytical relationship between compressive and tensile elastic moduli and the location of the neutral axis in bending, and (2) verify the analytical relationship experimentally using a material with unequal compressive and tensile elastic moduli. 2. Mathematical derivation The theoretical development below follows a well established derivation relating normal stress to bending loads such as is done in Hibbeler (2003). Two assumptions are required for this analysis: (1) the beam material is elastic in compression and in tension, and (2) cross-sectional areas of the beam remain planar and perpendicular to the neutral axis. 2.1. Strain and geometry Consider the beam in Fig. 1. Under the moment induced by force P, cross sections of the beam, such as the one pictured in Fig. 2, will be compressed on the top and elongated on the bottom. A graph of the strain along the height of the cross-section (Fig. 3) is a straight line with the maximum compressive strain at the top, and maximum tensile strain at the bottom. Note that the neutral axis is located at the point where the strain is zero. If the compressive elastic modulus is not equal to the tensile elastic modulus, the neutral axis will not pass through the geometric center of the cross-section. Because the cross-section remains planar, the strain will vary linearly as indicated in Eq. (1): εyc = −
yc εc max Cc
and
εyt = −
yt εt max Ct
(1)
where yc , distance from the neutral axis toward the compressive side of the beam, yt , distance from the neutral axis toward the tensile side of the beam, Cc , distance from the neutral axis to the compression edge of the beam, Ct , distance from the neutral axis to the tensile edge of the beam, εcmax , strain at the compressive edge of the beam, εtmax , strain at the tensile edge of the beam εyc strain in the beam at a distance yc from the neutral axis, εyt strain in the beam at a distance yt from the neutral axis.
Fig. 1. Beam in bending.
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2.2. Elastic modulus and geometry If we consider the stress distribution along the cross section in Fig. 3, it becomes apparent that for the cross-section to remain stationary in space, the internal compressive and tensile forces resulting from the respective stresses must offset each other. Substituting stress for the forces, and introducing the cross-sectional area yields a relationship between stress and geometry. First the internal tensile and compressive forces must be equal: dFc = dFt (4) Ac
At
where Fc , differential compressive force; Ft , differential tensile force; At , cross-sectional area under tension; Ac , cross-sectional area under compression. Substituting F = σA into Eq. (4); σyt dAt = σyc dAc (5) At
We can introduce the maximum compressive and tensile stress by substituting Eq. (2) into Eq. (5): −yc −yt σc max dAc = σt max dAt (6) cc ct Ac At
Fig. 2. Deformation in a beam element under bending.
Noting that the elastic moduli Ec and Et are constant, and σ = Eε, we can substitute stress (σ) for strain (ε): σyc = −
yc σc max Cc
and σyt = −
yt σt max Ct
(2)
where σyc , the compressive stress at a distance y from the neutral axis, σyt , the tensile stress at a distance y from the neutral axis. In Fig. 3 the graph of the strains in the cross-section and the dimensions of the cross-section form similar triangles that define a relationship between the distance from the neutral axis to the surface (cc and ct ) and the amount of compressive and tensile strain (εc and εt ): εc cc = ct εt
Ac
(3)
Because σ cmax , σ tmax , cc , and ct are constant across their respective areas, we can move them outside the integration: σcmax σt max yc dAc = yt dAt (7) cc ct Ac At What remains within the integration is cross-section geometry, and is the first moment of area, Q. Eq. (7) can be simplified to: σc max σt max Qc = Qt (8) cc ct Eq. (8) is significant because it demonstrates that the ratio of compressive to tensile stress can be determined if we can determine the location of the neutral axis. Eq. (8) also is a starting point for relating the elastic modulus to the cross-section geometry. Substituting the basic definition between normal stress and strain (σ = Eε) into Eq. (8): Ec εc Qc Et εt Qt = cc ct
(9)
Gathering the strain terms to one side of the equation, substituting Eq. (3) into Eq. (9), and simplifying the equation yields a relationship between E and Q: Ec Qc εc Et Qt = c c εt ct cc εc subst for ct εt Ec Qc = Et Qt Ec Qt = Et Qc Fig. 3. Strain variation along the beam cross section.
(10)
This equations states that the ratio of the compressive to tensile elastic modulus is equal to the ratio of the first moment of
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area for the tensile to compressive areas. This tells us that if we can identify the areas under compression and under tension, and measure Ec using simple compression testing, we can derive Et . 3. Materials and methods Experiments were conducted to test the validity of the derived equation for materials with different elastic moduli in tension and compression. The test compared the value for the tensile elastic modulus measured by a bending test to the value of tensile elastic modulus measured by a traditional tensile test. Manufactured cork wine closures (ALTOP closures manufactured by Oeneo Closures USA, Napa CA) were selected as test material because the product has substantially different compressive and tensile elastic moduli (tensile E is higher than compressive E), because of the consistency between material used in different closures, and desirable small pore size in these closures. Samples were tested in compression, bending and tension to compute the compressive elastic modulus (required in the bending technique), the location of the neutral axis of samples placed in bending, and the tensile elastic modulus.
the sample’s stress versus strain curve. The mean compressive elastic modulus for the eight replicates was used in later computations.
3.1.1. Compressive test
3.1.2. Tensile test
Eight replicates (closures) were deformed in a parallel plate testing device (Instron Model 1350 with a 400 N capacity load cell). The closures were shaped into cylinders 20 mm in diameter and 40 mm in length. The ends of the samples were sanded to ensure that ends were parallel to each other and perpendicular to the cylinder sides (Fig. 4). The diameter and length of each sample was measured prior to testing. The cylinders were deformed 24 mm (nominally 60% strain) at a rate of 0.485 mm/s. One thousand data points (deformation and force) were collected for each sample. From the deformation and force data, compressive engineering stress and strain was computed. Because cork’s Poisson’s Ratio is very small (about 0.01), engineering stress and strain are very close to true stress and strain values. The compressive elastic modulus for each sample was computed from the linear region of
Nine replicates (straight-sided closures 20 mm in diameter and 40 mm in length) were placed in a lathe and the middle 30 mm were turned down to a nominal diameter of 10 mm. A 20 mm fillet was turned into the transition areas between the middle 30 mm section and the ends of the cork to minimize stress concentration in the transitional region. The diameter and length of the turned section of each sample was measured and recorded for use in computing tensile stress and strain. The ends of the sample were glued to the top and bottom plates of the testing device with a cyanoacrylate-based glue (Fig. 5). The Instron 1350 device used a hydraulic piston to move the lower plate, ensuring that only an axial force (no rotational or bending loading) was applied to the sample. The cork sample was pulled to failure at a rate of 0.485 mm/s. Deformation and applied force was recorded at 0.0125 mm increments of deformation. From the deformation and force data, engineering stress and strain were computed using the diameter and length of the turned down section of each sample. Tensile elastic modulus for each sample was computed from the linear region of the sample’s stress vs. strain curve. The mean tensile elastic modulus of the nine replicates was used in later computations.
Fig. 5. Tensile testing with hour-glass shaped cork.
3.1.3. Bending test
Fig. 4. Compression testing with cylinder of cork.
Ten replicates of cork samples (blocks 10 mm × 10 mm × 50 mm) were tested in bending. Each sample was laid on a bending jig (Fig. 6) on the lower plate of the Instron device. A 10 mm diameter horizontal rod was attached to the load cell on the fixed upper crossbeam. The bending jig was located so that the rod was perpendicular to the long axis of the cork sample and came in contact with the center of the sample (Fig. 7). A digital camera (Olympus 2100C, resolution 320 × 240 pixels) was mounted on
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Fig. 6. A block laid on the bending jig.
the lower plate, and moved with the sample. The center of the sample was deformed 5 mm by the horizontal rod at a rate of 0.485 mm/s. During the deformation, the camera recorded the deformation in a 15 frame/s movie. The amount of deformation and force was recorded at deformation intervals of 0.005 mm. Following each test, two frames were selected from the movie; one prior to deformation, the second at about 3.5 mm of deflection. Using image processing software (Adobe Photoshop) the images were inverted and then the deformed image was subtracted from the undeformed image (Fig. 7). The vertical edges of the sample in the subtracted image formed a white and a black triangle, joined at a point. This point is the pivot point of the vertical side, and thus the point is on the sample’s neutral axis. The ratio of the number of pixels from the bottom of the sample to the point on the neutral axis to the number of pixels from the neutral axis point to the top of the sample is the same as the ratio of ct to cc . 4. Results 4.1.1. Compression test A typical stress versus strain curve is shown in Fig. 8. The compressive elastic modulus for each sample was computed as the average tangent elastic modulus between the strain values of 5–20% strain. The average compressive elastic modulus of the 8 replicates was 2.40 MPa, with a standard deviation of 0.268 MPa. A high compressive strain rate was chosen because the material underwent more compressive strain than tensile strain during the bending. This conclusion falls out directly from the observation that the neutral axis is closer to the tensile side of the test specimen. If the cross-sections of the specimen are to remain planar (one of the assumptions for the derivation), and the neutral axis is displaced toward the tensile side of the specimen, then the compressive strain at the compression side of the specimen has to be greater than the tensile strain at the tension side. In this situation, it was appropriate to use a larger range of compressive strain to compute the average compressive elastic modulus.
Fig. 7. Typical image sequence used to determine the neutral axis. The right image is the difference between the left and middle images.
4.1.2. Tensile test A typical stress versus strain curve for tension is shown in Fig. 9. The tensile elastic modulus for each sample was computed as the average tangent elastic modulus between the strain values of 0–10% strain. The average tensile elastic modulus of the nine replicates was 14.36 MPa, with a standard deviation of 3.846 MPa. The ratio of the first moment of area for the tensile to compressive areas (Qc /Qt ) for each slab in the bending test was computed from the difference images (such as the one in Fig. 7(c), by first counting the number of pixels from the bottom of the slab to the point of the white triangle (pt ) and counting the pixels from the point of the white triangle to the top of the beam (pc ). A pixel-
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Fig. 8. Typical stress vs. strain curve for a cork sample in compression.
to-mm conversion factor was computed from the sum of pt and pc and the measured height (in mm) of the slab. The values of pt and pc were multiplied by this pixel-to-mm conversion factor to determine ct and cc , respectively. The first moments of area for the tensile and compressive regions were computed using Eq. (11): Q = c2
(width of the slab) 2
(11)
Finally, the ratio of the first moment of area for the tensile to compressive areas (Qc /Qt ) was computed by dividing the respective first moments of area. The average value among the nine slabs tested for this ratio was 0.166, with a standard deviation of 0.00313. The tensile elastic modulus computed from the bending test was determined by dividing the average compressive elastic modulus (2.40 MPa) by the average ratio of the first moments of compressive and tensile areas (0.166). The tensile elastic modulus computed from the bending test was 14.47 MPa. The average tensile elastic modulus measured from tensile loading of the cork was 14.36 MPa. The experimental error in the two methods of computing tensile elastic modulus was 0.67%. 5. Discussion The variation in the elastic modulus results was quite low for a biological material. The wine closure material used in this test was unusually homogenous for a biological material because the
Fig. 9. Typical stress vs. strain curve for a cork sample in tension.
Fig. 10. Bending procedure applied to apple tissue (left) and pear tissue (right).
stopper was manufactured by gluing ground cork into the stopper shape. The manufactured closure mimicked the compressive properties of biological materials well in that it had a low compressive elastic modulus, a large compressive failure strain, and exhibited plastic failure in compression. The material also simulated the characteristics of the tensile properties of biological materials in that the material had a high tensile elastic modulus, low failure strain and exhibited brittle failure in tension. The cork material worked well to verify the derivation described in this paper, but it is not a suitable surrogate for fruit or vegetable tissue. The cork material has a very low Poisson’s Ratio, and a very high failure strain in both tension and compression. These characteristics are not typical of fruit or vegetable tissue. Successful use of this technique requires a clear indication of the neutral axis position as seen in the difference of the two images. The more the material is deformed, the easier it will be to identify the point of the triangles (pivot point, or location of the neutral axis). But the deformation must not be so large to violate the assumptions made at the beginning of the derivation: (1) the beam material is elastic in compression and in tension, and (2) cross-sectional areas of the beam remain planar and perpendicular to the neutral axis). Even if the material remains elastic (not bruised) at excessive deformations, the cross-sectional areas will no longer remain planar and perpendicular to the neutral axis. For tissue tested to date (apple, pear), adjusting the length and/or height of the tissue sample was sufficient to clearly identify the point of the triangles (Fig. 10). For materials with higher elastic modulus (tension or compression), or a lower compressive or tensile yield strength, locating the point of the triangles
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will be more difficult. As a point of reference, the wine closure material was less stiff and weaker than most reported elastic modulus and compressive yield values for apples (Mohsenin, 1996). The precision of this method is dependent on the overall resolution of the imaging hardware. The bending samples used in this study were about 87 pixels in height. The effective resolution of the imaging system was 1/87 or about 1%. Newer digital cameras are likely to contain more memory, and as a result more vertical pixel resolution, but the overall resolution is a combination of the number of pixels in the vertical direction and the camera optics. The number of vertical pixels in the edge of a sample is also dependent on the telephoto, zoom and macro capabilities of the camera. A digital video camera rather than a digital camera will make a better video record of the bending (frames every 30 s during the deflection of the sample), but most commercial digital video cameras have only 240 pixels in the vertical direction. If a video camera is used to record the sample deflection, it is very important that the sample contain as many vertical pixels as possible. 6. Conclusion Tensile mechanical properties of fruit and vegetable tissue are likely to differ from their compressive mechanical properties, and properties in both load directions are likely to affect fruit and vegetable quality evaluations. Sample preparation for uniaxial tensile testing of fruit and vegetable tissue is difficult due to fixture limitations. This study has shown that bending loading in conjunction with a uniaxial compression loading can be used to determine tensile mechanical properties. The experimental errors in measuring tensile elastic modulus using this procedure derive from camera resolution, and are much lower percentages than random variations in biological material properties typically reported. These results indicate that the methods used in this paper merit wide adoption by researchers seeking to measure elastic properties of biological materials in both tension and compression. References Abbott, J.A., 1994. Firmness measurement of freshly harvested ‘Delicious’ apples by sensory methods, sonic transmission, Magness-Taylor, and compression. J. Amer. Soc. Hort. Sci. 119, 510–515. Abbott, J.A., Liljedahl, L.A., 1994. Relationship of sonic resonant frequency to compression tests and Magness-Taylor firmness of apples during refrigerated storage. Trans. ASAE 37, 1211–1215. Abbott, J.A., 1999. Quality measurement of fruits and vegetables. Postharvest Biol. Technol. 15, 207–225. Abbott, J.A., 2004. Textural quality assessment for fresh fruits and vegetables. Qual. Fresh Process. Foods 542, 265–279. Abbott, J.A., Affeldt, H.A., Liljedahl, L.A., 1992. Firmness measurement of stored ‘Delicious’ apples by sensory methods, Magness-Taylor, and sonic transmission. J. Amer. Soc. Hort. Sci. 117, 590–595 .
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Abbott, J.A., Lu, R., Upchurch, B.L., Stroshine, R.L., 1997. Technologies for nondestructive quality evaluation of fruits and vegetables. Hort. Rev. 20, 1–120. Armstrong, P., Zapp, H.R., Brown, G.K., 1990. Impulsive excitation of acoustic vibrations in apples for firmness determination. Trans. ASAE 33, 1353–1359. Armstrong, P.R., Brown, G.K., 1991. Apple firmness sorting using a nondestructive acoustic technique. ASAE Paper No. 916044. Belie, N., Schotte, S., Coucke, P., de Baerdemaeker, J., 2000. Development of an automated monitoring device to quantify changes in firmness of apples during storage. Postharvest Biol. Technol. 18, 1–8. Chen, H., de Baerdemaeker, J., 1993. Finite-element-based modal analysis of fruit firmness. Trans. ASAE 36, 1827–1833. Cooke, J.R., Rand, R.H., 1973. A mathematical study of resonance in intact fruits and vegetables using a 3-media elastic sphere model. J. Agric. Eng. Res. 18, 141–157. Delwiche, M., Sarig, Y., 1991. A probe impact sensor for fruit firmness measurement. Trans. ASAE 34, 187–192. Finney, E.E., 1970. Mechanical resonance within Red Delicious apples and its relation to fruit texture. Trans. ASAE 13, 177–180. Finney, E.E., 1971. Dynamic elastic properties and sensory quality of apple fruit. J. Texture Stud. 2, 62–74. Finney, E.E., 1972. Vibration techniques for testing fruit firmness. J. Texture Stud. 3, 263–283. Garrett, R.E., Furry, R.B., 1972. Velocity of sonic pulses in apples. Trans. ASAE 15, 770–774. Harker, F.R., Hallett, I.C., 1992. Physiological changes associated with development of mealiness of apple fruit during storage. HortScience 27, 1291– 1294. Harker, F.R., Hallett, I.C., 1997. Texture of parenchymatous plant tissue: a comparison between tensile and other instrumental and sensory measurements of tissue strength and juiciness. Postharvest Biol. Technol. 11, 63–72. Harker, F.R., Maindonald, J.H., Jackson, P.J., 1996. Penetrometer measurement of apple and kiwifruit firmness: Operator and instrument differences. J. Amer. Soc. Hort. Sci. 121, 927–936. Harker, F.R., Maindonald, J., Murray, S.H., Gunson, F.A., Hallett, I.C., Walker, S.B., 2002. Sensory interpretation of instrumental measurements. 1. Texture of apple fruit. Postharvest Biol. Technol. 24, 225–239. Hibbeler, R.C., 2003. Mechanics of Materials, fifth ed. Prentice Hill, Pearson Education Inc., Upper Saddle River, NJ, ISBN: 0-13-008181-7. Hung, Y., Prussia, S., Ezeike, G.O.I., 2001. Chapter 7 Firmness-measurement methods. In: Nondestructive Food Evaluation: Techniques to Analyze Properties and Quality. Marcel Dekker, Inc, New York, pp. 243–285. Lu, R., Abbott, J.A., 2004. Chapter 5 Force/deformation techniques for measuring texture. In: Texture in Food: vol. 2: Solid Foods. Woodhead Publishing Limited, England, 109–145. Lu, R., Srivastava, A.K., Ababneh, H.A.A., 2006. Finite element analysis and experimental evaluation of bioyield probes for measuring apple fruit firmness. Trans. ASAE 49, 123–131. Mizrach, A., Galili, N., Rosenhouse, G., 1989. Determination of fruit and vegetable properties by ultrasonic excitation. Trans. ASAE 32, 2053–2058. Mizrach, A., Flitsanov, U., Fuchs, Y., 1997. An ultrasonic non-destructive method for measuring maturity of mango fruit. Trans. ASAE 40, 1107– 1111. Mohsenin, N.N., 1996. Physical Properties of Plant and Animal Materials. Gordon and Breach Publishers, Amsterdam, The Netherlands. ISBN: 0-67721370-0. Sarkar, N., Wolfe, R.R., 1983. Potential of ultrasonic measurements in food quality evaluation. Trans. ASAE 26, 624–629.