YOUNG'S MODULUS, POISSON'S RATIO, AND LAME'S COEFFICIENTS OF GOLDEN DELICIOUS APPLE

ABSTRACT

Three measurement methods (falling impact, resonance impulse method and puncture test) are applied to determine some mechanical properties of Golden Delicious apple. Hertz' theory modified by Kozma and Cunningham for impact between two elastic bodies are applied in this work. This allows Young's modulus E, Poisson's ratio ν and Lame's coefficients µ and λ calculation from elasticity coefficients measured by puncture test and acoustic impulse method. Young's modulus and Lame's coefficients decreased with storage time and with degree of fruit ripening. Variation of Poisson's ratio during ripening appears to occurred into three steps: a moderate increase at the beginning of storage, a virtually constant value close to 0.170 up to the fourth storage month and an increase close to 25% up to 186 days cold storage. At the beginning of storage, increasing of Poisson's ratio linked to the decrease of percent contribution of the flesh to the overall firmness. Relationship between Young's modulus and Lame's coefficient µ is linear. Lame's coefficient µ and percent contribution of the flesh to the overall firmness decreased in the same extent (about 43%) during a 186 days cold storage.

INTRODUCTION

Biological materials are biomechanical systems with complex structure. Their mechanical behaviors can not be explained from a single physical constant like many materials (metals and alloys). Because of this complexity, it is necessary to put many hypotheses when studying their mechanical properties. In the same way, results are only valid with precisely defined conditions of measurement. Empirical methods are used to describe the phenomenon insofar, as theory rarely results an accurate indication of fact. Variety of testing methods makes difficult if not uncertainly all comparison between results, and gives often inconsistencies.1-5

Compressive, tensile, shear and impact tests are usually applied to study mechanical properties of tissue structure. They are applied to the samples with defined size (cylinder for example) or to intact plant. The use of test pieces allows minimising geometrical and heterogeneous effects.6-14 Currently, results are displayed like force-time, deformation-time or force-deformation relations from whom mechanical parameters are deduced. Some mechanical models applied to data explanation gave satisfactorily results and allow to explain some phenomenon like turgidity effect, cell shape influence, cell breaking mechanism.15-20 With intact fruit, data explanation is more complicated and various parameters are needed to take in account. Indeed, the deformation extent is related to the applied energy, the number of loads, the surface properties, the size and the physiological status of the fruit.

Various physical parameters are deduced from acoustical test or from force-deformation curves issued from compression test and impact test.[4] In theory, Young's modulus and Poisson's ratio of a sphere submitted to a free vibration are closely connected to its resonance frequency, apparent density and mass.21-22 In the same way, relations between deformation, contact surface and elastic modulus are inferable from Hertz' theory related to contact between solid elastic spheres.

Three methods previously described23-27 are applied to determine mechanical properties (Young's modulus, Poisson's ratio and Lame's coefficients) of Golden Delicious apples as:

• an acoustical test realised on intact fruit giving a global estimate of firmness,

• a free falling of the fruit on a plane and rigid surface giving absorbed energy, maximum impact force and maximum deformation,

• a puncture test giving total firmness (skin+flesh) and flesh firmness when skin bursting.

There is little published information on Young modulus and Lame's coefficients of apples, and factors that might affect these properties. The objectives of the present work were (i) to calculate these physical properties from impact, penetrometric and acoustic data, (ii) to study their variations during cold storage and (iii) to validate measurement methods of the sonic and penetrometric elasticity coefficients. Knowledge of these physical properties is useful to explain rheological behavior of apple.

MATERIAL AND METHODS

Apple Samples

Golden Delicious apples were hand-harvested in an experimental orchard at INRA (Montfavet, France) in the month of August–September 1995. Immediately after each harvest, a sorting machine graded defect free fruit, before placing them in cold storage at 2°C and 96% relative humidity for a 186 days period. In order to achieve a wide range of maturity and to follow their ripening under conventional cold storage conditions, lots of 60 fruits with size from 60 to 85 mm were randomly selected after various cold storage period. Each fruit was characterized by next criteria: mass (weighing), apparent density (weighing), force-displacement curve (puncture test), sonic firmness (acoustical test) and impact curve (falling device). All measurements were made after equilibration of fruit at constant temperature (20±1°C) in a ventilated storage room.

Apparent Density Measurement

Apparent density is measured by weighing from the buoyancy of the fruit submerged in water.[27]

Local Puncture Test

Firmness measurements were performed using a firmness tester controlled and programmed using an internal PC card.[23] Measurements and data explanation were carried out following previously described procedure.[[24],[28]] A cylindrical probe, 4 mm in diameter with a hemispherical tip is used for puncture. A computer driven stepping motor moves the probe at a constant speed of 20 cm/min. Local measurements with intact fruit were made on the equator, perpendicularly to the apex-calyx axis.

Impact Test

The falling device developed in our laboratory was intended to measured the effect of a fruit falling without initial speed and from an exactly known drop height on a force transducer.[26] The impact signal is captured with a rate of 2 µs per point. Full-scale amplitude is 4096 points, each point corresponding to 1.831×10⁻² N for 1 V full scale. Drop height ranges from 0.015 to 0.35 m and absorbed energy from 0.01 to 0.45 J was used in this experiment.

Acoustic Test

Acoustic impulse method developed by Yamamoto et al.[29] was applied for measurement. The apple was placed with its equatorial region close to a microphone. It is tapped at a point diametrically opposite to the microphone, using a small hand-held rod. The acoustic signal captured by the microphone is amplified by a sound meter and stored in a digital form using a digital oscilloscope. The frequency with the greatest amplitude is determined using a Fast Fourier Transform (FFT). An elasticity coefficient proportional to the Young's modulus is calculated from this frequency, the apparent density and the mass of the fruit.

SIMPLIFYING ASSUMPTIONS AND EQUATIONS

Fruit Sphericity

In first approximation, measurement of its mass (m) and its apparent density (ρ) allows to compute its theoretical radius (R) considering spheroid shape apple:

where R is the theoretical radius (m), m is the mass (kg), ρ is the apparent density (kg·m⁻³).

Mechanical Assumptions

Hertz' theory applied to contact between solid elastic bodies requires to lay down the following assumptions:

• the fruit is homogeneous, elastic and spherical,

• the loads applied are static and remain below the fruit elasticity limits,

• the loads developed during compression are located and affect only a small part of the fruit volume,

• the radii of curvature of contacting bodies are large when compared with the size of the contact area,

• the contact surfaces (fruit or indenter) are smooth and are not subjected to tangential load.

For dynamic impact measurement (falling device), following assumptions are set up:

• the centre of gravity and the geometrical centre remain identical during compression and during impact,

• the displacement of the centre of gravity during impact fits with the squashing of a portion of a sphere; fruit remaining its global shape,

• the internal vibrations are in practice insignificant.

For static measurements (puncture tests), following assumptions are set up:

• the fruit radius R₁ (a few centimetres) is infinite related to the indenter radius R₂ (2 mm),

• the fruit deformation at the opposite side in contact with the plate is insignificant.

Elasticity Coefficient Measurement

From Impact Test

Hertz's equation allows to compute contact surface of a sphere with a rigid plane when it is subject to a force F. Although this computation was developed only for static experiment, many authors30-38 have applied them to dynamical impact. Hertz' theory modified by Kozma and Cunningham[39] allows computing of fruit Young's modulus. When two spherical bodies with respectively radii R₁ and R₂, Young's modulus E₁ and E₂ and Poisson's ratio ν ₁ and ν ₂ come into contact and are subject to a compressive force F directed to their centre directions, the centres of the sphere bring together for a length d such as:

where, d normal approach of the two bodies or deformation (m), F force (N), ν Poisson's ratio, E Young's modulus (Pa), R radius (m), ϵ: angle between the normal planes related to the radii of curvature (degree), φ modular angle (degree), K complete first order elliptical integral, M complete second order elliptic integral as:

As load is applied in the direction of the axis of the centers, then:

Index 2 is related to the stainless steel probe (indenter or plane). Young's modulus E₂ of stainless steel is infinite in comparison with the fruit Young's modulus E₁ and so, Eq. 2 reduces to:

From Puncture Test

Fruit radius R₁ is infinite with respect to the probe's radius R₂ and this allows setting:

where, R^′ ₂ indenter radius of curvature (m). Owing to the fact that E₂=∞, we can set E=E₁ and ν=ν ₁ and Eq. 2 reduces to:

and

Poisson's ratio being not measured for each fruit, E* is put equal to E/(1−ν2) and called elasticity coefficient. If the force F is equal to the force developed to break the skin (Ft) by puncture, E* is put equal to Et* and called penetrometric elasticity coefficient (skin+flesh):

where Et is the penetrometric Young's modulus (Pa), Et* is the penetrometric elasticity coefficient (Pa).

From Acoustic Test

Elasticity coefficient measured by the acoustic impulse method is compu ted by means of Cooke's Equation.[21] This coefficient is proportional to the Young's modulus Es.[[21],[22],[40]]

where Es is the sonic Young's modulus (Pa), Es* is the sonic elasticity coefficient (Pa), Hz is frequency with the greatest amplitude (Hz).

Determination of Maximum Deformation During Impact

At the beginning of the contact and if air friction are neglected, initial speed depends on the drop height h:

where, g is gravitational acceleration (9.81 mċ s−2), h is drop height (m), v_i is initial speed (mċs−1). The instantaneous acceleration [avec](t) is computed from the instantaneous force [Fvec](t) measured by the transducer:

where, m fruit mass (kg), [avec](t) instantaneous acceleration (mċs−2). Successive integration of the acceleration function gives the instantaneous speed [vvec](t) and the instantaneous displacement [dvec](t) of the fruit centre of gravity, ranked as deformation.

Maximum deformation and maximum force are reached when the speed is equal to zero. In this case, maximum force F_max is equal to:

Maximum deformation is obtained by substituting F for F_max in Eq. 3:

RESULTS AND DISCUSSION

Maximum Theoretical Deformation

Penetrometric elasticity coefficient (Et*) and sonic elasticity coefficient (Es*) constitute fruit firmness indicators. Curves giving the evolution of Et* and Es* with cold storage duration have a good regularity (Fig. 1). Data are sufficiently precise to prove their use instead of E/(1−ν ²) in computation of the maximum deformation by means of Eq. 6.

Figure 1. Evolution of sonic elasticity coefficient (Es*) and penetrometric elasticity coefficient (Et*) during cold storage (2°C, 96% rh, 186 days) of Golden Delicious apples. Harvest 09/05/1995. Mean and standard error.

Figure 2 displays great adequacy between the maximum theoretical deformation d_max calculated from impact data and those computed by means of Hertz' Eq. 4 using on the one hand the sonic elasticity coefficient Es* (Fig. 2a) and on the other hand the penetrometric elasticity coefficient Et* (Fig. 2b). Using sonic elasticity coefficient overestimates about 4.5% the deformation while using the penetrometric elasticity coefficient underestimates about 1.5%. When Et* is used instead of E, substitution is accurate [Et*=Et/(1−ν2)]. When Es* is used, a factor equal to (1+ν) is neglected [Es*=Es · (1+ν)] and the formula to take in account is:

Figure 2. Figure 2. Relationship between maximum theoretical deformation computed from Hertz' equation, and maximum theoretical deformation (d_max) deduced from impact data. Golden Delicious apples (1357 fruits). 2a: d_max(Es*) – sonic elasticity coefficient 2b: d_max(Et*) – penetrometric elasticity coefficient.

Poisson's Ratio

Assuming that d_max(Et*)=d_max(Es*) =k · d_max and admitting that Young's moduli are equal, Et=Es, allow calculation of ν value from the sonic elasticity coefficient Es* and from the penetrometric elasticity coefficient Et*:

and

The same procedure applied to bruise volumes deduced from impact parameters (absorbed energy, maximum impact force and maximum deformation) gives identical values of ν. In this case applied formula is:

Young's Modulus

Young's modulus of each fruit is estimated from Poisson's ratio and from sonic elasticity coefficient Es* or penetrometric elasticity coefficient Et*, following:

These two elasticity coefficients give perfectly identical values for Young's moduli Es and Et:

Lame's Coefficients λ and µ

Lame's coefficients λ and µ are derived from Young's modulus and Poisson's ratio following equations:

Contribution of the Apple Flesh to the Overall Firmness

Percent contribution of the apple flesh to the overall firmness is the ratio of the flesh strength at skin rupture (Fc) to the overall force at skin bursting (Ft):

Table 1 gives mean values obtained for each parameter after various cold storage duration. Chappell and Hamann[41] and Finney[42] report following values (Table 2) of ν for some apple varieties. For Golden Delicious, values reported in literature are close to our results.

Table 1. Mean Value of Young's Modulus (E), Poisson's Ratio (ν), Lame's Coefficients (λ and µ), and Percent Contribution of the Flesh to Overall Firmness (y) After Various Cold Storage Duration (2°C, 96% hr) (Golden Delicious Apples)

Time (days)	E (MPa)	ν	λ (MPa)	μ (MPa)	y (%)
4	3.33	0.167	0.839	1.43	66.56
17	3.09	0.184	0.847	1.31	62.25
47	2.79	0.175	0.824	1.19	53.04
61	2.61	0.181	0.771	1.11	44.64
78	2.50	0.162	0.623	1.08	43.68
91	2.49	0.188	0.756	1.05	45.23
109	2.33	0.176	0.635	0.993	38.90
112	2.28	0.185	0.706	0.966	41.83
120	2.33	0.150	0.492	1.01	38.34
139	2.25	0.160	0.533	0.970	37.47
152	2.10	0.191	0.619	0.882	36.88
172	2.09	0.198	0.683	0.874	38.53
182	2.01	0.229	0.779	0.819	38.08
186	2.00	0.221	0.763	0.823	37.93

Table 2. Reported Values of Poisson's Ratio ν for Some Apple Varieties41-42

Variety	ν
Golden delicious	0.169–0.244
Red delicious	0.155–0.207
Rome beauty	0.020–0.236
Turley	0.081–0.134
York	0.040–0.105

Evolution During Cold Storage

Variations of Young's modulus, Poisson's ratio and Lame's coefficients with cold storage time are shown on Fig. 3. Young's modulus (E) and Lame's coefficients (µ and λ) decrease respectively 39.8, 42.4 and 9.0% during a 186 days cold storage. All these physical properties have the same size (MPa). Mean value of Poisson's ratio (ν) is 0.173 during the first 139 days cold storage, afterwards this mean value increases to reach 0.221 after 186 days.

Figure 3. Evolution of Young's modulus (E), Poisson's ratio (ν) and Lame's coefficients (λ and μ) during a 186 days cold storage (2°C, 96% hr). Golden Delicious apples: mean value.

Relationship between µ and E is straightforward for Young's modulus between 1.63 and 3.94 MPa (1357 apples):

Relationship between Lame's coefficient λ and Poisson's ratio is exponential (1357 apples):

During a 186 days cold storage, percent contribution of the flesh to the overall firmness (Fig. 4) decreases for more than 43% and Lame's coefficient µ for 42.3%. Young's modulus and percent contribution of the flesh to the overall firmness vary nearly concurrently: their mean values are linearly correlated

Figure 4. Evolution of Young's modulus (E), Lame's coefficients (λ and μ) and percent contributi on of the flesh to overall firmness (y) during a 186 days cold storage (2°C, 96% hr). Golden Delicious apples: mean value.

CONCLUSION

Hertz' theory modified by Kozma and Cunningham allows Young's modulus E, Poisson's ratio ν and Lame's coefficients µ and λ calculation from elasticity coefficients measured by puncture test and acoustic impulse method. Poisson's ratio values obtained has the same magnitude that those earlier published and measured on test piece of apple (tensile test or dynamical compression test). Taking into account Poisson's ratio in Young's modulus calculation validates measurement methods of the sonic elasticity coefficient and of the penetrometric elasticity coefficient. Percent contribution of the apple flesh to the overall firmness, Young's modulus and Lame's coefficients decrease with storage time. Variation of Poisson's ratio during ripening appears to occur into three steps: a light increase at the beginning of storage, a virtually constant value up to the fourth storage month and an increase up to 186 days cold storage. At the beginning of storage, increasing of Poisson's ratio appears linked to the decrease of percent contribution of the flesh to the overall firmness (y). Later on, no relationship is found between these two parameters. Relationship between Young's modulus and Lame's coefficient µ is linear. Lame's coefficient µ and percent contribution of the flesh to the overall firmness decrease in the same extent (about 43%) during a 186 days cold storage.

Acknowledgments