**2.2 Viscoelastic materials**

308 Biomaterials – Physics and Chemistry

can be seen in the figure that correlations of the liquid rheological properties with its acoustic properties measured by the quality factor *Q* are strong and that the quality evaluated at the first peak provides a better representation of the liquid viscosity. This is also enhanced due to the presence of two peaks for low and moderate solid contents (measured as Brix), which provides more experimental points to establish the correlation

Fig. 6. Frequency response spectra of tomato concentrates at different soluble solids

0 400 800 1200 1600

Frequency (Hz)

23 Brix

9.5 Brix

8.15 Brix

3.5 Brix

concentrations, measured as Brix. Curves were shifted up for clarity.

0

3

Abs(1/Za0) x 104 (m/Pa.s)

6

9

(see Figure 7 – and compare first peak and second peak data).

#### **2.2.1 Semifluid materials**

Squeezing flow is a well-known technique that has been applied to characterize the properties of various biomaterials ranging from liquids to semisolids. The traditional method involves measuring the force required to squeeze a sample between two cylindrical disks either at a constant velocity or by applying a constant force or stress (Campanella and Peleg 2002).

The oscillatory squeezing low method (OSF) uses the same geometry as the standard squeezing flow method but it involves the application of small amplitude oscillations at random frequencies up to 20 kHz (Mert and Campanella, 2008). The method allows one to calculate both the viscous and elastic components of the sample viscoelasticity by measuring the response of the material in terms of force and acceleration to those oscillations. Transformation of the measured force and acceleration to the frequency domain yields a frequency spectrum for the sample and, ultimately, its resonance frequency. From analysis of this frequency response, two important viscoelastic properties of the samples, the loss modulus G" (viscous) and the storage modulus G' (elastic), can be obtained.

The application of acoustic principles to the squeezing flow method is a novel technique, which convert the squeezing flow method into OSF methos can measure the rheological properties of materials that range from pure liquids to solids. Advantages of this method include it being non-invasive, little to no sample preparation, and its ability to monitor rapid changes during dynamic processes.

A schematic of the OSF testing apparatus is illustrated in Figure 8. The design uses a piezoelectric crystal stack attached to an impedance head. Upon the application of voltage, the upper plate oscillates, and the force and acceleration at the oscillating plate are measured

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 311

the plate (*h*o) is noted. A load cell is attached above the oscillating plate to control the squeezing force applied to the sample prior the application of the oscillation. The squeezing force is simply applied to make sure that a good contact is established between the oscillating plate and the sample. It will be shown below, however, that the results are

The frequency response data is obtained using software that interfaces the results obtained with the signal generator device (see also details in Figure 1). The transformed mechanical

they have been transformed into the frequency domain, they, as well as the mechanical impedance, are complex variables. The measured complex mechanical impedance at the

*<sup>u</sup>* is then calculated from the force and velocity measured on the oscillating

*F* and *u*ˆ are the Fourier transformed variables and since

ˆˆ ˆ *ZZ Z meas instrument sample* (11)

*Zinstrument* is the instrument impedance that can be

because the instrument does not have any spring

*Zsample* can be

independent of that squeezing force.

ˆ *<sup>F</sup> <sup>Z</sup>*

driving point can be defined as:

calculated simply as ˆ

*Zmeas* is the measured impedance, <sup>ˆ</sup>

sample can be described by Equation (12) below:

stiffness and the mass of the system by Equation (13):

with concentration (see Equation 13).

*Z im instrument plate*

*sample sample*

*Z R im*

*res*

*f*

biopolymer that produces viscoelastic suspensions, is illustrated in Figure 10.

mechanism or internal damping. From Equation (11) the sample impedance ˆ

obtained by subtracting the instrument impedance from the measured impedance.

The rheological behavior of the sample can be described in terms of the viscous component (*R*), which provides the damping of the oscillation and the elastic component (*S*), which provides the sample elasticity (Figure 9). The relationship between the mechanical impedance and the damping (viscous component) and stiffness (elastic component) of the

ˆ ( ) *sample*

The mobility of the sample, can be plotted as a function of frequency to provide the resonance spectrum of the sample. The resonance frequency, *fres* of the sample, which is obtained as the frequency at which the mobility is a maximum is directly related to the

is the angular frequency of the oscillation, *m* is the mass of the system, and *i* is

*sample*

*S*

A typical plot of Mobility versus frequency for different concentrations of xanthan gum, a

The higher is the concentration of xanthan gum the higher is its elasticity, which is clearly illustrated in the Figure 10 by a shifting to the right of the resonance frequency. That shifting of the frequency is a clear indication on increase in the stiffness of elasticity of the sample

*S*

(12)

*<sup>m</sup>* (13)

plate with the impedance head. ˆ

impedance <sup>ˆ</sup> <sup>ˆ</sup>

where ˆ

where

1 .

through the impedance head and transformed into the frequency domain using a Fast Fourier Transformation (FFT) routine. This transformation is very useful because from the inspection of the frequency response of the measurement it is possible to identify a characteristic resonance frequency for the sample.

Fig. 8. Schematic of the OSF testing apparatus

Since the samples are considered viscoelastic they can be represented by a combination of elastic component, with a stiffness *S*, and viscous components with a damping *R*. A schematic of the elastic-viscous system used for analysis of the measurements is illustrated in Figure 9. For the testing, the sample is placed between the oscillating plate and a fixed plate. The oscillating plate is then brought down to touch the sample and the gap between

Fig. 9. Schematic of the spring-dashpot system, used for the analysis of the data. *ho*, *R*, and *S*  are the height, the viscous resistance (damping), and the stiffness of the sample respectively.

through the impedance head and transformed into the frequency domain using a Fast Fourier Transformation (FFT) routine. This transformation is very useful because from the inspection of the frequency response of the measurement it is possible to identify a

> Load Cell

Impedance Head

Oscillating

Plate

Rigid

Surface

Sample

Since the samples are considered viscoelastic they can be represented by a combination of elastic component, with a stiffness *S*, and viscous components with a damping *R*. A schematic of the elastic-viscous system used for analysis of the measurements is illustrated in Figure 9. For the testing, the sample is placed between the oscillating plate and a fixed plate. The oscillating plate is then brought down to touch the sample and the gap between

Fig. 9. Schematic of the spring-dashpot system, used for the analysis of the data. *ho*, *R*, and *S*  are the height, the viscous resistance (damping), and the stiffness of the sample respectively.

characteristic resonance frequency for the sample.

Stack of Piezoelectric Crystals

Oscillating Plate

*R h s <sup>o</sup>*

Fig. 8. Schematic of the OSF testing apparatus

Fixed Plate

the plate (*h*o) is noted. A load cell is attached above the oscillating plate to control the squeezing force applied to the sample prior the application of the oscillation. The squeezing force is simply applied to make sure that a good contact is established between the oscillating plate and the sample. It will be shown below, however, that the results are independent of that squeezing force.

The frequency response data is obtained using software that interfaces the results obtained with the signal generator device (see also details in Figure 1). The transformed mechanical

impedance <sup>ˆ</sup> <sup>ˆ</sup> ˆ *<sup>F</sup> <sup>Z</sup> <sup>u</sup>* is then calculated from the force and velocity measured on the oscillating

plate with the impedance head. ˆ *F* and *u*ˆ are the Fourier transformed variables and since they have been transformed into the frequency domain, they, as well as the mechanical impedance, are complex variables. The measured complex mechanical impedance at the driving point can be defined as:

$$
\hat{Z}\_{m\text{cas}} = \hat{Z}\_{\text{instrument}} + \hat{Z}\_{\text{sample}} \tag{11}
$$

where ˆ *Zmeas* is the measured impedance, <sup>ˆ</sup> *Zinstrument* is the instrument impedance that can be calculated simply as ˆ *Z im instrument plate* because the instrument does not have any spring mechanism or internal damping. From Equation (11) the sample impedance ˆ *Zsample* can be obtained by subtracting the instrument impedance from the measured impedance.

The rheological behavior of the sample can be described in terms of the viscous component (*R*), which provides the damping of the oscillation and the elastic component (*S*), which provides the sample elasticity (Figure 9). The relationship between the mechanical impedance and the damping (viscous component) and stiffness (elastic component) of the sample can be described by Equation (12) below:

$$\hat{Z}\_{sample} = R\_{sample} + \mathbf{i} \cdot (o \cdot m - \frac{S\_{sample}}{o}) \tag{12}$$

where is the angular frequency of the oscillation, *m* is the mass of the system, and *i* is 1 .

The mobility of the sample, can be plotted as a function of frequency to provide the resonance spectrum of the sample. The resonance frequency, *fres* of the sample, which is obtained as the frequency at which the mobility is a maximum is directly related to the stiffness and the mass of the system by Equation (13):

$$f\_{res} = \sqrt{\frac{S\_{sample}}{m}}\tag{13}$$

A typical plot of Mobility versus frequency for different concentrations of xanthan gum, a biopolymer that produces viscoelastic suspensions, is illustrated in Figure 10.

The higher is the concentration of xanthan gum the higher is its elasticity, which is clearly illustrated in the Figure 10 by a shifting to the right of the resonance frequency. That shifting of the frequency is a clear indication on increase in the stiffness of elasticity of the sample with concentration (see Equation 13).

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 313

storage modulus *G'* by Equation (16) is required to know the mass of the system, which consist of an effective mass that includes the mass of the sample and the squeezing force imposed to achieve good contact between the oscillating plate and the sample. If Equation (12) is rewritten in terms of those masses we can obtain an equation for the measured

> *Z R im m S meas sample plate effective sample* (

However, to estimate the inertia produced by the effective mass *meffective* some additional calculations are required. If the imaginary part of the sample impedance is divided by the

<sup>2</sup> <sup>ˆ</sup> Im( ) ( / ) *sample*

applied to the sample, *S* equals to meffective . That value can then be used to calculate the storage modulus by Equation (16). Results for a xanthan gum suspension of concentration 2%, and whose mobility versus frequency data is shown in Figure 10, are shown in Figure 11 in terms of the viscoelastic storage and loss moduli defined by Equations (16) and (17). As shown in the figure the data compares well with those obtained with a conventional rheometer at comparable frequencies. It must be also noted that the range of frequencies of the OSF method is significantly higher than those applied by conventional rheological

Physical properties of solid viscoelastic foods and other biological products are very important in food production, storage, handling, and processing. The importance of the knowledge of the physical properties of biomaterials is demonstrated in the case of fruits. One of the most important quality parameter of fruits is its texture. Texture is the first judgment a purchaser makes about the quality of a fruit, before sweetness, sourness, or flavor. Since fruit texture is such an important attribute, one would expect the changes in texture during maturation and cool-storage to be well understood, however this is not the case. One of the main limitations to the study of fruit texture is the accurate and precise

Traditionally, the texture of fruits is measured by a Magness Taylor pressure tester (Magness and Taylor, 1925). This simple device measures the force required to insert a metal

*m S*

*effective sample*

*effective* . Results of this calculation show that the parameter

in Equation (19) approaches to zero as the frequency is very high,

reaches an asymptotic value, which is independent of the squeezing force

(19)

The inertia produced by the plate instrument can be easily estimated as ˆ

*Z*

methods where inertia may play an important role.

measurement of texture as perceived by a consumer.

*<sup>h</sup> G Re Z*

<sup>ˆ</sup>

*sample Im Z* and Re <sup>ˆ</sup>

complex mechanical impedance ˆ

complex mechanical impedance:

The term <sup>2</sup> *Ssample* /

<sup>ˆ</sup> Im( ) / *Zsample*

ˆ

frequency, the following equation is obtained:

which results in <sup>ˆ</sup> Im( ) / *Z m sample*

**2.2.2 Semisolid materials** 

2 ˆ

3 4

*a* 

*Zsample* are the imaginary and real part, respectively, of the sample

(17)

 

/ ) (18)

*Z im instrument plate* .

*Zsample* . It is important to note that for the calculation of the

3 *<sup>o</sup> sample*

Fig. 10. Mobility plots obtained with the OSF method for xanthan gum dispersions of different concentrations

Although the mobility and the elastic and viscous components of a viscoelastic sample are often useful as quality and processing parameters (Gonzalez et al., 2010), for modeling purposes it is necessary to get quantitative and fundamental rheological information of a sample in terms of elastic and viscous modulus. Phan-Thien (1980) demonstrated that for viscoelastic materials subjected to squeezing flow under an oscillating plate, the squeezing flow force can be calculated as:

$$F\_o \cos(\alpha t) = \frac{3\pi a^4}{2h\_o^3} \eta^"\ u\_o \cos(\alpha t) \tag{14}$$

where *a* is the radius of the top plate, *ho* is the distance between plates and \* ' " *i* is the complex dynamic viscosity of the sample. " ' *<sup>G</sup>* and ' " *<sup>G</sup>* are the viscous and elastic components of the sample, respectively. Application of FFT to Equation (15) yields:

$$\hat{Z}\_{sample} = \frac{3\pi a^4}{2h\_o^3} \eta^\* \tag{15}$$

Expressions can be rearranged and using the definition of the complex viscosity \* , expressions for the viscoelastic moduli, i.e. the storage and loss modulus, *G'* and *G"*, can be obtained as:

$$\mathbf{G}' = \left[ \operatorname{Im} \left( \hat{Z}\_{sample} \right) + \frac{3 \, m\_{effective} \, o \, a^2}{20 h\_o^2} \right] \cdot \frac{2 h\_o^3 \, o}{3 \pi a^4} \tag{16}$$

$$\mathbf{G''} = \text{Re}\left(\hat{Z}\_{sample}\right) \cdot \frac{2h\_o^3 \alpha}{3\pi a^4} \tag{17}$$

 <sup>ˆ</sup> *sample Im Z* and Re <sup>ˆ</sup> *Zsample* are the imaginary and real part, respectively, of the sample complex mechanical impedance ˆ *Zsample* . It is important to note that for the calculation of the storage modulus *G'* by Equation (16) is required to know the mass of the system, which consist of an effective mass that includes the mass of the sample and the squeezing force imposed to achieve good contact between the oscillating plate and the sample. If Equation (12) is rewritten in terms of those masses we can obtain an equation for the measured complex mechanical impedance:

$$\hat{Z}\_{\text{meas}} = R\_{\text{sample}} + \text{i}\{o\,m\_{\text{plate}} + o\,m\_{\text{effective}} - S\_{\text{sample}} / \,\alpha\} \tag{18}$$

The inertia produced by the plate instrument can be easily estimated as ˆ *Z im instrument plate* . However, to estimate the inertia produced by the effective mass *meffective* some additional calculations are required. If the imaginary part of the sample impedance is divided by the frequency, the following equation is obtained:

$$\frac{\text{Im}(\hat{Z}\_{sample})}{\alpha} = \left(m\_{\text{effective}} - S\_{sample} \,/\,\alpha^2\right) \tag{19}$$

The term <sup>2</sup> *Ssample* / in Equation (19) approaches to zero as the frequency is very high, which results in <sup>ˆ</sup> Im( ) / *Z m sample effective* . Results of this calculation show that the parameter <sup>ˆ</sup> Im( ) / *Zsample* reaches an asymptotic value, which is independent of the squeezing force applied to the sample, *S* equals to meffective . That value can then be used to calculate the storage modulus by Equation (16). Results for a xanthan gum suspension of concentration 2%, and whose mobility versus frequency data is shown in Figure 10, are shown in Figure 11 in terms of the viscoelastic storage and loss moduli defined by Equations (16) and (17). As shown in the figure the data compares well with those obtained with a conventional rheometer at comparable frequencies. It must be also noted that the range of frequencies of the OSF method is significantly higher than those applied by conventional rheological methods where inertia may play an important role.

#### **2.2.2 Semisolid materials**

312 Biomaterials – Physics and Chemistry

Fig. 10. Mobility plots obtained with the OSF method for xanthan gum dispersions of

where *a* is the radius of the top plate, *ho* is the distance between plates and \*

components of the sample, respectively. Application of FFT to Equation (15) yields:

3 ˆ 2 *sample*

*<sup>a</sup> <sup>Z</sup>*

Expressions can be rearranged and using the definition of the complex viscosity \*

*m a <sup>h</sup> G Im Z*

 

expressions for the viscoelastic moduli, i.e. the storage and loss modulus, *G'* and *G"*, can be

complex dynamic viscosity of the sample. " ' *<sup>G</sup>*

0.00

0.01

0.02

Mobility Abs(1/Zsample) - m/N.s

0.03

0.04

0.05

Although the mobility and the elastic and viscous components of a viscoelastic sample are often useful as quality and processing parameters (Gonzalez et al., 2010), for modeling purposes it is necessary to get quantitative and fundamental rheological information of a sample in terms of elastic and viscous modulus. Phan-Thien (1980) demonstrated that for viscoelastic materials subjected to squeezing flow under an oscillating plate, the squeezing

0 20000 40000

 1% Xanthan Gum 2% Xanthan Gum 3% Xanthan Gum

Resonance Freq1% (rad/s)

<sup>4</sup> \* 3 <sup>3</sup> cos( ) cos( ) <sup>2</sup> *o o o <sup>a</sup> Ft u t h* 

<sup>4</sup> \* 3

*o*

*h* 

<sup>3</sup> <sup>2</sup> <sup>ˆ</sup>

*effective <sup>o</sup> sample o*

 

 and ' " *<sup>G</sup>* 

2 3 2 4

20 3

*h a*

(14)

 ' " *i* is the

are the viscous and elastic

(16)

,

(15)

different concentrations

flow force can be calculated as:

obtained as:

Physical properties of solid viscoelastic foods and other biological products are very important in food production, storage, handling, and processing. The importance of the knowledge of the physical properties of biomaterials is demonstrated in the case of fruits. One of the most important quality parameter of fruits is its texture. Texture is the first judgment a purchaser makes about the quality of a fruit, before sweetness, sourness, or flavor. Since fruit texture is such an important attribute, one would expect the changes in texture during maturation and cool-storage to be well understood, however this is not the case. One of the main limitations to the study of fruit texture is the accurate and precise measurement of texture as perceived by a consumer.

Traditionally, the texture of fruits is measured by a Magness Taylor pressure tester (Magness and Taylor, 1925). This simple device measures the force required to insert a metal

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 315

stiffness coefficient measured by vibration was closely associated with sensory panel subjective evaluations of Red Delicious apples (Finney, 1970, 1971). Garrett (1970) argued that the stiffness factor should be proportional to the frequency squared times mass to the two-thirds power. Cooke (1972) confirmed Garrett's theory using elastic theory relating the stiffness factor to the shear modulus of the fruit flesh. Both Finney's and Garrett's stiffness factors have been shown to correlate well with the fruit modulus (Clark and Shackelford, 1973; Yamamoto et al., 1980; and De Bardemaeker (1989); Abbott, 1994; Abbott and Liljedahl, 1994). Differences in shear moduli between green and ripe fruits can be detected non-

Through a few decades of development, a vibration-based characterization method called Experimental Modal Analysis (EMA) has advanced to become a very efficient tool for obtaining the dynamic properties such as natural frequency, damping and mode shapes in aerospace and mechanical engineering. For fruits, natural frequency is related to the shear modulus of the tissue (Cooke, 1972), which in turn determines the firmness of the fruit. For instance, in apples, high damping prevents them from giving a nice them from giving a nice crispy ringing sound when tapped (De Baerdemaeker and Wouters, 1987). Mode shapes are an important indicator of the whole fruit's conditions such as ripeness, bruises, or defects (Cherng, 2000). It is obvious that there is abundant information on the texture and quality of fruits that could be associated with their acoustic parameters. This section of the chapter is meant to give an idea of the direction that researchers have taken in the last decade or so. To give specific examples, a few details are given on the measurement of Young's modulus,

An instrument that measures force as a function of displacement, such as a UTM, can be used to obtain force-displacement curves of the samples. For a melon, the flesh and the rind should be tested differently. Melons have a rind that is significantly stiffer than the flesh. The cylindrical samples of the flesh can be tested with a compression test. The rind samples can be tested using a three point jig, and treated as a simply supported beam for

For modeling vibrations, where only small displacements are of interest, the tests should be limited to small displacements and changes in the cross-section and in the length can be ignored in the data processing. Also, the mechanical properties of biological materials in general are not constant. Because of the structures at the micro scale, the Young's modulus is a function of strain. The following method can be used to facilitate the computation of the Young's modulus at zero deformation. The stress-strain curve can be

> 3 2 3 2 10

 

The coefficients *a*0 through *a*3 can be obtained by cubic regression analysis. The Young's

2 3 21 3 2 *<sup>d</sup> E a aa*

 

> 

*a a aa* (20)

(21)

*d* 

modulus is the derivative of the stress with respect to strain

destructively, and relatively easily, by vibration testing.

finite element modeling, and experimental modal analysis.

**2.2.2.1 Measuring modulus of elasticity** 

analysis.

fit to a cubic function

Fig. 11. Storage and loss modulus of a xanthan gum suspension of 2% concentration obtained with the OSF and conventional rheological methods

cylindrical plunger for a given distance into the fruit flesh. Although this device is cheap, quick, and easy to use, it has some disadvantages. Firstly, it is a destructive test, and it is not possible to have repeated measurements on a single fruit. Due to extremely high fruit-tofruit variability, repeated measures on the same fruit would be highly desirable, especially from a research perspective. Secondly, there is high variability between operators, depending on the speed and force that the plunger is inserted into the fruit. However the greatest limitation of the Magness Taylor fruit pressure tester is that it measures the compression or crushing force that the cells of the fruit cortex are able to withstand. This force is very different from the force that is exerted on a fruit by the teeth of a consumer or the pressure that the consumer can exert with the fingers to assess the fruit texture. Biting an apple, for instance, involves cleaving cells apart, the exact opposite of the force measured by pressure testers. Universal Testing Machines (UTM), like for example the Instron™ instrument, are devices that can measure cleaving force, but they are slow, expensive, and again the method used with these machines is a destructive test.

Throughout the literature, there is abundant evidence that the ripeness or softness of intact fruits is related to their vibration properties. A device that uses vibrations to characterize intact fruits was patented in 1942 (Clark et al., 1942). Vibration properties of some fruits are correlated with firmness and ripeness. Finney (1970) explored methodologies for measuring and characterizing the vibration response of many fruits, and concluded that Young's modulus and shear modulus of apple flesh were correlated with the product between the resonance frequency squared and the mass of the whole fruit. It was also found that the stiffness coefficient measured by vibration was closely associated with sensory panel subjective evaluations of Red Delicious apples (Finney, 1970, 1971). Garrett (1970) argued that the stiffness factor should be proportional to the frequency squared times mass to the two-thirds power. Cooke (1972) confirmed Garrett's theory using elastic theory relating the stiffness factor to the shear modulus of the fruit flesh. Both Finney's and Garrett's stiffness factors have been shown to correlate well with the fruit modulus (Clark and Shackelford, 1973; Yamamoto et al., 1980; and De Bardemaeker (1989); Abbott, 1994; Abbott and Liljedahl, 1994). Differences in shear moduli between green and ripe fruits can be detected nondestructively, and relatively easily, by vibration testing.

Through a few decades of development, a vibration-based characterization method called Experimental Modal Analysis (EMA) has advanced to become a very efficient tool for obtaining the dynamic properties such as natural frequency, damping and mode shapes in aerospace and mechanical engineering. For fruits, natural frequency is related to the shear modulus of the tissue (Cooke, 1972), which in turn determines the firmness of the fruit. For instance, in apples, high damping prevents them from giving a nice them from giving a nice crispy ringing sound when tapped (De Baerdemaeker and Wouters, 1987). Mode shapes are an important indicator of the whole fruit's conditions such as ripeness, bruises, or defects (Cherng, 2000). It is obvious that there is abundant information on the texture and quality of fruits that could be associated with their acoustic parameters. This section of the chapter is meant to give an idea of the direction that researchers have taken in the last decade or so. To give specific examples, a few details are given on the measurement of Young's modulus, finite element modeling, and experimental modal analysis.

#### **2.2.2.1 Measuring modulus of elasticity**

314 Biomaterials – Physics and Chemistry

G' OSF Method

G'' OSF Method

Fig. 11. Storage and loss modulus of a xanthan gum suspension of 2% concentration

cylindrical plunger for a given distance into the fruit flesh. Although this device is cheap, quick, and easy to use, it has some disadvantages. Firstly, it is a destructive test, and it is not possible to have repeated measurements on a single fruit. Due to extremely high fruit-tofruit variability, repeated measures on the same fruit would be highly desirable, especially from a research perspective. Secondly, there is high variability between operators, depending on the speed and force that the plunger is inserted into the fruit. However the greatest limitation of the Magness Taylor fruit pressure tester is that it measures the compression or crushing force that the cells of the fruit cortex are able to withstand. This force is very different from the force that is exerted on a fruit by the teeth of a consumer or the pressure that the consumer can exert with the fingers to assess the fruit texture. Biting an apple, for instance, involves cleaving cells apart, the exact opposite of the force measured by pressure testers. Universal Testing Machines (UTM), like for example the Instron™ instrument, are devices that can measure cleaving force, but they are slow, expensive, and

10-1 <sup>10</sup><sup>1</sup> <sup>10</sup><sup>3</sup> <sup>105</sup> <sup>100</sup>

Frequency (rad/s)

G'' Conventional Rheology

Throughout the literature, there is abundant evidence that the ripeness or softness of intact fruits is related to their vibration properties. A device that uses vibrations to characterize intact fruits was patented in 1942 (Clark et al., 1942). Vibration properties of some fruits are correlated with firmness and ripeness. Finney (1970) explored methodologies for measuring and characterizing the vibration response of many fruits, and concluded that Young's modulus and shear modulus of apple flesh were correlated with the product between the resonance frequency squared and the mass of the whole fruit. It was also found that the

obtained with the OSF and conventional rheological methods

G' Conventional Rheology

102

G' or G" (Pa)

104

again the method used with these machines is a destructive test.

An instrument that measures force as a function of displacement, such as a UTM, can be used to obtain force-displacement curves of the samples. For a melon, the flesh and the rind should be tested differently. Melons have a rind that is significantly stiffer than the flesh. The cylindrical samples of the flesh can be tested with a compression test. The rind samples can be tested using a three point jig, and treated as a simply supported beam for analysis.

For modeling vibrations, where only small displacements are of interest, the tests should be limited to small displacements and changes in the cross-section and in the length can be ignored in the data processing. Also, the mechanical properties of biological materials in general are not constant. Because of the structures at the micro scale, the Young's modulus is a function of strain. The following method can be used to facilitate the computation of the Young's modulus at zero deformation. The stress-strain curve can be fit to a cubic function

$$
\sigma = a\_3 \varepsilon^3 + a\_2 \varepsilon^2 + a\_1 \varepsilon + a\_0 \tag{20}
$$

The coefficients *a*0 through *a*3 can be obtained by cubic regression analysis. The Young's modulus is the derivative of the stress with respect to strain

$$E = \frac{d\sigma}{d\varepsilon} = 3a\_3\varepsilon^2 + 2a\_2\varepsilon + a\_1 \tag{21}$$

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 317

Fig. 12. Stress-strain curve of a core sample from compression test.

quality of fruits during consumption, storage and processing.

experimental modal analysis results.

**2.2.2.2 Numerical modal analysis with finite element modeling (FEM)** 

The application of stresses and strains in biological materials not only has a component associated to consumer evaluation but also it is related to quality control. Many of the stresses and strains are applied locally either during processing or consumption of the fruits. But they are distributed over the entire surface, thus it is necessary to use methods that can estimate the distribution of stresses over the product. Finite Element Modeling (FEM) is a well developed and proven mathematical and simulation tool to apply to the study of

Cherng (2000) stated that the lowest natural frequency for hollow fruit like melons corresponded to the elongation along the major axis. Cooke (1972) concluded that the most important mode was the first twisting mode because it corresponds to the shear modulus of the flesh, which in turn corresponds to the fruit ripeness. Finite element models published in Cherng (2000) did not model the rind separately but assumed that the rind has the same properties as the flesh. Nourain et al (2004) compared a finite element model to

As an illustration, a finite element mesh used for a melon fruit is shown in Figure 13, with an octant removed to show the inside. The idealized melon-like geometry is a prolate ellipsoid with the major axis in the vertical direction. The major radius of the flesh is 77.5mm. The minor radius is 75mm. The hollow radii are 52.5mm (major) and 50mm (minor). The rind thickness is 2.5mm. For simulation purposes assumptions for the rind Young's modulus was 4.0 MPa whereas values used for the flesh Young's modulus were 1.0MPa, 1.5MPa, 2MPa, and 3MPa. The modal frequencies will be given in the order corresponding to those Young's moduli. The Young's moduli selected above are somewhat arbitrary and do not represent the values shown in Figure 12 from the test applied to a

For vibration analysis and modeling, the most important value of the Young's modulus is at zero strain, which is simply *a*1. Therefore, the constitutive relation for vibration models is

$$
\sigma = \mathbb{E}\mathfrak{x} \tag{22}
$$

where *E* = *a1* from the test above.

In the bending test where the rind is treated as a simply supported beam, the equation for the young's modulus can be derived accordingly. If the force is applied in the middle of the span, then the deflection at that point is

$$\infty = \frac{FL^3}{48EI} \tag{23}$$

If the material is linear, the constant Young's modulus is

$$E = \frac{L^3}{48I} \frac{F}{\varkappa} \tag{24}$$

Where *L* is the span, and *I* is the moment of inertia of the rind sample, which is given by

$$I = \frac{\text{base.height}^3}{3} \tag{25}$$

For nonlinear materials, *E* can be obtained by taking the derivative of the force with respect to *x.* In particular, the Young's modulus at zero strain is

$$E(\mathbf{x}) = \frac{L^3}{48I} \frac{dF}{d\mathbf{x}}\bigg|\_{\mathbf{x}=0} \tag{26}$$

The modulus of elasticity of a melon was measured statically using an UTM machine, which can record the force as a function of deflection. For the flesh, cylindrical core samples were cut out of the melon; whereas for the rind, beam-like samples were cut out (Ehle, 2002). The forcedisplacement curve for the cylindrical core compression test was transformed into a stressstrain curve for small deflections by dividing the force by the cross-section area and dividing the deflection by the length. The small changes in cross-section area and the length were neglected because the displacement was small. Cubic regression of the stress-strain curve obtained the coefficients in Equation 20. Then Equation 21 was used to calculate the slope, which was the Young's modulus. The value at zero strain is the value used for modeling the vibration properties of the melon. Figure 12 shows that a cubic function fits the data very well, and that the slope at zero strain can be obtained accurately. In the above compression test of the flesh samples, the Young's modulus could also be obtained from the load-displacement curve without transforming into the stress-strain curve numerically. A little algebra gives:

$$E(\mathbf{x}) = \frac{L}{A} \left. \frac{df}{d\mathbf{x}} \right|\_{\mathbf{x}=0} \tag{27}$$

For the bending test, the force-displacement data were processed according to Eq. 24 to obtain the Young's modulus of the rind. The result shows characteristics similar to those illustrated in Figure 12.

For vibration analysis and modeling, the most important value of the Young's modulus is at zero strain, which is simply *a*1. Therefore, the constitutive relation for vibration models is

In the bending test where the rind is treated as a simply supported beam, the equation for the young's modulus can be derived accordingly. If the force is applied in the middle of the

> 3 48 *FL*

3 48 *L F <sup>E</sup>*

> <sup>3</sup> . 3

For nonlinear materials, *E* can be obtained by taking the derivative of the force with respect

3

( ) <sup>48</sup> *<sup>x</sup> L dF E x*

The modulus of elasticity of a melon was measured statically using an UTM machine, which can record the force as a function of deflection. For the flesh, cylindrical core samples were cut out of the melon; whereas for the rind, beam-like samples were cut out (Ehle, 2002). The forcedisplacement curve for the cylindrical core compression test was transformed into a stressstrain curve for small deflections by dividing the force by the cross-section area and dividing the deflection by the length. The small changes in cross-section area and the length were neglected because the displacement was small. Cubic regression of the stress-strain curve obtained the coefficients in Equation 20. Then Equation 21 was used to calculate the slope, which was the Young's modulus. The value at zero strain is the value used for modeling the vibration properties of the melon. Figure 12 shows that a cubic function fits the data very well, and that the slope at zero strain can be obtained accurately. In the above compression test of the flesh samples, the Young's modulus could also be obtained from the load-displacement curve without transforming into the stress-strain curve numerically. A little algebra gives:

0

0

*x*

*A dx*

For the bending test, the force-displacement data were processed according to Eq. 24 to obtain the Young's modulus of the rind. The result shows characteristics similar to those

( )

*<sup>L</sup> df E x*

*I dx*

Where *L* is the span, and *I* is the moment of inertia of the rind sample, which is given by

(22)

*EI* (23)

*I x* (24)

*base height <sup>I</sup>* (25)

(26)

(27)

 *E*

*x*

where *E* = *a1* from the test above.

illustrated in Figure 12.

span, then the deflection at that point is

If the material is linear, the constant Young's modulus is

to *x.* In particular, the Young's modulus at zero strain is

Fig. 12. Stress-strain curve of a core sample from compression test.

#### **2.2.2.2 Numerical modal analysis with finite element modeling (FEM)**

The application of stresses and strains in biological materials not only has a component associated to consumer evaluation but also it is related to quality control. Many of the stresses and strains are applied locally either during processing or consumption of the fruits. But they are distributed over the entire surface, thus it is necessary to use methods that can estimate the distribution of stresses over the product. Finite Element Modeling (FEM) is a well developed and proven mathematical and simulation tool to apply to the study of quality of fruits during consumption, storage and processing.

Cherng (2000) stated that the lowest natural frequency for hollow fruit like melons corresponded to the elongation along the major axis. Cooke (1972) concluded that the most important mode was the first twisting mode because it corresponds to the shear modulus of the flesh, which in turn corresponds to the fruit ripeness. Finite element models published in Cherng (2000) did not model the rind separately but assumed that the rind has the same properties as the flesh. Nourain et al (2004) compared a finite element model to experimental modal analysis results.

As an illustration, a finite element mesh used for a melon fruit is shown in Figure 13, with an octant removed to show the inside. The idealized melon-like geometry is a prolate ellipsoid with the major axis in the vertical direction. The major radius of the flesh is 77.5mm. The minor radius is 75mm. The hollow radii are 52.5mm (major) and 50mm (minor). The rind thickness is 2.5mm. For simulation purposes assumptions for the rind Young's modulus was 4.0 MPa whereas values used for the flesh Young's modulus were 1.0MPa, 1.5MPa, 2MPa, and 3MPa. The modal frequencies will be given in the order corresponding to those Young's moduli. The Young's moduli selected above are somewhat arbitrary and do not represent the values shown in Figure 12 from the test applied to a

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 319

Fig. 15. Mode 2, elongation along the major axis, 79.4Hz, 91.5Hz, 102.0Hz, 120Hz

Fig. 16. Twisting about vertical axis, 101.4Hz, 125.1Hz, 135.0Hz, 152.3Hz

melon. The purpose of this model is to give insight into how the vibration modes change with the stiffness of the flesh.

Fig. 13. A finite element mesh of a melon model

Figure 14 shows the elastic mode with the lowest natural frequency, which is elongation in the minor axis direction. Figure 15 shows the second mode, which is elongation in the major axis direction. Figure 16 shows the twisting mode. When the Young's modulus is 1.5MPa or lower, this mode changes shapes into the shape shown in Figure 17

Fig. 14. Mode 1, "sideways elongation", 75.2Hz, 90.4Hz, 97.4Hz, 115.0Hz

melon. The purpose of this model is to give insight into how the vibration modes change

Figure 14 shows the elastic mode with the lowest natural frequency, which is elongation in the minor axis direction. Figure 15 shows the second mode, which is elongation in the major axis direction. Figure 16 shows the twisting mode. When the Young's modulus is 1.5MPa or

with the stiffness of the flesh.

Fig. 13. A finite element mesh of a melon model

lower, this mode changes shapes into the shape shown in Figure 17

Fig. 14. Mode 1, "sideways elongation", 75.2Hz, 90.4Hz, 97.4Hz, 115.0Hz

Fig. 15. Mode 2, elongation along the major axis, 79.4Hz, 91.5Hz, 102.0Hz, 120Hz

Fig. 16. Twisting about vertical axis, 101.4Hz, 125.1Hz, 135.0Hz, 152.3Hz

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 321

Fig. 18. Measurement points for the above described testing

Fig. 19. A long elastic cord and rubber bands supporting the melon.

Fig. 17. Mode replacing the twisting mode when flesh Young's modulus is 1.5MPa
