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Use of the generalized Maxwell model for describing the stress relaxation behavior of solid-like foods
Journal of Food Engineering 78 (2007) 978–983 www.elsevier.com/locate/jfoodeng
Use of the generalized Maxwell model for describing the stress relaxation behavior of solid-like foods M.A. Del Nobile b
a,b,*
, S. Chillo b, A. Mentana b, A. Baiano
b
a Department of Food Science, University of Foggia, Via Napoli, 25-71100 Foggia, Italy Istituto per la Ricerca e le Applicazioni Biotecnologiche per la Sicurezza e la Valorizzazione dei Prodotti Tipici e di Qualita`, Universita` degli Studi di Foggia, Via Napoli, 25-71100 Foggia, Italy
Received 5 March 2005; accepted 2 December 2005 Available online 2 February 2006
Abstract A comprehensive study on the ability of the generalized Maxwell model to describe the stress relaxation behavior of food matrices is presented in this work. Five different food matrices were chosen as representative of a wide range of foods: agar gel, meat, ripened cheese, ‘‘mozzarella’’ cheese and white pan bread. The above foods matrices were selected on the base of their macro-structure and compressive stress strain behavior. Stress relaxation tests were run on the above food matrices at room temperature. The relaxation times distribution function used in this investigation to describe the viscoelastic behavior of the investigated foods was obtained by summing two normal distribution functions with the means equal to zero. Results show that the proposed model satisfactorily fits the experimental data. Moreover, it was also found a substantial difference between the relaxation times distribution curves of the investigated bulky and spongy foods. 2005 Elsevier Ltd. All rights reserved. Keywords: Modeling; Generalized maxwell model; Stress relaxation; Mechanical properties
1. Introduction Food products are complex matrices showing a wide range of different mechanical behaviors (Gunasekaran & Mehmet, 2000) even if some of them could be described with a certain approximation on the base of simplified models such as the ideal solid (elastic), the ideal liquid (viscous), the ideal plastic (elastic properties below a specific applied stress and viscous behavior above this value) and combination of two or more of them. Most food products are neither pure liquids nor pure solids, simultaneously showing viscous and elastic behaviors. For instance, when a force is applied to a viscoelastic food matrix it needs a certain time to acquire its new dimensions and, when the * Corresponding author. Address: Department of Food Science, University of Foggia, Via Napoli, 25-71100 Foggia, Italy. Tel.: +39 881 589 242; fax: +39 881 740 211. E-mail address: [email protected] (M.A. Del Nobile).
0260-8774/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2005.12.011
force is removed, the material could remain permanently deformed. To quantitatively determine the viscoelastic behavior of a matrix, transient (quasi-static) and dynamic tests can be performed. The more typical transient tests are represented by creep and stress relaxation experiments. In creep tests, a constant stress is applied to a specimen and the corresponding strain is measured as a function of time. The considered parameters are the ‘‘compliance’’ (that is the strain–stress ratio) and the ‘‘relaxation time’’. In stress relaxation tests, a constant strain is applied and the stress required to maintain the deformation is measured as a function of time. When a stress relaxation test is performed, different behaviors can be observed: ideal elastic materials do not relax whereas ideal viscous materials instantaneously show a relaxation. Viscoelastic solids gradually relax and reach an equilibrium stress greater than 0, whereas for viscoelastic fluids, instead, the residual stress vanishes to zero (Steffe, 1992). The dynamic experiments
M.A. Del Nobile et al. / Journal of Food Engineering 78 (2007) 978–983
differ from transient tests because a sinusoidal stress or strain is applied instead of a constant stress or strain. In order to describe the viscoelastic behavior of a material, massless mechanical models are used. These models are composed of springs (considered ideal solids, they account for the elastic behavior of viscoelastic materials) and dashpots (representing ideal fluids, they account for the viscous behavior) combined in many different ways. Three are the most commonly mechanical analogs used: the Kelvin–Voigt model, the Maxwell model (Bruno & Moresi, 2004; Correia & Mittal, 2002), and the standard linear solid one. The Kelvin–Voigt model, composed of a spring and a dashpot in parallel, represents the start point for the development of mechanical analogs describing the creep behavior. In fact, the Kelvin model is not sufficient to describe creeps in biological materials that are better modeled by the Burgers model (a Kelvin and a Maxwell element in series) (Steffe, 1992). A model consisting of one Maxwell element in series with two Kelvin–Voight elements is able to describe a liquid-like viscoelastic behavior (Mitchell & Blanshard, 1976). Stress relaxation data are very importance since they supply information about phenomena involving food products such as fruit ripening (Hassa, Alhamdan, & Elansari, 2005), fruit firmness (Blahovec, 1996), staling of cereal products (Limanond, Castell-Perez, & Moreira, 2002), checking phenomenon (Kim & Okos, 1999). The Maxwell model, consisting of a Hookean spring and a Newtonian dashpot in series (Mohsenin & Mittal, 1977), is suitable for understanding stress relaxation data, but does not consider the equilibrium stress. For this reason, the viscoelastic behavior of food can be better described by using a generalized Maxwell model consisting of several elements in parallel with a spring (Steffe, 1992). In a similar model, if the system is subjected to a constant strain, the total stress is the sum of the stress of each element. Since each element may have a different relaxation time, a relaxation spectra can be obtained for a viscoelastic material. Models containing more exponential components and a residual terms have been used to describe the stress relaxation behavior of Cheddar cheese (Hort, 1997) and ‘‘pasta filata’’ cheese (Masi, 1989). Mancini, Moresi, and Rancini (1999) were able to describe the viscoelastic behavior of several alginate gels differing for the effective alginate concentration by means of a generalized Maxwell model consisting of five elements. A model including one spring and two Maxwell elements was successfully use to describe the stress relaxation of lipids such as beeswax, candelilla wax, carnauba wax and a high-melting milkfat fraction (Shellhammer, Rumsey, & Krochta, 1997). Limanond et al. (2002) modeled the kinetics of corn tortilla staling through stress relaxation data and found that a sevenelement generalized Maxwell model fits the data better than the three and five ones. The viscoelastic properties of some pork ham muscles were better fitted to a generalized Maxwell model consisting of a parallel coupling of a
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Hookes body and two Maxwell’s bodies (Lachowicz, Sobczak, Gajowiecki, & Zych, 2003). The compression stress relaxation data obtained for potatoes exposed to pulsified electric field were modeled with a five parameter generalized Maxwell model (Finca & Dejmek, 2003). Hassa et al. (2005) compared three popular stress relaxation models, namely the generalized Maxwell, Nussinovitch, and Peleg model, in order to describe the viscoelastic properties of eight date cultivars at their khalal (balah) and rutab stages of maturity. They found that all considered models were valid for quantifying the relaxation behavior of the products but the generalized Maxwell model was the best in predicting experimental data. The standard linear solid model, also called Zener model, can consist in two different mechanical analogs: a spring in series with a Kelvin model or a spring in parallel with a Maxwell model. In addition to the used model, the equilibrium modulus, the decay modulus, time of relaxation, and specific viscosity in the relaxation model are affected by the specimen orientation and its location with the product (Wang, 2003). In this study the possibility to use the generalized Maxwell model for describing the stress relaxation behavior of food matrices is addressed. Five different food matrices were selected with the criteria to be representative of a wide class of foods. Stress relaxation tests were run at room temperature and the posed model was fit to the experimental data to establish if the generalized Maxwell model can be used to describe the viscoelastic behavior of foods. 2. Materials and methods Five different matrices were selected as representative of a wide class of food products: agar gel and four commercially available foods such as ‘‘mozzarella’’ cheese, meat, ripened cheese and white pan bread. The mean composition for each of the considered products (Carnovale & Marletta, 1997) is reported below: ‘‘mozzarella cheese’’—water 58.8%, proteins 18.7%, lipids 19.5%, sugars 0.7%, mineral salts and vitamins; meat (beef)—76.9% water, proteins 20.7%, lipids 1.0%, mineral salts; ripened cheese—water 31.5%, proteins 33.9%, lipids 28.5%, sugars 3.7%, mineral salts and vitamins; white pan bread—water 33.5%, proteins 9%, lipids 8.7%, sugars 48.2%, mineral salts and vitamins. The preparation of the food matrices, performed immediately before dynamic-mechanical analyses, consisted in cutting 13 mm-diameter specimens by means of a cylindrical mould. The external portion of each cylindrical sample was covered with some grease to avoid dehydration during the dynamic-mechanical analysis. Agar gel was prepared by dissolving agar (Oxoid, Milan, Italy) in distilled water (final concentration 1.0% w/v). The obtained solution was then heat treated at 121 C for
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15 min. Cylindrical specimens (11 mm-diameter) were obtained by injection in a mould and cooling at room temperature. Before analyses, these specimen were refrigerated at 4 C for 2 h. Gels represent good systems for basic rheological studies since they are usually isotropic and homogeneous. 2.1. Elastic modulus determination The investigated food matrices were submitted to compression tests by means of a dynamic mechanical analyser (DMA-Q 800, TA Instruments, New Castle, DE, USA) equipped with parallel plates (B15 mm) to obtain the stress–strain curves. Tests were carried out at room temperature according to the following conditions. Preload force: 104 N for agar; 103 N for mozzarella, ripened cheese and white pan bread; 102 N for meat. Crosshead speed: 0.5 N/min for agar and mozzarella; 1 N/min for meat, ripened cheese and white pan bread. The elastic modulus could be evaluated from the initial slope of the stress–strain curve. However, since the samples surfaces are not perfectly parallel, the estimation of the storage modulus is generally affected by an error that might be quite significant. For this reason, a mathematical model capable to describe the entire stress–strain curve was used: rT ðeT Þ ¼ EC eT expðeT KÞ
ria of choosing food matrices representative of a wide range of foods. In fact, three food matrices were selected as representative of bulky foods (that is, agar, meat and ripened cheese); whereas, the other two were chosen as representative of spongy foods (that is, white pan bread and ‘‘mozzarella’’ cheese). Fig. 1 shows the stress–strain curves relative to the investigated bulky foods, whereas Fig. 2 shows the stress–strain curves relative to the investigated spongy foods. The curves shown in the figures are the best fit of Eq. (1) to the experimental data. As expected the concavity of the stress–strain curve is positive for the bulky food matrices whereas a downward concavity was found for the stress–strain curves of the spongy foods. In fact, spongy foods are more elastic than the bulky ones, due to the presence of pores containing gas instead of interstitial spaces containing fluids that, for the effects of the compression, definitively leave the sample. The values of the elastic modulus obtained according to the procedure reported above are listed in Table 1. As can be inferred
ð1Þ
where eT and rT (MPa) are the true strain and the true stress, respectively (Mancini et al., 1999), EC (MPa) is the elastic modulus (i.e., the tangent to the stress strain curve at the origin), K is a constant and have to be regarded as fitting parameter. 2.2. Stress relaxation tests Compression tests were also performed in order to evaluate the stress relaxation behavior of the chosen five matrices. Tests were carried out at room temperature according to the following conditions. Preload force: 104 N for agar; 103 N for mozzarella, meat, ripened cheese and white pan bread. Instantaneous strain: 6.0% for mozzarella and white pan bread; 6.5% for agar; 7.5% for meat and ripened cheese. Relaxation time: 10 min. The stress relaxation value was calculated as the time-dependent stress divided by the constant strain. All the tests were replicated three times, thus the mean values and the corresponding standard deviations are reported in the following.
Fig. 1. Compressive stress strain test conducted at room temperature: (s) agar; (n) meat; (h) cheese. The curves shown in the figure were obtained by fitting Eq. (1) to the experimental data. (—–) agar; (- - - -) meat; (— —) ripened cheese.
3. Results and discussion 3.1. Food matrices selection As reported above the suitability of the generalized Maxwell model to be used to describe the stress relaxation behavior of foods was addressed by using five different food matrices. The selection was carried out with the crite-
Fig. 2. Compressive stress strain test conducted at room temperature: () ‘‘mozzarella’’ cheese; (j) white pan bread. The curves shown in the figure were obtained by fitting Eq. (1) to the experimental data. (—–) ‘‘mozzarella’’ cheese; (- - - -) white pan bread.
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Table 1 Elastic modulus and relaxation times distribution curve parameters of the investigated food matrices EC (MPa) Agar Meat Ripened cheese ‘‘Mozzarella’’ cheese White pan bread
3
G1 (MPa)
d1 4
13.7 · 10 (5.66 · 10 ) 6.49 · 103 (5.69 · 105) 0.102 (0.0293) 11.9 · 103 (1.44 · 103) 19.6 · 103 (3.39 · 103)
80.6 45.0 30.4 17.2 5.57
(34.4) (7.11) (9.82) (6.10) (2.11)
from this table, the elastic modulus values of the investigated food span in a wide range of values, corroborating the idea that the selected group of foods can be indeed considered as representative of a wide range of food matrices. 3.2. Stress relaxation tests As reported in the literature (Rosen, 1982) the generalized Maxwell model is given by the following expression: Z 1 t rðtÞ EðtÞ ¼ ¼ EðkÞ exp dk ð2Þ e0 k 0
3
G2 (MPa)
d2 3
12.9 · 10 (7.52 · 10 ) 1.18 · 103 (4.95 · 104) 0.464 (0.0469) 12.7 · 103 (2.17 · 103) 5.44 · 103 (2.12 · 103)
1820 1490 1190 197 315
(614) (628) (262) (42.3) (47.4)
20.1 · 103 (2.47 · 103) 2.01 · 103 (6.99 · 104) 0.740 (0.164) 15.3 · 103 (2.73 · 103) 20.6 · 103 (6.45 · 103)
decrease in the stress after which the stress level off to constant value. This behavior was in accord with that found by Mancini et al. (1999), who pointed out an asymptotically decaying trend for the alginate gels tested. As can be inferred from the above figures, the final stress values obtained for spongy matrices (‘‘mozzarella’’ cheese and white pan bread) were lower than those calculated for bulky materials. The curves shown in the above figure were obtained by fitting Eq. (4) to the experimental data. As can be inferred from the data shown in Figs. 3–5 the generalized Maxwell model excellently fits the experimental data.
where E(t) (MPa) is the elastic modulus at time t(s), r(t) (MPa) is the stress at time t(s), e0 is the imposed strain, E(k) (MPa) is the continuous distribution function of relaxation times, k(s) is the relaxation time. In principle any expression can be used for describing the continuous distribution function of relaxation times, in this investigation the following expression was used: ( " 2 #) 1 1 k pffiffiffiffiffiffiffiffiffi exp EðkÞ ¼ G1 2 d d1 2 p 1 ( " 2 #) 1 1 k pffiffiffiffiffiffiffiffiffi exp þ G2 ð3Þ 2 d2 d2 2 p Eq. (3) was obtained by summing two normal distribution functions with the means equal to zero and a standard deviations equal to di, each of which was multiplied by a constant (Gi). The parameters appearing in Eq. (3) account for the height (Gi) and wideness (di) of the relaxation time distribution curves. By substituting Eq. (3) in Eq. (2) the following expression is obtained: ( " 2 #) Z 1 1 1 k pffiffiffiffiffiffiffiffiffi exp EðtÞ ¼ G1 2 d d 2 p 1 0 1 ( Z t 1 1 pffiffiffiffiffiffiffiffiffi exp dk þ G2 k d2 2 p 0 " 2 #) 1 k t exp ð4Þ exp dk 2 d2 k The quadrature of a function such as that appearing in Eq. (4) was made by using the extended Simpson’s rule (Press, Flannery, Teukolsky, & Vetterling, 1989, Chapter 4). Figs. 3–5 show the stress relaxation texts for the five investigated foods. As expected there is first a sharp
Fig. 3. Compressive stress plotted as a function of time in a stress relaxation test conducted at room temperature. ( ) agar, (}) ‘‘mozzarella’’ cheese, (j) white pan bread.
Fig. 4. Compressive stress plotted as a function of time in a stress relaxation test conducted at room temperature on meat samples.
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Fig. 5. Compressive stress plotted as a function of time in a stress relaxation test conducted at room temperature on ripened cheese samples.
The values of the parameters appearing in Eq. (4) are listed in Table 1. Bulky and spongy foods particularly differed each other for the di values that were higher for the first ones. The values of the generalized Maxwell model’s parameters listed in Table 1 were used to calculate the relaxation time distribution curve for the investigated foods, the results are shown in Figs. 6–9. As can be inferred from the above figures for each of the investigated foods the continuous distribution curve of relaxation times is given by the superposition of two well districted contributions: the first part describe the short time response of the food matrices, and is narrow and high; the second describe the long time response of the food matrices, and is wide and low. As one would expect the relaxation times distribution curve strongly depends on the food structure. In fact, for spongy foods, such as ‘‘mozzarella’’ cheese and white pan bread, the relaxation times distribution curve vanish to zero at relaxation time ranging between 600 and 1000 s; whereas, for bulky food matrices, such as meat, ripened cheese and agar, the relaxation times distribution curve vanish to zero at relaxation time ranging between 2000 and
Fig. 7. The relaxation times distribution curve of meat.
Fig. 8. The relaxation times distribution curve of ripened cheese.
Fig. 9. The relaxation times distribution curve of (—–) ‘‘mozzarella’’ cheese and (- - - -) white pan bread.
Fig. 6. The relaxation times distribution curve of agar.
5000 s. Since spongy foods are more elastic than the bulky ones, the release of the stress is faster in the former than in the latter. If the applied strain is not higher than the critical strain, spongy foods recover the initial shape whereas bulky foods show a permanent deformation. Independently from food structure, stress relaxation of high moisture foods such as ‘‘mozzarella’’ cheese, meat and agar gel
M.A. Del Nobile et al. / Journal of Food Engineering 78 (2007) 978–983
would seem associated with water loss from the matrix pores (Tang, Tung, & Zeng, 1998). 4. Conclusions The possibility of using the gneralized Maxwell model to describe the stress relaxation behavior of food matrices was addressed in this work. Three food matrices representative of bulky foods and two representative of spongy foods were used. Their compressive stress strain behavior was determined. Results show that the elastic modulus of the examined samples span in wide range of values corroborating the idea that the selected group of foods can be considered representative of a wide range of food matrices. Stress relaxation tests were run on the selected food matrices. The posed model was fit to the experimental data. Results show that the proposed model satisfactorily fits the experimental data suggesting that it can be advantageously used to determine the viscoelastic behavior of many foods. It was also found that the relaxation times distribution curve is given by the superposition of two districted contributions: the first describe the short time response of the food matrices; the second describe the long time response of the food matrices. Moreover, a substantial difference between the relaxation times distribution curves of the investigated bulky and spongy foods was found. References Blahovec, J. (1996). Stress relaxation in cherry fruit. Biorheology, 33(6), 451–462. Bruno, M., & Moresi, M. (2004). Viscoelastic properties of Bologna sausages by dynamic methods. Journal of Food Engineering, 63, 291–298. Carnovale, E., & Marletta, L. (1997). Tables of food composition. Rome, Italy: Istituto Nazionale della Nutrizione. Correia, L. R., & Mittal, J. S. (2002). Viscoelastic properties of meat emulsions. In M. A. Rao & J. F. Steffe (Eds.), Viscoelastic properties of foods (pp. 185–204). London: Elsevier Applied Science.
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Finca, M., & Dejmek, P. (2003). Effect of osmotic pretreatment and pulsed electric field on the viscoelastic properties of potato tissue. Journal of Food Engineering, 59, 169–175. Gunasekaran, S., & Mehmet, M. Ak. (2000). Dynamic oscillatory shear testing of foods—selected applications. Trends in Food Science and Technology, 11, 115–127. Hassa, B. H., Alhamdan, A. M., & Elansari, A. M. (2005). Stress relaxation of dates at khalal and rutab stages of maturity. Journal of Food Engineering, 66, 439–445. Hort, J. (1997). Cheddar cheese: its texture, chemical composition and rheological properties. Ph.D. Thesis, Sheffield Hallam University. Kim, M. H., & Okos, M. R. (1999). Some physical, mechanical, and transport properties of crackers related to the checking phenomenon. Journal of Food Engineering, 40, 189–198. Lachowicz, K., Sobczak, M., Gajowiecki, L., & Zych, A. (2003). Effects of massaging time on texture, rheological properties, and structure of three pork ham muscles. Meat Science, 63, 225–233. Limanond, B., Castell-Perez, M. E., & Moreira, R. G. (2002). Modeling the kinetics of corn tortilla staling using stress relaxation data. Journal of Food Engineering, 53, 237–247. Mancini, M., Moresi, M., & Rancini, R. (1999). Mechanical properties of alginate gels: empirical characterization. Journal of Food Engineering, 39, 369–378. Masi, P. (1989). Characterization of history-dependent stress relaxation behaviour of cheeses. Journal of Texture Studies, 19, 373–388. Mitchell, J. R., & Blanshard, J. M. V. (1976). Rheological properties of nalginate gels. Journal of Texture Studies, 7, 219–234. Mohsenin, N. N., & Mittal, J. P. (1977). Use of rheological terms and correlation of compatible measurements in food texture research. Journal of Texture Studies, 8, 395–408. Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. (1989). Numerical recipes in Pascal Cambridge. University Press, p. 122. Rosen, L. H. (1982). Fundamental principles of polymeric materials. New York: John Wiley and Sons, p. 250. Shellhammer, T. H., Rumsey, T. R., & Krochta, J. M. (1997). Viscoelastic properties of edible lipids. Journal of Food Engineering, 33, 305–320. Steffe, J. F. (1992). Rheological methods in food process engineering. East Lansing, MI, USA.: Freeman Press. Tang, J., Tung, M. A., & Zeng, Y. (1998). Characterization of gellan gels using stress relaxation. Journal of Food Engineering, 38, 279– 295. Wang, J. (2003). Anisotropic relaxation properties of pear. Biosystems Engineering, 85(1), 59–65.
Use of the generalized Maxwell model for describing the stress relaxation behavior of solid-like foods M.A. Del Nobile b
a,b,*
, S. Chillo b, A. Mentana b, A. Baiano
b
a Department of Food Science, University of Foggia, Via Napoli, 25-71100 Foggia, Italy Istituto per la Ricerca e le Applicazioni Biotecnologiche per la Sicurezza e la Valorizzazione dei Prodotti Tipici e di Qualita`, Universita` degli Studi di Foggia, Via Napoli, 25-71100 Foggia, Italy
Received 5 March 2005; accepted 2 December 2005 Available online 2 February 2006
Abstract A comprehensive study on the ability of the generalized Maxwell model to describe the stress relaxation behavior of food matrices is presented in this work. Five different food matrices were chosen as representative of a wide range of foods: agar gel, meat, ripened cheese, ‘‘mozzarella’’ cheese and white pan bread. The above foods matrices were selected on the base of their macro-structure and compressive stress strain behavior. Stress relaxation tests were run on the above food matrices at room temperature. The relaxation times distribution function used in this investigation to describe the viscoelastic behavior of the investigated foods was obtained by summing two normal distribution functions with the means equal to zero. Results show that the proposed model satisfactorily fits the experimental data. Moreover, it was also found a substantial difference between the relaxation times distribution curves of the investigated bulky and spongy foods. 2005 Elsevier Ltd. All rights reserved. Keywords: Modeling; Generalized maxwell model; Stress relaxation; Mechanical properties
1. Introduction Food products are complex matrices showing a wide range of different mechanical behaviors (Gunasekaran & Mehmet, 2000) even if some of them could be described with a certain approximation on the base of simplified models such as the ideal solid (elastic), the ideal liquid (viscous), the ideal plastic (elastic properties below a specific applied stress and viscous behavior above this value) and combination of two or more of them. Most food products are neither pure liquids nor pure solids, simultaneously showing viscous and elastic behaviors. For instance, when a force is applied to a viscoelastic food matrix it needs a certain time to acquire its new dimensions and, when the * Corresponding author. Address: Department of Food Science, University of Foggia, Via Napoli, 25-71100 Foggia, Italy. Tel.: +39 881 589 242; fax: +39 881 740 211. E-mail address: [email protected] (M.A. Del Nobile).
0260-8774/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2005.12.011
force is removed, the material could remain permanently deformed. To quantitatively determine the viscoelastic behavior of a matrix, transient (quasi-static) and dynamic tests can be performed. The more typical transient tests are represented by creep and stress relaxation experiments. In creep tests, a constant stress is applied to a specimen and the corresponding strain is measured as a function of time. The considered parameters are the ‘‘compliance’’ (that is the strain–stress ratio) and the ‘‘relaxation time’’. In stress relaxation tests, a constant strain is applied and the stress required to maintain the deformation is measured as a function of time. When a stress relaxation test is performed, different behaviors can be observed: ideal elastic materials do not relax whereas ideal viscous materials instantaneously show a relaxation. Viscoelastic solids gradually relax and reach an equilibrium stress greater than 0, whereas for viscoelastic fluids, instead, the residual stress vanishes to zero (Steffe, 1992). The dynamic experiments
M.A. Del Nobile et al. / Journal of Food Engineering 78 (2007) 978–983
differ from transient tests because a sinusoidal stress or strain is applied instead of a constant stress or strain. In order to describe the viscoelastic behavior of a material, massless mechanical models are used. These models are composed of springs (considered ideal solids, they account for the elastic behavior of viscoelastic materials) and dashpots (representing ideal fluids, they account for the viscous behavior) combined in many different ways. Three are the most commonly mechanical analogs used: the Kelvin–Voigt model, the Maxwell model (Bruno & Moresi, 2004; Correia & Mittal, 2002), and the standard linear solid one. The Kelvin–Voigt model, composed of a spring and a dashpot in parallel, represents the start point for the development of mechanical analogs describing the creep behavior. In fact, the Kelvin model is not sufficient to describe creeps in biological materials that are better modeled by the Burgers model (a Kelvin and a Maxwell element in series) (Steffe, 1992). A model consisting of one Maxwell element in series with two Kelvin–Voight elements is able to describe a liquid-like viscoelastic behavior (Mitchell & Blanshard, 1976). Stress relaxation data are very importance since they supply information about phenomena involving food products such as fruit ripening (Hassa, Alhamdan, & Elansari, 2005), fruit firmness (Blahovec, 1996), staling of cereal products (Limanond, Castell-Perez, & Moreira, 2002), checking phenomenon (Kim & Okos, 1999). The Maxwell model, consisting of a Hookean spring and a Newtonian dashpot in series (Mohsenin & Mittal, 1977), is suitable for understanding stress relaxation data, but does not consider the equilibrium stress. For this reason, the viscoelastic behavior of food can be better described by using a generalized Maxwell model consisting of several elements in parallel with a spring (Steffe, 1992). In a similar model, if the system is subjected to a constant strain, the total stress is the sum of the stress of each element. Since each element may have a different relaxation time, a relaxation spectra can be obtained for a viscoelastic material. Models containing more exponential components and a residual terms have been used to describe the stress relaxation behavior of Cheddar cheese (Hort, 1997) and ‘‘pasta filata’’ cheese (Masi, 1989). Mancini, Moresi, and Rancini (1999) were able to describe the viscoelastic behavior of several alginate gels differing for the effective alginate concentration by means of a generalized Maxwell model consisting of five elements. A model including one spring and two Maxwell elements was successfully use to describe the stress relaxation of lipids such as beeswax, candelilla wax, carnauba wax and a high-melting milkfat fraction (Shellhammer, Rumsey, & Krochta, 1997). Limanond et al. (2002) modeled the kinetics of corn tortilla staling through stress relaxation data and found that a sevenelement generalized Maxwell model fits the data better than the three and five ones. The viscoelastic properties of some pork ham muscles were better fitted to a generalized Maxwell model consisting of a parallel coupling of a
979
Hookes body and two Maxwell’s bodies (Lachowicz, Sobczak, Gajowiecki, & Zych, 2003). The compression stress relaxation data obtained for potatoes exposed to pulsified electric field were modeled with a five parameter generalized Maxwell model (Finca & Dejmek, 2003). Hassa et al. (2005) compared three popular stress relaxation models, namely the generalized Maxwell, Nussinovitch, and Peleg model, in order to describe the viscoelastic properties of eight date cultivars at their khalal (balah) and rutab stages of maturity. They found that all considered models were valid for quantifying the relaxation behavior of the products but the generalized Maxwell model was the best in predicting experimental data. The standard linear solid model, also called Zener model, can consist in two different mechanical analogs: a spring in series with a Kelvin model or a spring in parallel with a Maxwell model. In addition to the used model, the equilibrium modulus, the decay modulus, time of relaxation, and specific viscosity in the relaxation model are affected by the specimen orientation and its location with the product (Wang, 2003). In this study the possibility to use the generalized Maxwell model for describing the stress relaxation behavior of food matrices is addressed. Five different food matrices were selected with the criteria to be representative of a wide class of foods. Stress relaxation tests were run at room temperature and the posed model was fit to the experimental data to establish if the generalized Maxwell model can be used to describe the viscoelastic behavior of foods. 2. Materials and methods Five different matrices were selected as representative of a wide class of food products: agar gel and four commercially available foods such as ‘‘mozzarella’’ cheese, meat, ripened cheese and white pan bread. The mean composition for each of the considered products (Carnovale & Marletta, 1997) is reported below: ‘‘mozzarella cheese’’—water 58.8%, proteins 18.7%, lipids 19.5%, sugars 0.7%, mineral salts and vitamins; meat (beef)—76.9% water, proteins 20.7%, lipids 1.0%, mineral salts; ripened cheese—water 31.5%, proteins 33.9%, lipids 28.5%, sugars 3.7%, mineral salts and vitamins; white pan bread—water 33.5%, proteins 9%, lipids 8.7%, sugars 48.2%, mineral salts and vitamins. The preparation of the food matrices, performed immediately before dynamic-mechanical analyses, consisted in cutting 13 mm-diameter specimens by means of a cylindrical mould. The external portion of each cylindrical sample was covered with some grease to avoid dehydration during the dynamic-mechanical analysis. Agar gel was prepared by dissolving agar (Oxoid, Milan, Italy) in distilled water (final concentration 1.0% w/v). The obtained solution was then heat treated at 121 C for
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15 min. Cylindrical specimens (11 mm-diameter) were obtained by injection in a mould and cooling at room temperature. Before analyses, these specimen were refrigerated at 4 C for 2 h. Gels represent good systems for basic rheological studies since they are usually isotropic and homogeneous. 2.1. Elastic modulus determination The investigated food matrices were submitted to compression tests by means of a dynamic mechanical analyser (DMA-Q 800, TA Instruments, New Castle, DE, USA) equipped with parallel plates (B15 mm) to obtain the stress–strain curves. Tests were carried out at room temperature according to the following conditions. Preload force: 104 N for agar; 103 N for mozzarella, ripened cheese and white pan bread; 102 N for meat. Crosshead speed: 0.5 N/min for agar and mozzarella; 1 N/min for meat, ripened cheese and white pan bread. The elastic modulus could be evaluated from the initial slope of the stress–strain curve. However, since the samples surfaces are not perfectly parallel, the estimation of the storage modulus is generally affected by an error that might be quite significant. For this reason, a mathematical model capable to describe the entire stress–strain curve was used: rT ðeT Þ ¼ EC eT expðeT KÞ
ria of choosing food matrices representative of a wide range of foods. In fact, three food matrices were selected as representative of bulky foods (that is, agar, meat and ripened cheese); whereas, the other two were chosen as representative of spongy foods (that is, white pan bread and ‘‘mozzarella’’ cheese). Fig. 1 shows the stress–strain curves relative to the investigated bulky foods, whereas Fig. 2 shows the stress–strain curves relative to the investigated spongy foods. The curves shown in the figures are the best fit of Eq. (1) to the experimental data. As expected the concavity of the stress–strain curve is positive for the bulky food matrices whereas a downward concavity was found for the stress–strain curves of the spongy foods. In fact, spongy foods are more elastic than the bulky ones, due to the presence of pores containing gas instead of interstitial spaces containing fluids that, for the effects of the compression, definitively leave the sample. The values of the elastic modulus obtained according to the procedure reported above are listed in Table 1. As can be inferred
ð1Þ
where eT and rT (MPa) are the true strain and the true stress, respectively (Mancini et al., 1999), EC (MPa) is the elastic modulus (i.e., the tangent to the stress strain curve at the origin), K is a constant and have to be regarded as fitting parameter. 2.2. Stress relaxation tests Compression tests were also performed in order to evaluate the stress relaxation behavior of the chosen five matrices. Tests were carried out at room temperature according to the following conditions. Preload force: 104 N for agar; 103 N for mozzarella, meat, ripened cheese and white pan bread. Instantaneous strain: 6.0% for mozzarella and white pan bread; 6.5% for agar; 7.5% for meat and ripened cheese. Relaxation time: 10 min. The stress relaxation value was calculated as the time-dependent stress divided by the constant strain. All the tests were replicated three times, thus the mean values and the corresponding standard deviations are reported in the following.
Fig. 1. Compressive stress strain test conducted at room temperature: (s) agar; (n) meat; (h) cheese. The curves shown in the figure were obtained by fitting Eq. (1) to the experimental data. (—–) agar; (- - - -) meat; (— —) ripened cheese.
3. Results and discussion 3.1. Food matrices selection As reported above the suitability of the generalized Maxwell model to be used to describe the stress relaxation behavior of foods was addressed by using five different food matrices. The selection was carried out with the crite-
Fig. 2. Compressive stress strain test conducted at room temperature: () ‘‘mozzarella’’ cheese; (j) white pan bread. The curves shown in the figure were obtained by fitting Eq. (1) to the experimental data. (—–) ‘‘mozzarella’’ cheese; (- - - -) white pan bread.
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Table 1 Elastic modulus and relaxation times distribution curve parameters of the investigated food matrices EC (MPa) Agar Meat Ripened cheese ‘‘Mozzarella’’ cheese White pan bread
3
G1 (MPa)
d1 4
13.7 · 10 (5.66 · 10 ) 6.49 · 103 (5.69 · 105) 0.102 (0.0293) 11.9 · 103 (1.44 · 103) 19.6 · 103 (3.39 · 103)
80.6 45.0 30.4 17.2 5.57
(34.4) (7.11) (9.82) (6.10) (2.11)
from this table, the elastic modulus values of the investigated food span in a wide range of values, corroborating the idea that the selected group of foods can be indeed considered as representative of a wide range of food matrices. 3.2. Stress relaxation tests As reported in the literature (Rosen, 1982) the generalized Maxwell model is given by the following expression: Z 1 t rðtÞ EðtÞ ¼ ¼ EðkÞ exp dk ð2Þ e0 k 0
3
G2 (MPa)
d2 3
12.9 · 10 (7.52 · 10 ) 1.18 · 103 (4.95 · 104) 0.464 (0.0469) 12.7 · 103 (2.17 · 103) 5.44 · 103 (2.12 · 103)
1820 1490 1190 197 315
(614) (628) (262) (42.3) (47.4)
20.1 · 103 (2.47 · 103) 2.01 · 103 (6.99 · 104) 0.740 (0.164) 15.3 · 103 (2.73 · 103) 20.6 · 103 (6.45 · 103)
decrease in the stress after which the stress level off to constant value. This behavior was in accord with that found by Mancini et al. (1999), who pointed out an asymptotically decaying trend for the alginate gels tested. As can be inferred from the above figures, the final stress values obtained for spongy matrices (‘‘mozzarella’’ cheese and white pan bread) were lower than those calculated for bulky materials. The curves shown in the above figure were obtained by fitting Eq. (4) to the experimental data. As can be inferred from the data shown in Figs. 3–5 the generalized Maxwell model excellently fits the experimental data.
where E(t) (MPa) is the elastic modulus at time t(s), r(t) (MPa) is the stress at time t(s), e0 is the imposed strain, E(k) (MPa) is the continuous distribution function of relaxation times, k(s) is the relaxation time. In principle any expression can be used for describing the continuous distribution function of relaxation times, in this investigation the following expression was used: ( " 2 #) 1 1 k pffiffiffiffiffiffiffiffiffi exp EðkÞ ¼ G1 2 d d1 2 p 1 ( " 2 #) 1 1 k pffiffiffiffiffiffiffiffiffi exp þ G2 ð3Þ 2 d2 d2 2 p Eq. (3) was obtained by summing two normal distribution functions with the means equal to zero and a standard deviations equal to di, each of which was multiplied by a constant (Gi). The parameters appearing in Eq. (3) account for the height (Gi) and wideness (di) of the relaxation time distribution curves. By substituting Eq. (3) in Eq. (2) the following expression is obtained: ( " 2 #) Z 1 1 1 k pffiffiffiffiffiffiffiffiffi exp EðtÞ ¼ G1 2 d d 2 p 1 0 1 ( Z t 1 1 pffiffiffiffiffiffiffiffiffi exp dk þ G2 k d2 2 p 0 " 2 #) 1 k t exp ð4Þ exp dk 2 d2 k The quadrature of a function such as that appearing in Eq. (4) was made by using the extended Simpson’s rule (Press, Flannery, Teukolsky, & Vetterling, 1989, Chapter 4). Figs. 3–5 show the stress relaxation texts for the five investigated foods. As expected there is first a sharp
Fig. 3. Compressive stress plotted as a function of time in a stress relaxation test conducted at room temperature. ( ) agar, (}) ‘‘mozzarella’’ cheese, (j) white pan bread.
Fig. 4. Compressive stress plotted as a function of time in a stress relaxation test conducted at room temperature on meat samples.
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Fig. 5. Compressive stress plotted as a function of time in a stress relaxation test conducted at room temperature on ripened cheese samples.
The values of the parameters appearing in Eq. (4) are listed in Table 1. Bulky and spongy foods particularly differed each other for the di values that were higher for the first ones. The values of the generalized Maxwell model’s parameters listed in Table 1 were used to calculate the relaxation time distribution curve for the investigated foods, the results are shown in Figs. 6–9. As can be inferred from the above figures for each of the investigated foods the continuous distribution curve of relaxation times is given by the superposition of two well districted contributions: the first part describe the short time response of the food matrices, and is narrow and high; the second describe the long time response of the food matrices, and is wide and low. As one would expect the relaxation times distribution curve strongly depends on the food structure. In fact, for spongy foods, such as ‘‘mozzarella’’ cheese and white pan bread, the relaxation times distribution curve vanish to zero at relaxation time ranging between 600 and 1000 s; whereas, for bulky food matrices, such as meat, ripened cheese and agar, the relaxation times distribution curve vanish to zero at relaxation time ranging between 2000 and
Fig. 7. The relaxation times distribution curve of meat.
Fig. 8. The relaxation times distribution curve of ripened cheese.
Fig. 9. The relaxation times distribution curve of (—–) ‘‘mozzarella’’ cheese and (- - - -) white pan bread.
Fig. 6. The relaxation times distribution curve of agar.
5000 s. Since spongy foods are more elastic than the bulky ones, the release of the stress is faster in the former than in the latter. If the applied strain is not higher than the critical strain, spongy foods recover the initial shape whereas bulky foods show a permanent deformation. Independently from food structure, stress relaxation of high moisture foods such as ‘‘mozzarella’’ cheese, meat and agar gel
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would seem associated with water loss from the matrix pores (Tang, Tung, & Zeng, 1998). 4. Conclusions The possibility of using the gneralized Maxwell model to describe the stress relaxation behavior of food matrices was addressed in this work. Three food matrices representative of bulky foods and two representative of spongy foods were used. Their compressive stress strain behavior was determined. Results show that the elastic modulus of the examined samples span in wide range of values corroborating the idea that the selected group of foods can be considered representative of a wide range of food matrices. Stress relaxation tests were run on the selected food matrices. The posed model was fit to the experimental data. Results show that the proposed model satisfactorily fits the experimental data suggesting that it can be advantageously used to determine the viscoelastic behavior of many foods. It was also found that the relaxation times distribution curve is given by the superposition of two districted contributions: the first describe the short time response of the food matrices; the second describe the long time response of the food matrices. Moreover, a substantial difference between the relaxation times distribution curves of the investigated bulky and spongy foods was found. References Blahovec, J. (1996). Stress relaxation in cherry fruit. Biorheology, 33(6), 451–462. Bruno, M., & Moresi, M. (2004). Viscoelastic properties of Bologna sausages by dynamic methods. Journal of Food Engineering, 63, 291–298. Carnovale, E., & Marletta, L. (1997). Tables of food composition. Rome, Italy: Istituto Nazionale della Nutrizione. Correia, L. R., & Mittal, J. S. (2002). Viscoelastic properties of meat emulsions. In M. A. Rao & J. F. Steffe (Eds.), Viscoelastic properties of foods (pp. 185–204). London: Elsevier Applied Science.
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