RHEOLOGY of SOLID FOODS M. Anandha Ra Foods May Be

RHEOLOGY OF SOLID FOODS M. Anandha Ra

Foods may be classified in terms of their rheological properties and sensory attributes as liquids, semi-solids, soft solids and hard-solids (Van Vliet et al., 2009; Foegeding et al., 2011). It may be noted that while there is no universal definition for absolute transitions from one state to another, based on chewing criteria, it can generally be considered that: (1) fluid foods flow and do not require any chewing during oral processing (e.g., milk); (2) semisolid foods are processed in the mouth by squeezing the tongue and palate (e.g., pudding); (3) soft-solid foods require chewing but do not have “crispy” attributes (e.g., cheese); and (4) hard-solid foods are crispy and associated with an acoustic emission (e.g., crackers).

We have learned earlier that with many liquid foods, we work with their viscosity or apparent viscosity because of their shear-dependent nature. Foods that are semisolids, soft solids, and hard solids are viscoelastic materials, with varying levels of elastic (solid-like) behavior being predominant. One definition of a solid is that it is a state of matter characterized by particles arranged such that their shape and volume are relatively stable. Further, the constituents of a solid tend to be packed together much closer than the particles in a gas or liquid. In contrast, a liquid is one of the states of matter. The particles in a liquid are free to flow and while a liquid has a definite volume, it does not have a definite shape (http://chemistry.about.com/od/chemistryglossary/a/liquiddef.htm).

In many studies on solid foods, the results are expressed in force and deformation units, e.g., slope of force-deformation curve (often, erroneously referred to as Young’s modulus), fracture force, etc. As noted by various authors (Baltsavias et al., 1999; Dobraszczyk et al., 1987), such an approach does not allow comparison of results because the force-deformation relationship is strongly affected by specimen dimensions. Fundamental properties, such as Young’s modulus, derived from stress vs. strain data, are independent of specimen size. Fundamental mechanical parameters of solids include fracture stress and strain, yield stress, elastic modulus, Poisson’s ratio, coefficient of friction and fracture toughness.

Lillford (2001) reviewed the fracture properties of solid foods with particular emphasis on the role of microstructure. It was pointed out that the microscopic structure, and particularly 2

the examination of fracture surfaces, give significant insight into the origin of the materials’ properties. He noted that, “the microscopic structure, and particularly the examination of fracture surfaces, give significant insight into the origin of the materials’ properties.” In addition, Lillford (2001) pointed out several interesting relationships, outlined next, between mechanical properties and structural characteristics of solid foods. Bourne (2002) described fracture as occurring under Types 1, 2 and 3. Type 1 is described as a simple fracture, where the imposed stress has exceeded the strength of the material and the body has separated into 2 or more pieces or occasionally the fracture may be partial and the body may not separate into pieces. Type 2 is described as brittle fracture, where there is little or no deformation before fracture and the original un-deformed body may result in many pieces. Type 3 is described as ductile fracture where there is substantial plastic deformation and low energy absorption prior to fracture.

In a pioneering study, Griffith (1921) showed that the specific fracture energy (fracture toughness), �!, the elastic modulus, E, the fracture stress, �!, and the critical crack length, l, are related by:

� = !!!! (9.1) ! !"

�! can be measured directly from the experiment, and an estimate of the modulus, E, can be made, but l and �! are unknown, (the work of fracture can also be estimated from the area under the force-displacement curve, and is referred to as "fracture toughness"). Fracture toughness is a property of a material that describes the ability of the material containing a crack to resist fracture. The fracture toughness, to be discussed later, is determined from the stress intensity factor. Ashby (1983) has presented relations for the moduli and fracture stresses in air filled foams, obtaining: ! ! !! = �! (9.2) !! !!

Where, E = foam modulus, �! = cell wall modulus, �! = density of cell wall material, �! = bulk density of the foam, and C1 = a constant. 3

The crushing stress of a brittle foam, �!", and the bending fracture stress of cell wall, �!, are related according to:

! !!" !! ! = �! (9.3) !! !!

Where �! is a constant. For the elastic buckling stress, �!", the relationship is: ! !!" !! = �! (9.4) !! !!

Where �! is a constant. In agreement with the above relationship, the modulus of a cake, E, was related to the bulk density, �!: ! � = �!�! (9.5)

The constant, �!, varies with water activity. At a given water activity, assuming that the modulus of the cell wall modulus, �!, and the density of the cell wall material, �!, do not change significantly over a range of bulk densities:

!!!! �! = ! (9.6) !! The above equation predicts that the modulus and fracture stress decrease as air content is increased, in agreement with the softer texture of aerated bread and cake, loss of brittleness (jaggedness in stress-strain curves) with increased humidity (Lillford, 2001).

SOME RHEOLOGICAL TESTS ON SOLID FOODSÍ

The rheological tests on solid foods can be classified based on the type of strain used, such as: (1) extension/compression, (2) flexure, (3) torsion, and (4) small-amplitude oscillatory (dynamic). Commercial instruments capable of measuring force and deformation simultaneously can be used for conducting tests of types 1 and 2. For conducting a torsion strain test, a sample must be shaped like a capstan. Commercially available dynamic mechanical analyzers can BE used for conducting dynamic rheological tests.

Fracture and crack propagation are important in quality assessment of many foods, such as fruits and vegetables, biscuits, and cheeses; the fracture mechanics approach using concepts of energy is valid after the onset of failure (Lillford, 2001; Luyten et al., 1992). Systematic studies have been conducted to understand the relationship between composition and/or microstructure on fracture and crack propagation, as well as on the mechanical properties of a number of foods. Vincent et al. (1991) noted that a crack can be propagated by torsion (giving out-of plane tearing), by tension (giving crack-opening) or shear (giving edge-sliding), as well as the wedge fracture test. The choice of a fracture technique depends on material limitations and other constraints, such as specimen size, shape and homogeneity of the sample and its preparation, as well as additional preparation of the sample to fit in the chosen experimental set up (Foegeding et al., 2011). Studies conducted on specific foods should be consulted in order to examine the unique details of the experimental techniques.

Poisson's Ratio

Poisson's Ratio, ν, is a basic rheological property of ideal elastic solids and viscoelastic solids. It is defined as the ratio of lateral strain to axial strain in a specimen subjected to axial deformation. For very small deformation of an incompressible homogeneous material (rubbery), i.e., a material that does not change its volume when subjected to stress or strain, the Poisson’s ratio tends to 0.5. For compressible materials that show a certain reduction of volume under stress or strain are characterized by 0 < � < 0.5; specifically, for porous materials � tends to zero. It is the constant that relates modulus of rigidity to Young's modulus in the equation:

� = 2� � + 1 5

where, E is the Young's modulus; G, is the shear modulus (modulus of rigidity); and �, Poisson's ratio. The formula is valid only within the elastic limit of a material. For incompressible materials, E=3G.

Rohm et al. (1997) used a video-based method to monitor lateral specimen expansion during compression of selected solid foods continuously and to establish a procedure for the calculation of average compressional stress, based on actual values of a specimen’s cross- section. The test specimens were compressed to failure at constant crosshead speed of 10 mm/min between parallel stainless steel plates; the plate-food interfaces were generously lubricated with a low-viscosity paraffin oil.

The apparent Poisson’s ratio �!, which refers to the ratio of lateral expansion to uniaxial compressional strain, was calculated at specific values of deformation according to:

�!=�� !/�! �� ℎ! ℎ! ] (9.7)

Where the subscripts o and t refer to the initial and actual values. Values of the Poisson’s ratio for various foods are shown in Table 9.1 (Rohm et al., 1997). 6

Table 9.1. Values of the apparent Poisson ratio, �!, of selected food materials at different values of deformation (Rohm et al., 1997).

______

Sample Apparent Poisson ratios at different values of axial Hencky strain, ϵH:

ϵH~0.05 ϵH~0.12 ϵH~0.30 ϵH~0.65 ϵH~0.90 ______

Apples Red Delicious 0.21 0.25 f Jonagold 0.17 0.22 f Bread Rye bread A 0.28 0.22 0.21 0.19 0.20 Rye bread B 0.30 0.23 0.19 0.19 0.21 White bread C 0.17 0.14 0.11 0.07 0.07 Butter Sample A 0.42 0.44 0.43 f Sample B 0.44 0.45 0.45 f Potatoes Raw 0.38 0.43 0.46 f Steamed 0.40 0.42 f

f, beyond specimen fracture 7

EXTENSIONAL STRAIN STUDIES ON FILMS AND SKINS

The rheological properties of a number of solid, thin, ductile, foods, such as edible films and fruit skins can be determined in extension. First, we consider tests in which a sample is tested as-is, as opposed to tests in which a notch is made. In addition, extensional strain has been utilized to study fracture in foods that are brittle and ductile and these procedures are described later. Figure 9.1 is a typical example of the stress-strain data obtained on an apple skin subjected to extensional strain. In the figure, one can see a limited, initial, linear region of strain from the origin to point A. The apparent modulus can be calculated as: � = ∆� ∆�. With further increase in strain, the stress increases non-linearly with the stress response being concave up indicating the ductile nature of the material. At point B, the material yields and at point C, the material fractures. The fracture stress is also called ultimate tensile stress or break strength.

Protein films have found use as packaging materials. Values of the tensile properties of protein films (Sothornvit et al., 2007) are given in Table 9.2; the % elongation is the maximum % change in length before breaking. 8

2.5 Yield stress B 2.0

Fracture stress, point C 1.5

1.0 Stress, Stress, MPa

Δx 0.50 A Modulus of elasticity, E=Δy/Δx Δy 0.0 0 2 4 6 8 10 12 14 16 Strain, %

Figure 9.1. Stress vs. strain diagram for apple skin in extensional deformation: A is end of linear region, B is the yield stress; some foods may exhibit a more pronounced dip after yield stress, and C indicates the break strength. Protein films An Instron Universal Testing Machine was used to determine tensile strength (TS), elastic modulus (EM or Young’s modulus) and % elongation (%E) according to the ASTM standard method D882 of protein films (Sothornvit et al., 2007). Maximum TS is the largest stress (force/area) that a film or sheet is able to sustain before break. EM, calculated from the 9

slope of the initial linear region of the force–deformation curve, reflects stiffness. The %E is the maximum % change in length before breaking. Values of the rheological properties of several protein films reported by Hernandez- Izquierdo and Krochta (2008) are given in Table 9.2: [Hernandez-Izquierdo, V. M. and Krochta, J. M. 2008. Thermoplastic processing of proteins for film formation—A Review. Journal of Food Science 73, Nr. 2, R30-R39.] For comparison, the breaking stress of high-density polyethylene and poly vinyl chloride films are 26 and 93 MPa, respectively.

Table 9.2. Tensile properties of compression-molded and extruded protein films.

Film formulation Formation method Fracture stress Elastic modulus Elongation (MPa) (MPa) (%)

40% Gly-WPI Compression molding 8 144 85 50% Gly-WPI Compression molding 4 60 94 46% Gly-WPI Extrusion 4 46.5 127 49% Gly-WPI Extrusion 3.5 37 121 Oleic acid-zein Extrusion 4.2 96.4 96.3 40% Gly-SPI Compression molding 2.6 --- 74.5 50Gly:100 SPI Extrusion 7.1 144 ---

aAbbreviations: Gly = glycerol; WPI = whey protein isolate; SPI = soy protein isolate; Table based on: Hernandez-Izquierdo, V. M. and Krochta, J. M. 2008. Thermoplastic processing of proteins for film formation—A Review. Journal of Food Science 73, Nr. 2, R30-R39.

Rao and Brown (2011) studied the mechanical properties of samples of skins of several varieties of apples; specifically, they examined the modulus from stress-strain data of apple skin in tension and its breaking stress. The breaking stress of the skin of McIntosh, CA stored, apples,

2.51 MPa, was higher than that of the cold temperature stored apples, 2.03 MPa; however, these values were not significantly different (P≥0.05). The corresponding moduli showed an opposite trend: 26.2 MPa for the CA stored versus 34.1 MPa for the cold temperature stored apples (Table

9.3). The modulus of the skin of cold-stored McIntosh apple was significantly higher (P<0.05) than the moduli of the skins of the CA stored apples: McIntosh, Red Delicious, and Empire; 11

however, the breaking stress was not significantly different (P≥0.05). The moduli of the skins of

CA stored McIntosh, Red Delicious, and Empire apples were not substantially different (Table

9.3): 26.2 MPa, 25.1 MPa, and 27.1 MPa, respectively; values of the breaking stress of the skins of the different apple cultivars were also not significantly different (P≥0.05). The average values of the breaking stress and apparent modulus of the skins of light and heavy crop Honey Crisp and Crimson Gala apples were not significantly different (P≥0.05). However, the moduli of the skins of the crispy apples: Honey Crisp and Crimson Gala, skin were significantly lower

(P<0.05) than the skins of the other apples: McIntosh, Red Delicious, and Empire.

Table 9.3. Fracture stress and apparent modulus of the skins of McIntosh, cold and CA, and CA stored Red Delicious, and Empire applesa.

______

Cultivar, Fracture stress, MPa Apparent Modulus, MPa storage condition ______

McIntosh, CA, n=4 2.51±0.17c 26.2±6.3e

Red Delicious, CA, n=4 1.74±0.15c 25.1±5.9e

Empire, CA, n=5 2.66±0.52c 27.1±5.9e

Honey Crisp, light crop, n=9 1.48±0.26c 19.8±2.2f

Honey Crisp, heavy crop, n=9 1.44±0.25c 18.2±2.1f

Crimson Gala, light crop, n=8 1.90±0.31c 24.3±3.3f

Crimson Gala, heavy crop, n=8 1.85±0.31c 19.4±2.5f

______an refers to number of tests. In each column, values with the same letter are not significantly (p<0.05) different. bCold temperature: storage without control of atmosphere; these apples were not used in puncture tests. CA: controlled atmosphere storage 13

Three-point Bending Test

The three-point bending test, shown schematically in Figure 9.X, has been used extensively. For a beam of rectangular cross section in a three-point bending test (Baltsavias 1997), a number of parameters can be derived (http://en.wikipedia.org/wiki/Three_point_flexural_test).

Figure 9.2. Schematic diagram of a three-point bending test. 14

The tensile or compressive stress, σ is given by:

� = !!" (9.8) !!!!

Where, where F is the applied force (N), L the distance between supports, also known as span length (m), d the width of the test-piece (m), and b the thickness of the test-piece (also referred to as depth) (m). Denoting the maximum deflection at the center of the beam as y, the tensile or compressive strain (-), can be calculated from:

� = !!" (9.9) !!

Fracture properties of biscuits For dense solid materials, such as a biscuit, the fracture is not directly related to the intrinsic properties of the material but rather the presence of microscopic defects (cracks) (Lillford, 2001). The fracture properties of biscuits were studied by means of three-point bending tests; they were carried out at room temperature (Baltsavias et al., 1999). In these tests, the span length, L, was 4 cm and the Instron crosshead speed was set at 5 mm/min; the biscuits were placed on supports with their top surface down.

For a beam of rectangular cross section in a three-point bending test (Baltsavias et al., 1999; http://en.wikipedia.org/wiki/Three_point_flexural_test), the tensile or compressive stress, σ is given by:

� = !!" (9.10) !!!!

� = !!" (9.11) !!

Table 9.4. Influence of fat content and fat type on Young’s modulus (E), fracture stress (�!) and fracture strain (�!). Also shown are values of air volume fraction (�!"#$) and fat volume fraction

(�!"#) (Baltsavias et al., 1999). ______

3 Fat Fat �!"# E �! �!�10 �!"#$ type (%, w/w) (%) (MPa) (kPa) (-) (%) ______Standard 37·0 18·4 33 151 6·2 63 28·4 15·0 47 240 6·5 62 20·9 16·2 97 446 5·8 47 16·4 13·2 100 807 11·6 46 Firm 28·5 15·4 65 404 9·0d 62 20·9 19·3 92 671 8·3 36 liquid oil 20·5 13·1 125 829 8·6 56 16·3 11·7 149 961 7·9 52 ______

Baltsavias et al. (1999) suggested that reduction of fat content increased the modulus and the fracture stress of biscuits with the magnitude of the effect depending on the type of fat. With respect to sweeteners, sucrose syrup increased slightly the brittleness of biscuits, compared with crystalline sucrose; from this observation, authors suggested that sucrose crystals were not the fracture-inducing defects. Further, sugar-free biscuits had a significantly lower modulus and fracture stress. Substituting starch for part of the flour had relatively little effect on the mechanical properties.

Overall, the mechanical properties of biscuits were governed by: (1) �!"#$, �!"#, and the geometrical characteristics of the cellular materials, and (2) size of the inhomogeneities. The former influence the modulus which, together with the latter, influence the fracture stress and fracture strain. The state of the studied material, such as glassy or rubbery, and if fat is continuously ditributed, also play important roles. 16

Lillford (2001) noted that before swallowing, we must not only deform the food, but also break it into fragments and lubricate them with saliva. This requires us to understand the mechanism of their breakdown of composites in the mouth and therefore the physics of fracture mechanics becomes immediately relevant. Two major approaches that can be found in the study of fracture properties of foods; they include: (1) the approach used in the evaluation of polymers, especially the use of the single-edge notched bend (SENB) test (Williams and Cawood, 1990;

Duizer et al. 2011), and (2) the torsion test developed by Hamann and his colleagues (Diehl et al.

1979; Foegeding et al. 1994) in which a capstan shaped food sample is used.

The single-edge notched bend (SENB) geometry

Fracture mechanics was originally developed to describe fracture processes in stiff, linear and highly brittle solids. Such, linear elastic fracture mechanics (LEFM) describes the fracture resistance of a material either in terms of the stress distribution around a crack tip (Stress Intensity Factor) or in terms of the release of strain energy at which crack growth occurs. Both the Stress Intensity Factor and the strain energy released can be expressed in terms of stress, stiffness and crack length. It is noted that the Stress Intensity Factor, often expressed as � ! or

� !" is also called the Critical Stress Intensity Factor. The single-edge notched bend (SENB) geometry is essentially a rectangular body with a notch, as shown in Fig. 9.X, is used to evaluate the fracture properties of materials. This geometry was described in detail in Williams and Cawood (1990).

Figure 9.2. The single-edge notched bend (SENB) geometry (Williams and Cawood, 1990).

The SENB geometry is a well-established test method, that has been extensively used in research on metals and polymeric materials (Williams and Cawood, 1990). The SENB geometry eliminates the need for gripping the specimen; further, the test specimens could be prepared and tested in a minimal amount of time (Alvarez et al., 2000). The dimensions of each specimen were: length, S 40 mm, depth, W 8 mm and breadth, B 4 mm. The distance, L between the supports was 32 mm. Very importantly, an artificial crack, a in the form of a notch, was introduced into each specimen by using a razor blade; the notch-to-depth ratio, a/W was between

0.45 and 0.55 as recommended by Williams and Cawood (1990). Each specimen, in turn, was loaded in three-point bending and deformed at a crosshead displacement rate of 120 mm min-1.

After fracture the crack length was measured using a travelling microscope.

The fracture toughness, �! and the fracture energy, �!, can be evaluated assuming linear elastic behavior:

� = !!! � ! (9.12) ! !!!/! !

Where Fc, is the load at the onset of crack propagation, L the distance between the supports, B is the specimen breadth, W is the specimen depth, and f(a/W) is a geometric factor which takes into account the finite width effects and is given by (Alvarez et al., 2000; Duizer et al. 2011).

! ! ! ! ! ! !.!!! !! !.!"!!.!" ! ! !!.! !! !! � ! = ! ! ! (9.13) ! ! ! !! !! ! !! ! ! !

� �! = �� (9.14) 18

Where U is the energy represented by the area under the load displacement curve, from zero load to the crack initiation load, Fc, � is a calibration factor dependent on the ratios: a/W and L/W, and have been tabulated in Williams and Cawood (1990). When all the lengths are in meters, the

-2 unit for Gc is J m (Duizer et al. 2011).

Young’s Modulus, E for these materials was also determined using the SENB geometry, but without the notches; they were also deformed in a similar manner as for the fracture tests.

Young’s Modulus, E was determined from the following expression:

� = �! 4��! � � (9.15)

Where, � � is the initial slope of the load-displacement curve.

Recognizing, that �! can be written as �!, the fracture stress, the stress intensity factor,

�!", can be written as:

!.! K!" = C! σ!a (9.16)

In addition, Alvarez et al. (2000) performed uniaxial compression tests to determine the yield stress, �!, of the produce from the peak load of the compression load-displacement curve:

� = !! (9.17) !!!!

Where F is the load, H the original height, R the initial radius and h the current height (i.e., H-

δh).

The SENB geometry was used in fracture tests on vegetables and fruits by Alvarez et al.

(2000). Typically, the load-displacement curves were approximately linear up to a critical load

(marked by the arrow) where a crack propagated from the introduced notch in the specimen 19

resulting in fracture. Except for cucumber, an abrupt drop in the load was seen with the fractures.

In the case of celery and apple, saw-tooth type force-displacement curves were seen. The load corresponding to the onset of fracture, F, was used to determine the fracture toughness, Kc, using

Eq. 9.12. The fracture toughness and fracture energy values are shown in Table 9.5.

TABLE 9.5.

Fracture data for selected fruit and vegetables (Alvarez et al., 2000). ______

Produce Fracture toughness, � !" Fracture energy, �!, Yield stress, �! �� !/! ��!! �� ______

Carrot 48.6 3979 1.8

Celery l7.9 80.0 0.9

Cucumber 10.0 109.3 0.7

Granny Smith apple 10.1 39.6 0.3 ______

With respect to �!", the Roman numeral I refers to Mode I tensile or cleavage mode of crack growth. Mode I crack growth is the type most frequently encountered and most likely to result in failure. A crack can be made to propagate under other conditions, for example, in shear (mode II) or in tear (mode III) [shear (mode II) or tear (mode III)]. In order for the fracture toughness, KIC, values to be considered valid or a plane-strain value (i.e., a minimum value) they have to satisfy certain geometric criteria as detailed elsewhere (e.g., in the ASTM E-399-83 standard or in Williams and Cawood 1990):

!!" �! ≥ 2.5 (9.17) !!

Where �! is the minimum specimen breath and �! is the compressive yield stress. In the study of

Alvarez et al. (2000), values of �! were less than the specimens’ width. The experimental values of the Young’s modulus (MPa) for carrot, celery, cucumber, and apple, were: 6.7 ± 0.8, 4.1 ± 0.6, 1.3 ± 0.3, and 3.4 ± 0.3, respectively. Values of the Young’s modulus (MPa) estimated using 21

! Equation 9.X, � = !! , assuming a value of 0.37 for Poisson’s ratio, and the above values of ! ! !!!!

Gc and Kc were: 6.9, 4.6, 1.1, and 3.0 for carrot, celery, cucumber, and apple, respectively.

Duizer et al. (2011) determined values of fracture toughness, fracture energy, and brittleness of a number of foods also using the SENB geometry (Table 9.6).

Brittleness was estimated from the equation:

� � = �! (9.18)

Where � and �’ are the areas under the force–displacement curve before and after crack propagation, respectively

Table 9.6. Values of fracture toughness, fracture energy, and brittleness of several foods (Duizer et al., 2011). ______

Food Fracture toughness, � !" Fracture energy, �! Brittleness ! �� ! ��!! ______Beef Jerky 27.54 0.776 0.64 Cheese, Edam 4.37 0.072 -0.09 Cheese, Romano 23.99 0.269 -0.19 Biscuits, Ginger nut 100.00 0.214 0.55 Biscuits, Vanilla wine 20.42 0.085 -0.03 Dark chocolate 144.54 0.339 0.96 Egg whites, boiled 0.95 0.039 0.16 ______

Torsion Test The torsion test, developed by Don Hamann, has several advantages, such as: (1) torsion produces pure shear, a stress condition that does not significantly change the specimen volume, 22

(2) the specimen shape is maintained during the test, minimizing geometric considerations, (3) there is no restriction on the criterion for fracture, and the material can fail in shear, tension, compression or a combination mode; and (4) during the test, the internal pressure remains low so minimal fluids will be forced from the specimen during testing. The primary disadvantages are that specimen shaping and preparation are more complex than required for other procedures.

The typical capstan geometry (Foegeding et al., 1994) used is illustrated in Figure 9.X.

Figure 9.3. Capstan shape of specimen used for torsional fracture testing. Where, r is the radius of an arbitrary cross section of a torsion specimen; rmin is the radius of the smallest cross section of a torsion specimen; rc is the radius of curvature of a torsion specimen surface in a plane containing the longitudinal axis; Z is the Longitudinal axis of a torsion specimen; origin in the cross section of radius rmin; ZO is the one-half the effective length of a torsion specimen; and a is the distance from the z axis of a torsion 23

specimen to the center of curvature from which rc is measured (Diehl et al. 1979). Dimensions of the capstan-shaped specimens used by Foegeding et al. (1994) were: ZO = 6.36 mm, rc. = 9. 53 mm , a =

14.53 mm, length = 28.7 mm. The standard test conditions were a rotation rate of 2.5 r.p.m. (0.262 rad/s), full scale torque of 0.0288 N rn and an instrument spring constant of 1.39 radians at full scale torque ;

�!= 1580 �! (9.19)

�!= 0.150 t - 0.0079 �! (9.20)

Where, �! is shear stress at fracture (Pa), �!. is the torque digital reading (percent of scale), �! is the uncorrected shear strain at fracture (dimensionless), and t is time (s).

Based mostly on the work of Nadai in 1931on the shear stresses in permanently twisted bars an equation was presented to calculate the maximum shear stress at failure (Diehl et al.

1979):

! !!! �!"#= ! �! + 3�! (9.21) !!! !!!

Where, �! is angular deformation of the curved section (rad), Mt is the torque (N m), and the constant K is given by the equation:

! ! ! !! !"#!! !! � = ! (9.22) ! !!! !"#!! !!

It is assumed that the radial stress distribution in the torsion specimen is that of the moment- angle of twist curve and the shear stress is a monotonically increasing function of shear strain

(Diehl et al., 1979). In the event that the moment-angle of twist data are linear up to the occurrence of failure, then: 24

� = !!!! (9.23) !"# !!!

The maximum shear strain is given by:

�!!!"# = ��! (9.24)

!! Where, � = ! , and Q is given by: !!!"#!

! � = ! ! !" (9.25) ! ! ! ! ! !! !! !!

The shear modulus, G, is given by:

� = � !! (9.26) !!

Typical values of shear modulus of potato flesh reported were in the range, 1.26-1.82 MPa; for melon flesh the range was, 0.25-0.54 MPa, and for apple flesh, 0.43-0.84 MPa. Values of stress at failure of potato flesh reported were in the range, 0.52-0.66 MPa; for melon flesh the range was 0.033-0.097 MPa, and for apple flesh, 0.094-0.157 MPa (Diehl et al., 1979) [Diehl, K. C., Hamann, D. D., and Whitfield, J.

K. 1979. Structural failure in selected raw fruits and vegetables. Journal of Texture Studies 10:371-400].

From elastic theory, the relationship between the moduli E and G can be shown to be:

! = 2 1 + � (9.27) !

For potatoes, using the value 0.49 for Poisson’s ratio, the ratio of the uniaxial compression modulus to the shear modulus had an average value of 2.9, very close to the value 3.0 for an incompressible material.

The true strain is given by (Truong and Daubert, 2001): 25

! ! ! ! � = �� 1 + !!!!"# + � 1 + !!!!"# (9.28) !!!"# !"#$ ! !!!"# !

The torsion test was used to study the characteristics of fruits and vegetables (Diehl et al., 1979), polyacrylamide gels (Foegeding et al. 1994), the effect of temperature on less elastic materials, such as fish muscle gels (Howe et al. 1994), konjac gum gels (Case and Hamann 1994); natural and processed cheeses (Truong and Daubert, 2001), and soybean protein (tofu) and gellan gum gels (Truong and

Daubert, 2000).

A crack can be propagated in one of three ways: by torsion (giving out-of plane tearing), by tension (giving crack-opening) or shear (giving edge-sliding) (Vincent et al., 1991). The magnitude and distribution of these defects govern the strength of the material. Fracture occurs when these defects grow and traverse the solid creating new fracture surfaces. Mathematical relationships have been established between three critical variables: stress, flaw size and material toughness (energy required to fracture a material); and with knowledge of two parameters, the third can be determined; for linear elastic brittle materials, the strain energy release rate, GI, and the stress intensity factor are related by the Young’s modulus (Foegeding et al., 2011):

! � = !! (9.29) ! !

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