Rheological Properties of Solid Foods
6.2 QUASISTATIC TESTS FOR SOLID FOODS .1 Introduction
Before we discuss the fundamental rheological tests that have been used on solid foods, we present a short review of fundamental rheology. Two extremes of behavior may result (from a rheological viewpoint) when a force is applied to a material; the pure elastic defor-mation of a solid and the pure viscous flow of a liquid.
Pure elastic behavior is defined such that when a force is applied to a material, it will instantaneously and finitely deform; and when the force is released, the material will instantaneously return to its original form. Such a material is called a Hookean solid. The amount of deformation is proportional to the magnitude of the force. The rheological rep-resentation for this type of solid is a spring. A material of this nature can be given a rheo-logical constant, termed the elastic modulus. The elastic modulus is the ratio of stress to strain in a material, where stress is equal to force per unit area and strain is the observed deformation due to the force, divided by the original length of the material. Three types of moduli may be calculated for a Hookean solid, depending on the method of applying the force. The modulus calculated by applying a force perpendicular to the area defined by the stress is called the modulus of elasticity (e). The modulus calculated by applying a force parallel to the area defined by the stress, or a shearing force, is called the shear modulus or modulus of rigidity (g). If the force is applied from all directions (isotropically) and the change in volume per original volume is obtained, then one can calculate the bulk modulus (K). Thus, these are material constants, because the deformation is proportional to the applied force, and unit area and length are considered in the calculations.
A pure viscous flow of a liquid means that the liquid begins to flow with the slightest force and that the rate of flow is proportional to the magnitude of force applied. This liquid flows infinitely until the force is removed, and upon removal of the force, it has no ability to regain its original state. Such a material is called a newtonian liquid. The rheological rep-resentation for this type of liquid is a dashpot, which can be thought of as a piston inside a cylinder. When a force is applied to the piston, it moves in or out of the cylinder at constant velocity, the rate depending on the magnitude of the force. When the force is removed, the piston remains fixed and cannot return to its original position. A material of this nature has a rheological constant called the coefficient of viscosity (ή). The coefficient of viscosity is defined as the shearing stress applied divided by the resulting rate of strain. In this way, it is very similar to the modulus for Hookean solids.
If foods were either Hookean solids or Newtonian liquids, determination of their rheological constants would be simple. However, foodstuffs possess rheological proper-ties associated with both the elastic solid and the viscous fluid. Such materials are called viscoelastic (Mohsenin, 1978). The rheological representation of this type of material is a body incorporating at least one spring (representing the solid character) and at least one dashpot (representing the viscous character). The number of springs and dashpots in the body and the manner in which they are connected can be manipulated to represent dif-ferent types of viscoelastic materials, and to demonstrate how they will behave under a stress or strain. Thus, a viscoelastic material has several rheological constants, depending on the number of springs and dashpots that represent its behavior. There is no simple
constant for viscoelastic materials such as modulus, because the modulus will change over time. Thus, if one subjects a viscoelastic material to a constant stress, the manner in which the material is strained will change over time. The rheological constants for a vis-coelastic material are represented by an equation to give modulus as a function of time.
The theory of viscoelasticity is discussed in detail by Reiner (1960, 1971), Christensen (1971), and Flugge (1975).
Because foods are viscoelastic, both time-dependent and time-independent measure-ments are required. Alfrey (1957) lists three methods that use experimental curves to “map out” the viscoelastic character of a material:
1. The creep curve, showing strain as a function of time at constant stress,
2. The relaxation curve, showing stress as a function of time at constant strain, and 3. The dynamic modulus curve, consisting of the dynamic modulus as a function of
the frequency of the sinusoidal strain.
For linear viscoelastic materials, these three types of experimental curves should yield consistent results; that is, the moduli and coefficients of viscosity from the relaxation, creep, and dynamic tests should be interconvertible mathematically and should be independent of the magnitude of the imposed stress or strain.
The following review includes publications that have employed one or more of the three methods listed above as well as other studies that lend insight into the fundamental rheological behavior of solid foods.
6.2.2 Some Simple Tests
The simplest of all the quasistatic tests is perhaps the uniaxial compression/tension test.
In this test, a sample with a convenient geometry (e.g., cylinder or rectangular prism) is subjected either to a deformation or to a force, and the corresponding force or deformation is recorded. If the magnitudes of force and deformation are small, then the body may be assumed to be elastic. The resultant stress (σ) and strain (e) may be calculated as
σ = F
A (6.1)
and
e= ∆l
l (6.2)
where F is the force, A is the cross-sectional area of the body, Δl is the deformation, and l is the original length of the body (Figure 6.1). The modulus of elasticity (also called Young’s modulus), usually denoted by e or Y, can be computed as σ/e. In a similar manner, it is also possible to determine the shear modulus (g) or bulk modulus (K). The bulk modulus (K) can be evaluated as the ratio of isotropic stress (σ in Figure 6.1) to the volumetric strain.
The volumetric strain is defined as the change in volume divided by original volume. The essential point is that the deformations and forces have to be extremely small in order to assume elastic behavior.
RHeologiCAl PRoPeRties oF solid Foods
Uniaxial compression and tension tests provide the researcher with an extremely sim-ple test for the determination of material properties under conditions of both elastic and viscoelastic behavior. However, due to the effects of bonding and lubrication, such tests have certain serious problems with the computation of the elastic moduli. In most uniaxial compression tests, the effect of frictional forces between the loading plates and the mate-rial under test need to be considered in the computation of stress–strain relationships.
Extensive work in this area has been conducted by Bagley and Christianson (1987). They have shown that bonding the material to the test platen by adhesives leads to improved and consistent reproducibility of the results. The other alternative is to lubricate the test platens to eliminate or minimize the effect of friction. This approach has been used by many researchers (Montejano et al., 1983). In an extensive and pioneering study, Casiraghi et al. (1985) examined the effect of lubrication of the test platen and bonding of the material to the test platen in uniaxial compression tests using cheese. In bonded compression they recommend that strain be calculated as
e= −δh h/ (6.3)
where δh and h are the change in height and the height after deformation. (Note that this definition of strain is different from Equation 6.2, where the original undeformed dimen-sion was used.) The calculation of stress (σB) for bonded compression is as usual:
σB = /F Rπ 02
where R0 is the radius. For comparing the bonded response with the lubricated response, the correction for stress (σBC) was
σ
σ
BC
B
R h
= 1+ 02/2 2 (6.4)
LL−ΔL
L
θ
σ
σ
σ
σ σ σ ΔL
F F
F
A A
Uniaxial compression Shear Bulk compression
Figure 6.1 Uniaxial compression, shear, and isotropic (bulk) compression of an elastic solid.
For lubricated materials they recommend calculating stress (σl) as σl πR
F
= 2
where R is the radius of the deformed cylinder with an initial radius of R0 and height h0
(assuming that there is no change in volume). Using these equations, it appears that the agreement between results from lubricated and bonded compression of mozzarella cheese samples are valid at least until the strain levels correspond to 60% deformation. In conclu-sion, Casiraghi et al. (1985) recommend that all uniaxial compression of foods be carried out under all conditions (lubrication, bonding) before meaningful results are obtained.
This appears to be a good rule, especially if the strain levels are considerable (usually more than a few percent). This procedure of correcting for the effects for bonding and lubrica-tion is equally applicable to uniaxial tests for elastic and viscoelastic solid foods.
Another test that can be applied to foods that are fairly brittle is the bending test (Figure 6.2). The advantage of this method lies in the extremely small true deformations in the material in addition to measurable deflections. The calculations are as follows.
In a loading of a material with two symmetrical vertical supports, the bending moment (M) at any point x is
M = Px
2 (6.5)
where P is the force.
If the effects of shearing force and shortening of the beam axis are neglected, the expression for the curvature of the axis of beam is
eid y dx
Px
2
2 = −2 (6.6)
where i is the moment of inertia. Integrating Equation 6.6 twice yields y Px
ei Ax B
= 3 + +
12
δ P
L
F F
R1 R′1 d
Bending Plate on a sphere Plate on a convex body
Figure 6.2 Application of bending and Hertz’s equations to measure the modulus of elasticity (e).
RHeologiCAl PRoPeRties oF solid Foods
For this problem, the maximum deflection (δ, Figure 6.2) occurs in the center, that is, x = l/2.
y Pl
= =δ ei3 48 Hence, the modulus of elasticity (e) is
e Pl
= i3
48 δ (6.7)
For materials that cannot be modified to yield a sample possessing convenient geom-etry (e.g., some fruits, vegetables, grains), the application of Hertz’s equations is appropri-ate (Mohsenin, 1978). The necessary equations for a spherical and a convex body (Figure 6.2) are as follows:
For axial loading of a spherical sample between flat plates,
e F
where F is the force corresponding to deformation d, d is the diameter of the sphere, and µ is Poisson’s ratio.
For a plate on a convex body,
e F
The only problem in using the Hertz equations is the prior knowledge of the Poisson ratio (µ). However, the error introduced by assuming an approximate value would be min-imal (Mohsenin, 1978). The above technique has been widely used in the literature for a variety of convex-shaped foods.
6.2.3 Rheological Modeling
Food materials seem to behave as viscoelastic materials when they are exposed to vari-ous conditions of stress or strain (Chappell and Hamann, 1968; Chen and Fridley, 1972;
Clevenger and Hamann, 1968; Datta and Morrow, 1983; Hammerle and Mohsenin, 1970;
Hundtoft and Buelow, 1970; Mohsenin, 1978; Morrow and Mohsenin, 1966; Peleg, 1976a;
Skinner, 1983). Many researchers have designed experimental procedures that provide insight into the rheological modeling of these materials in order to characterize them and predict their behavior under specific physical conditions. These viscoelastic models con-tain various combinations of Hookean solid elements (springs) and Newtonian fluid ele-ments (dashpots), and show complex behavior that can represent various food materials.
If a material is found to be linearly viscoelastic, this property allows transformation of the individual element constants to fit different arrangements of these elements into other equivalent models. This linearity is guaranteed only at very small levels of strain. Thus, the moduli of viscoelastic elements is a function of time, not stress, at these small strains.
The rheological constants for viscoelastic materials are represented by mathematical equa-tions for different models where the modulus is expressed as a function of time.
If an accurate model is made to represent food material, it can be used to predict changes in the material that may occur during mechanical harvesting or handling, and perhaps further be used to reduce the risk of damage and other structural defects in raw agricultural commodities. Researchers such as Peleg (1976a) found rheological models to be extremely useful tools in predicting mechanical response of foods to specific stress–
strain conditions.
Peleg (1976a) provided a list of conditions to satisfy in constructing a rheological model to represent a food material:
1. The model must enable the prediction of a real material behavior under any force–
deformation history.
2. The model should be able to respond to both positive and negative forces and defor-mations (i.e., tension and compression). However, this is limited to the instance where the “physical structure” of the elements themselves would be under stress.
3. Changes and variations occurring in the behavior of the real material must be explained in terms of the model parameters.
Two of the most useful physical tests used in the determination of a rheological model and in the computation of the individual model constants incorporated into the chosen model are static creep and stress relaxation (Datta and Morrow, 1983; Gross, 1979; Skinner, 1983). It is therefore imperative to discuss these two tests in detail. The elementary models used to build more complex models are termed Maxwell (a spring and a dashpot in series) and Kelvin (a spring and a dashpot in parallel), after the inventors.
6.2.4 Creep
Static uniaxial normal creep is a condition in which the constant shear or dynamic forces involved are all parallel to the longitudinal axis of the specimen. This imposed stress must not be so great as to yield large sample deformations to the point where elastic limit of the material is exceeded and it no longer behaves as a linearly viscoelastic material. In such a case, representation of these materials by rheological models would no longer be valid.
Shama and Sherman (1973) state that for a material to be linearly viscoelastic, (1) the strain must be linearly related to the stress; (2) the stress–strain ratio must be a function of