Rheology stress-strain relationships

The main reasons for studying rheology (stress-strain relationships) in this chapter are (1) to determine the suitability of materials to serve specific applications and (2) to relate the results to polymer structure and form. Studying structure-property relationships allows a better understanding of the observed results on a molecular level resulting in a more knowledgeable approach to the design of materials. 

A plot of the stress-strain relationship for an ion-exchange ka-olinite also follows the Crossian rheology with very high low shear viscosities. In fact, the data show a yield stress at the lowest shear rate measured of y 1 sec and a high shear viscosity, which are both a function of the volume fraction of the clay, as shown in Figure... 

Many lipid-based foods show time-dependent rheological behavior and a combination of these, for example, viscous-elastic and viscous-plastic. In general, lipids, from a rheological point of view, can be classified as Newtonian and non-Newtonian, as depicted in Figure 4.1. To characterize lipid properties, such as Newtonian and non-Newtonian behavior, several approaches can be taken and the stress-strain relationships obtained. 

There are other classes of fluids, such as Herschel-Bulkley fluids and Bingham plastics, that follow different stress-strain relationships, which are sometimes useful in different drilling and cementing applications. For a discussion on three-dimensional effects and a rigorous analysis of the stress tensor, the reader should refer to Computational Rheology. For now, we will continue our discussion of mudcake shear stress, but turn our attention to power law fluids. The governing partial differential equations of motion, even for simple relationships of the form given in Equation 17-57, are nonlinear and therefore rarely amenable to simple mathematical solution. For example, the axial velocity v (r) in our cylindrical radial flow satisfies... 

A thermodynamic model was recently proposed to calculate the solubility of small molecules in assy polymers. This model is based on the assumption that the densiQr of the polymer matrix can be considered as a proper order parameter for the nonequilibrium state of the system (7). In this chapter, the fundamental principles of the model are reviewed and the relation of the model to the rheological properties of the polymeric matrix is developed. In particular, a unique relation between the equilibrium and non-equilibrium properties of the polymer-penetrant mixture can be obtained on the basis of a simple model for the stress-strain relationship. 

Chapter 3. In-plant measurement of flow behavior of fluid Foods. Using a vane-in-a-cup as a concentric cylinder system. The vane yield stress test can be used to obtain data at small- and large-deformations. Critical stress/strain from the non-linear range of a dynamic test. Relationships among rheological parameters. First normal stress difference and its prediction. 

FIGURE 10.1 Rheological properties of an ideal solid (a) depicts the rheological representation of an ideal solid, whereas (b) displays the relationship between stress, strain, and time for an ideal solid. 

Rheology is the study of flow of matter and deformation and these techniques are based on their stress and strain relationship and show behavior intermediate between that of solids and liquids. The rheological measurements of foodstuffs can be based on either empirical or fundamental methods. In the empirical test, the properties of a material are related to a simple system such as Newtonian fluids or Hookian solids. The Warner-Bratzler technique is an empirical test for evaluating the texture of food materials. Empirical tests are easy to perform as any convenient geometry of the sample can be used. The relationship measures the way in which rheological properties (viscosity, elastic modulus) vary under a... 

Rheology studies the relationship between force and deformation in a material. To investigate this phenomenon we must be able to measure both force and deformation quantitatively. Steady simple shear is the simplest mode of deforming a fluid. It allows simple definitions of stress, strain, and strain rate, and a simple measurement of viscosity. With this as a basis, we will then examine the pressure flow used in capillary rheometers. 

Although difficult, it is possible to measure stress vs. strain curves of PSAs. Examples of such work include that of Christenson et al. [.3J and Piau et al. [23J. One can do this at various elongation rates and temperatures and create a material response function. Of course, it is much easier to obtain rheological data at small strains than to obtain tensile stress-strain data. One can assume a shape of the stress vs. strain function (i.e. a constitutive relationship) and then use the small strain data to assign values to the parameters in such a function. In order for a predictive model of peel to be useful, one should be able to use readily obtained rheological parameters like those obtained from linear viscoelastic master curve measurements and predict peel force master curves. 

Rheology deals with deformation and flow and examines the relationship between stress, strain and viscosity. Most theological measurements measure quantities related to simple shear such as shear viscosity and normal stress differences. Material melt flows can be split into three categories, each behaving differently under the influence of shear as shown in Figure 10.9 Dilatent (shear thickening), Newtonian and Non-Newtonian pseudoplastic (shear thinning) behaviour. 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

Now, in rheological terminology, our compressibility JT, is our bulk compliance and the bulk elastic modulus K = 1 /Jr- This is not a surprise of course, as the difference in the heat capacities is the rate of change of the pV term with temperature, and pressure is the bulk stress and the relative volume change, the bulk strain. Immediately we can see the relationship between the thermodynamic and rheological expressions. If, for example, we use the equation of state for a perfect gas, substituting pV = RTinto a = /V(dV/dT)p yields a = R/pV = /Tand so for our perfect gas ... 

If you step back and think about it, the mechanical and rheological properties of many solids and liquids can be modeled fairly well by just two simple laws, Hooke s law and Newton s law. Both of these are what we call linear models, the stress is proportional to the strain or rate of strain. If we examine viscoelastic properties like creep, the variation of strain with time appears decidedly non-linear (see Figure 13-75). Nevertheless, it is possible to model this non-linear time dependence by the assumption of a linear relationship between stress and strain. By this we mean that if, for example, we measure the strain as a function of time in a creep experiment, then for a given time period (say 1 hour) the strain measured when the applied stress is 2o would be twice the strain measured when the stress was o. 

The relationships between stress and strain, and the influence of time on them are generally described by constitutive equations or rheological equations of state (Ferry, 1980). When the strains are relatively small, that is, in the linear range, the constitutive... 

If the solid does not shows time-dependent behavior, that is, it deforms instantaneously, one has an ideal elastic body or a Hookean solid. The symbol E for the modulus is used when the applied strain is extension or compression, while the symbol G is used when the modulus is determined using shear strain. The conduct of experiment such that a linear relationship is obtained between stress and strain should be noted. In addition, for an ideal Hookean solid, the deformation is instantaneous. In contrast, all real materials are either viscoplastic or viscoelastic in nature and, in particular, the latter exhibit time-dependent deformations. The rheological behavior of many foods may be described as viscoplastic and the applicable equations are discussed in Chapter 2. 

A fluid in which the shear stress is proportional to the shear velocity, corresponding to this law, is called an ideal viscous or Newtonian fluid. Many gases and liquids follow this law so exactly that they can be called Newtonian fluids. They correspond to ideal Hookeian bodies in elastomechanics, in which the shear strain is proportional to the shear. A series of materials cannot be described accurately by either Newtonian or Hookeian behaviour. The relationship between shear stress and strain can no longer be described by the simple linear rule given above. The study of these types of material is a subject of rheology. 

Certain phenomenological relationships exist between the temperature and rate dependences of the behavior of a material in mechanical and rheological tests. A very familiar example is the time-temperature superposition principle [46] which often holds. Consequently, similar curves can also be drawn for the failure stresses as functions of strain rate. For example, ay increases... 

The applied stress results in the shear strain of the cube, i.e. the top face becomes shifted with respect to the bottom one by distance y. This displacement is numerically equal to the tangent of a tilt angle of the side face, i.e. it is equal to the relative shear strain, y, and at small strains tany y. The relationship between shear stress, x, and shear strain, y, and the rates of change in these quantities with time, dx/dt=x, dy/dt=y, represent mechanical behavior, which is the main subject in rheology. One usually begins the description of mechanical behavior with three elementary models, namely elastic, viscous, and plastic behavior. 

In reality we find a superposition of elastic and viscous effects in the strain-stress relationship. In 3D-rheology properties of a given system turned out to be described by the combination of... 

In the cube shown in Fig. 3.5. the tensor components for the strain-stress relationship of a 3D-body can be seen. Neglecting the z-coordinate, the tensor reduces from a 3x4 to a 2x2 matrix. The use of the 3D-rheology for related surface problems is only valid if a 3D-analogue for the relaxation is introduced. This is the only way to learn about the surface state in the absence of ideally elastic behaviour of the adsorption layer. 

All these phenomena are contributions to the surface loss angle in the strain-stress relationship. 3D-rheology deals with a closed system. There is no mass transfer across the limits of this system. In contrast, the so-called surface phase with an adsorption layers can exchange matter with the bulk phase depending on the boundary conditions, such as adsorption after formation of a fresh surface or periodic adsorption and desorption due to periodic changes of the surface area. 

The relationship between an applied stress or strain and the response of the material, shear rate, or deformation is the aim of the rheology of suspensions. Normally, both the stress and the strain are tensors with each having nine components. In simple shear, which is the most common way of determining the rheological behavior, the shear stress oxy (some literature also uses the symbol r to stand for the shear stress) can be related to the shear rate y by... 

The measurement of rheological properties for non-Newtonian, lipid-based food systems, such as dilatant, pseudoplastic, and plastic, as depicted in Figure 4.1, are much more difficult. There are several measurement methods that may involve the ratio of shear stress and rate of shear, and also the relationship of stress to time under constant strain (i.e., relaxation) and the relationship of strain to time under constant stress (i.e., creep). In relaxation measurements, a material, by principle, is subjected to a sudden deformation, which is held constant and in many food systems structure, the stress will decay with time. The point at which the stress has decayed to some percentage of the original value is called the relaxation time. When the strain is removed at time tg, the stress returns to zero (Figure 4.8). In creep experi-... 

---