Stress-dilatancy relation

We introduce the stress-dilatancy relation presented by Taylor (1948), which is schematically shown in Fig. 6.6. Taylor visualized direct shearing along jagged surfaces, and obtained an increment of the work done due to a normal force P and shear force Q on the respective displacements, as follows ... 

In this case we have the following stress-dilatancy relation of Taylor ... 

Figure 4 illustrates the typical volume dilatation-strain behavior along with its first and second derivatives. Clearly these measures are realistic in that the derivatives do take on the character of cumulative and instantaneous frequency distributions. Similar models can be constructed to relate the loss in stiffness to the number of vacuoles that have formed resulting in very simple but accurate stress-strain relations (1). 

The work discussed here has related not only to structure relationships but also to means of protection of the macromolec-ular material and the protective functions of these materials. There are many modes of failure, by chemical reaction, failure by fracture, environmental stress cracking and creep. Further there are complicating interactions arising from chemical reaction during relaxation of polymer networks, and in multiphase polymer systems and cos osites, failure at interfaces by adhesive failxire or stress-stress dilation. 

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

Acute and chronic gastroenteritis, hyperchlorhydria, prolapsed stomach, dilatation of the stomach, peptic and duodenal ulcers, stress-related belching and vomiting, irritable bowel syndrome, allergies, hepatitis, cholecystitis, and the side effects of some medicines. 

In the case of fluids without yield stress, viscous and viscoelastic fluids can be distinguished. The properties of viscoelastic fluids lie between those of elastic solids and those of Newtonian fluids. There are some viscous fluids whose viscosity does not change in relation to the stress (Newtonian fluids) and some whose shear viscosity T] depends on the shear rate y (non-Newtonian fluids). If the viscosity increases when a deformation is imposed, we define the material as a shear-thickening (dilatant) fluid. If viscosity decreases, we define it as a shear-thinning fluid. 

There are two general types of constitutive equations for fluids Newtonian and non-Newtonian. For Newtonian fluids, the relation between the stress tensor, t, and the rate of deformation tensor or the shear stress is linear. For non-Newtonian fluids the relation between the stress tensor and the rate of deformation tensor is nonlinear. The various Newtonian and non-Newtonian rheologies of fluids are shown in Figure 12.2. There are four types of behavior (1) Newtonian, (2) pseudo-plastic, (3) Bingham plastic, and (4) dilatent. The reasons for these different rheological behaviors will also be discussed in subsequent sections of this chapter. But first it is necessary to relate the stress tensor to the rate of deformation tensor. 

The stress for pseudo-plastic and dilatent fluids is not a linear function of shear rate. For non-Newtonian fluids, the relation between t and A is not a simple proportionality because the viscosity is a function of A. For a Bingham plastic fluid, the following relationship holds ... 

To avoid these mathematical details and focus on the key concepts of tablet stress analysis this discussion will examine the simplest of viscoelastic models using the method outlined by Fluggie (97). To begin the analysis, the boundary conditions which apply to tablet compaction, will be used to set up the stress and strain tensors Equations (26) and (27). Then the dilation and distortion uations (28-31) will be used lo obtain dilation and distortion tensors. After obtaining the dilational and distortional stress and strain tensors, a Kelvin viscoelastic model will be used to relate the distortional stress to distortional strain and the dilational stress to dilational strain. 

Using the bulk and shear modulus to relate dilational stress to dilational strain and distortional stress to distortional strain yields ... 

Recalling the dilational Equationfi (28) and (29) and the distortional Equations (30) and (31). These equations can be adjusted for stress relaxation boundary conditions shown in Equations (73) and (74). Thus, using the tensor Equations (73) and (74) and Equations (28) and (29) to compute the dilational and distortional stresses and strains. Once the dilational and distortional stresses and strains are computed, the shear and bulk modulus Equations (63) and (64) can be used to relate stress to strain yielding ... 

The viscous properties of a liquid are defined in relation to the rate of distortion and dilatation of a local region R in response to stresses applied at the surface S which encloses it. In the analysis, we consider an element SS of the surface and construct a unit normal vector n directed outwardly from the surface. The applied stress can then be resolved into components tending to stretch the vector and components tending to rotate it. The force F per unit area can then be represented in terms of a tensor P(r) operating... 

The fundamental assumption of the classical rheological theories is that the liquid stmcture is either stable (Newtonian behavior) or its changes are well dehned (non-Newtonian behavior). This is rarely the case for flow of multiphase systems. For example, orientation of sheared layers may be responsible for either dilatant or pseudoplastic behavior, while strong interparticle interactions may lead to yield stress or transient behaviors. Liquids with yield stress show a plug flow. As a result, these liquids have drastically reduced extrudate swell, B = d/d (d is diameter of the extrudate, d that of the die) [Utracki et al, 1984]. Since there is no deformation within the plug volume, the molecular theories of elasticity and the relations they provide to correlate, for example either the entrance pressure drop or the extmdate swell, are not applicable. 

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