Stress-strain relationship simple shear flow

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... 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

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. 

The most important flow process in polymer liquids is shear flow. Polymer liquids differ from simple liquids, first in that the shear viscosity is invariably extremely large, and second in that Newton s empirical equation giving a linear relationship between shear stress r and shear strain rate y with constant shear viscosity ft... 

In more generalised flows, both the stress and the rate of deformation (strain rate) are tensor quantities, and the constitutive relationship between these may be very complex (Schowalter, 1978 Bird et a/., 1987a). The relationship between stress and shear rate frequently depends on the shear rate (flow rate), as is the case in a simple shear thinning fluid. However, for some fluids the stress may depend on the strain itself as well as on the rate of strain, and such fluids show some elasticity or memory behaviour, in that their stress at a given time depends on the recent strain history such fluids are... 

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. 

The variation in wall thickness and the development of cell wall rigidity (stiffness) with time have significant consequences when considering the flow sensitivity of biomaterials in suspension. For an elastic material, stiffness can be characterised by an elastic constant, for example, by Young s modulus of elasticity (E) or shear modulus of elasticity (G). For a material that obeys Hooke s law,for example, a simple linear relationship exists between stress, , and strain, a, and the ratio of the two uniquely determines the value of the Young s modulus of the material. Furthermore, the (strain) energy associated with elastic de-... 

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