Rate-of-strain

The equation of momentum conservation, along with the linear transport law due to Newton, which relates the dissipative stress tensor to the rate of strain tensor = 1 (y. 4, and which introduces two... 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

In an inelastic, time-independent (Stokesian) fluid the extra stre.ss is considered to be a function of the in.stantaneous rate of defomiation (rate of strain). Therefore in this case the fluid does not retain any memory of the history of the deformation which it has experienced at previous stages of the flow. 

In the differential models stress components, and their material derivatives, arc related to the rate of strain components and their material derivatives. 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

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... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

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) ... 

The coefficient Tj is termed the modulus of rigidity. The viscosities of thixotropic fluids fall with time when subjected to a constant rate of strain, but recover upon standing. This behavior is associated with the reversible breakdown of stmctures within the fluid which are gradually reestabflshed upon cessation of shear. The smooth sprea ding of paint following the intense shear of a bmsh or spray is an example of thixotropic behavior. When viscosity rises with time at constant rate of strain, the fluid is termed rheopectic. This behavior is much less common but is found in some clay suspensions, gypsum suspensions, and certain sols. 

A hardness indentation causes both elastic and plastic deformations which activate certain strengthening mechanisms in metals. Dislocations created by the deformation result in strain hardening of metals. Thus the indentation hardness test, which is a measure of resistance to deformation, is affected by the rate of strain hardening. 

A wide variety of nonnewtonian fluids are encountered industrially. They may exhibit Bingham-plastic, pseudoplastic, or dilatant behavior and may or may not be thixotropic. For design of equipment to handle or process nonnewtonian fluids, the properties must usually be measured experimentally, since no generahzed relationships exist to pi e-dicl the properties or behavior of the fluids. Details of handling nonnewtonian fluids are described completely by Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). The generalized shear-stress rate-of-strain relationship for nonnewtonian fluids is given as... 

A flowing fluid is acted upon by many forces that result in changes in pressure, temperature, stress, and strain. A fluid is said to be isotropic when the relations between the components of stress and those of the rate of strain are the same in all directions. The fluid is said to be Newtonian when this relationship is linear. These pressures and temperatures must be fully understood so that the entire flow picture can be described. 

A useful approximation of B for a conical hopper is B = 22f/a, where a is the bulk density of the stored product. The apparatus for determining the properties of solids has been developed and is offered for sale by the consulting firm of Jenike and Johansen, Winchester, Massachusetts, which also performs these tests on a contract basis. The flow-factor FF tester, a constant-rate-of-strain, direct-shear-type machine, gives the locus of points for the FF cui ve as well as ( ), the... 

W.R. Blumenthal and G.T. Gray III, Structure-Property Characterization of a Shock-Loaded Boron Carbide Aluminum Cermet, in 4th Oxford Conf. on Mech. Prop, of Mat. at High Rates of Strain, Int. Phys. Conf. Ser. 102, Oxford, 1989, 363 pp. 

Costin, L.S. and D.E. Grady (1984), Proc. 3rd Int. Conf. Mech. Properties of Materials at High Rates of Strain, Oxford (edited by J. Harding), p. 321. 

Figure 15.6. Effect of temperature on tensile strength of acrylic sheet (Perspex) at constant rate of strain (0.44% per second). (Reproduced by permission of ICl)...

At high rates of strain, or when complete disentanglement cannot occur when M > M, bond rupture occurs randomly in the network and the percolation parameter p becomes dominated by chain ends such that... 

In a perfectly viscous (Newtonian) fluid the shear stress, t is directly proportional to the rate of strain (dy/dt or y) and the relationship may be written as... 

The dashpot is the viscous component of the response and in this case the stress (72 is proportional to the rate of strain f2> ie... 

In a fluid under stress, the ratio of the shear stress, r. to the rate of strain, y, is called the shear viscosity, rj, and is analogous to the modulus of a solid. In an ideal (Newtonian) fluid the viscosity is a material constant. However, for plastics the viscosity varies depending on the stress, strain rate, temperature etc. A typical relationship between shear stress and shear rate for a plastic is shown in Fig. 5.1. 

It should be noted that test information would vary with specimen thickness, temperature, atmospheric conditions, and different speed of straining force. This test is made at 73.4°F (23°C) and 50% relative humidity. For brittle materials (those that will break below a 5% strain) the thickness, span, and width of the specimen and the speed of crosshead movement are varied to bring about a rate of strain of 0.01 in./in./min. The appropriate specimen size are provided in the test specification. 

A further important property which may be shown by a non-Newtonian fluid is elasticity-which causes the fluid to try to regain its former condition as soon as the stress is removed. Again, the material is showing some of the characteristics of both a solid and a liquid. An ideal (Newtonian) liquid is one in which the stress is proportional to the rate of shear (or rate of strain). On the other hand, for an ideal solid (obeying Hooke s Law) the stress is proportional to the strain. A fluid showing elastic behaviour is termed viscoelastic or elastoviseous. 

Shear stress is denoted by R in order to be consistent with other parts of the book r is frequently used elsewhere to denote shear stress. R without suffix denotes the shear stress acting on a surface in the direction of flow and Rq(= —R) denotes the shear stress exerted by the surface on the fluid. Rs denotes the positive value in the fluid at a radius, v and Ry the positive value at a distance y from a surface. Strain is defined as (he ratio (JU/ dy, where dr is the shear displacement of two elements a distance dy apart and is often denoted by y. The rate of strain or rate of shear is (dx/df)/dy or di /dy and is denoted by y. 

The rate of strain hardening, do /ds, at any given value of the true stain is given by the slope of the true stress-true strain plot at that strain and is called the modulus of strain hardening. 

We will now describe the basic hydrodynamic relationships applicable in the case of steady-state flow in which the Eulerian velocity field is time-independent and written as v(r). Here the rate of strain elements are given by [1]... 

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