Stress-strain relations fluids

Stress-Strain Relations for a Fluid. The medium is hydrostatic when the direct stresses in three orthogonal directions are equal and the shear stresses are zero a.. = a.. = (3A + 2p) e... 

B.Bemstein, E.A.Kearsley, L.J.Zapas, Elastic stress strain relations in perfect elastic fluids, Trans Soc Rheol. 2 (1965), 27-39. 

Pressure-cast bodies of ceramics also have problems, when they are cast from either a dispersed or a flocculated slurry. As the last portion of the sluny is consolidated, the pressure gradient across the cast becomes zero, so that the total applied pressure is transferred to the cast. Therefore, upon the applied pressure is removed, the cast expands, i.e., undergoes strain recovery, due to the stored elastic energy. However, the nature of the strain recovery is different from that in the die compaction of dry powders, in a way that the strain recovery for the compacts of the pressure casting is time dependent. This time-dependent strain recovery is attributed to the fact that the fluid, either liquid or gas, must flow into the compact to allow the particle network to expand and relieve the stored strain. The magnitude of the recovered strain increases with increasing consolidation pressure nonUnearly, which can be described by the following Hertzian elastic stress-strain relation ... 

First, when finite strains are imposed on solids (especially those soft enough to be deformed substantially without breaking), the stress-strain relations are much more complicated (non-Hookean deformation) similarly, in steady flow with finite strain rates, many fluids (especially polymeric solutions and undiluted uncross-linked polymers) exhibit marked deviations from Newton s law (non-Newtonian flow). The dividing line between infinitesimal and finite depends, of course, on the level of precision under consideration, and it varies greatly from one material to another. 

At finite deformations, equation 59 can be shown to be incorrect because it is not objective i.e., it predicts results which erroneously depend on the orientation of the sample with respect to laboratory coordinates. This error can be eliminated by replacing j/j in equation 59 by the components of a corotational rate-of-strain tensor or the components of one of several possible codeformational rate-of-strain tensors either of these replacements ensures that the unwanted dependence of cy on the instantaneous orientation of a fluid particle in space is removed. If the stress-strain relations are linear within the changing coordinate frame, equation 59 is modified only be replacing y,-j with a different strain rate tensor whose definition is complicated and beyond the scope of this discussion. The corresponding corotational model is that of Goddard and Miller and the codeformational models correspond to those of Lodge or Oldroyd, Walters, and Fredrickson. ... 

Differential Stress-Strain Relations and Solutions for a Maxwell Fluid... 

Nonlinear Mechanical Models It is possible to represent nonlinear behavior by introducing nonlinear spring and damper elements into the derivation of differential stress-strain relations. For example, for the four-parameter fluid shown in Fig. 10.5, the spring moduli, damper viscosities and relaxation times are functions of stress, i.e.,... 

Bernstein, B Kearsley, E. A., and Zapas, L. J., Elastic Stress-Strain Relations in Perfect elastic Fluids, Transactions of the Society of Rheology, 27-39 (1965). 

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) ... 

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. 

For elastic bodies, the shear stress is related to the shear strain by the shear modulus. For viscous fluids, the shear stress is related to the shear strain rate by the viscosity. We note that for laminar viscous flow in a Margules viscometer (Figure 10.7), radial fluid displacement is zero (gr = 0). Thus, differentiating with respect to time ... 

It has been shown that when a fluid is sheared with a shear stress t, its strain rate (or deformation rate) is proportional (for most fluids) to the shear stress. The proportionality constant is termed the fluid viscosity n. For a fluid sheared between two long parallel plates, the local velocity (at any height y) varies from zero at the fixed plate to v at the upper moving plate. As described above, the derivative of the local velocity (m) with respect to the height y (i.e., duldy) is termed the velocity gradient, strain rate, or deformation rate. The shear stress is related to duldy by the equation... 

To characterize Newtonian and non-Newtonian food properties, several approaches can be used, and the whole stress-strain curve can be obtained. One of the most important textural and rheological properties of foods is viscosity (or consistency). The evaluation of viscosity can be demonstrated by reference to the evaluation of creaminess, spreadability, and pourability characteristics. All of these depend largely on shear rate and are affected by viscosity and different flow conditions. If it is related to steady flow, then at any point the velocity of successive fluid particles is the same at successive periods of time for the whole food system. Thus, the velocity is constant with respect to time, but it may vary at different points with... 

To proceed formulating the momentum equation we need a relation defining the total stress tensor in terms of the known dependent variables, a constitutive relationship. In contrast to solids, a fluid tends to deform when subjected to a shear stress. Proper constitutive laws have therefore traditionally been obtained by establishing the stress-strain relationships (e.g., [11] [12] [13] [89] [184] [104]), relating the total stress tensor T to the rate of deformation (sometimes called rate of strain, i.e., giving the name of this relation) of a fluid element. However, the resistance to deformation is a property of the fluid. For some fluids, Newtonian fluids, the viscosity is independent both of time and the rate of deformation. For non-Newtonian fluids, on the other hand the viscosity may be a function of the prehistory of the flow (i.e., a function both of time and the rate of deformation). 

Employing the concepts of stress and conjugate strain, and their proper mathematical formulation as. second-rank tensors, now enables us to deal with mechanical work in a general anisotropic piece of matter. One realization of sudi a system are fluids in confinement to whidi this book is devoted. However, at the core of our subsequent treatment are thermal properties of confined fluids. In other words, we need to understand the relation between medianical work represented by stress-strain relationships and other forms of energy such as heat or chemical work. This relation will be formally... 

The principles of conservation of momentum, energy, mass, and charge are used to define the state of a real-fluid system quantitatively. The conservation laws are applied, with the assumption that the fluid is a continuum. The conservation equations expressing these laws are, by themselves, insufficient to uniquely define the system, and statements on the material behavior are also required. Such statements are termed constitutive relations, examples of which are Newton s law that the stress in a fluid is proportional to the rate of strain, Fourier s law that the heat transfer rate is proportional to the temperature gradient. Pick s law that mass transfer is proportional to the concentration gradient, and Ohm s law that the current in a conducting medium is proportional to the applied electric field. 

For time-dependent fluids (thixotropic or rheopectic) there are no simple relations now available for showing the stress-strain-rate-time. dependence. Figure 15.2 is, a typical stress-time curve for a thixotropic fluid, showing lines of... 

A Newtonian liquid has an approximately linear stress versus strain relation. Many fluids fall in this category including water, alcohol, and most hydrocarbon fuels. Due to their prevalence, the vast majority of secondary atomization studies have been conducted using Newtonian liquids. As a result, most of the available knowledge applies to them. 

Die swell appears when a polymer flows and the cqracity to flow is measured by the polymer s viscosity. Viscosity is nothing more than another modulus but, since it applies to a fluid instead of a solid, it has its own equations and nomenclature. Where moditlus usually relates force buildup in a solid, stress to distension of the solids dimensions, strain, viscosity applies to a fluid which can readily change dimensions under minimal stress. Yet different fluids water versus motor oil versus a polymer melt, change... 

The Maxwell Model. In the above development, discussion moves from elastic behavior to viscoelastic descriptions of material behavior. In a simple sense, viscoelasticity is the behavior exhibited by a material that has both viscous and elastic elements in its response to a deformation or load. In early days, this was often represented by elastic or viscous mechanical elements combined in different ways (9-12). The simplest models are two element models that contain a viscous element (dashpot) and an elastic element (spring). The dashpot is assumed to follow a Newtonian fluid constitutive law in which the stress is related directly to the strain rate by the following expression ... 

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. 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

Extensional flows occur when fluid deformation is the result of a stretching motion. Extensional viscosity is related to the stress required for the stretching. This stress is necessary to increase the normalized distance between two material entities in the same plane when the separation is s and the relative velocity is ds/dt. The deformation rate is the extensional strain rate, which is given by equation 13 (108) ... 

Viscoelasticity A combination of viscous and elastic properties in a plastic with the relative contribution of each being dependent on time, temperature, stress, and strain rate. It relates to the mechanical behavior of plastics in which there is a time and temperature dependent relationship between stress and strain. A material having this property is considered to combine the features of a perfectly elastic solid and a perfect fluid. 

Extensional flow describes the situation where the large molecules in the fluid are being stretched without rotation or shearing [5]. Figure 4.3.3(b) illustrates a hypothetical situation where a polymer material is being stretched uniaxially with a velocity of v at both ends. Given the extensional strain rate e (= 2v/L0) for this configuration, the instantaneous extensional viscosity r e is related to the extensional stress difference (oxx-OyY), as... 

Actually, some fluids and solids have both elastic (solid) properties and viscous (fluid) properties. These are said to be viscoelastic and are most notably materials composed of high polymers. The complete description of the rheological properties of these materials may involve a function relating the stress and strain as well as derivatives or integrals of these with respect to time. Because the elastic properties of these materials (both fluids and solids) impart memory to the material (as described previously), which results in a tendency to recover to a preferred state upon the removal of the force (stress), they are often termed memory materials and exhibit time-dependent properties. 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

Equation (5.5) is known as Hooke s Law and simply states that in the elastic region, the stress and strain are related through a proportionality constant, E. Note the similarity in form to Newton s Law of Viscosity [Eq. (4.3)], where the shear stress, r, is proportional to the strain rate, y. The primary differences are that we are now describing a solid, not a fluid, the response is to a tensile force, not a shear force, and we do not (yet) consider time dependency in our tensile stress or strain. 

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