Shear finite

The situation is more complex for rigid media (solids and glasses) and more complex fluids that is, for most materials. These materials have finite yield strengths, support shears and may be anisotropic. As samples, they usually do not relax to hydrostatic equilibrium during an experiment, even when surrounded by a hydrostatic pressure medium. For these materials, P should be replaced by a stress tensor, <3-j, and the appropriate thermodynamic equations are more complex. 

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. 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Rheology. The rheology of foam is striking it simultaneously shares the hallmark rheological properties of soHds, Hquids, and gases. Like an ordinary soHd, foams have a finite shear modulus and respond elastically to a small shear stress. However, if the appHed stress is increased beyond the yield stress, the foam flows like a viscous Hquid. In addition, because they contain a large volume fraction of gas, foams are quite compressible, like gases. Thus foams defy classification as soHd, Hquid, or vapor, and their mechanical response to external forces can be very complex. 

Inter-ply shear is prominently featured in cord—mbber composite laminates, and may relate to delamination-induced failures. Studies utilizing experimental, analytical, and finite element tools, with specific apphcation to tires, are significant in compliant cord—mbber composites (90—95). 

Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large paraU plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a sohd, which would undergo only a finite deformation. 

Non-Newtonian fluids include those for which a finite stress 1,. is reqjiired before continuous deformation occurs these are c ailed yield-stress materials. The Bingbam plastic fluid is the simplest yield-stress material its rheogram has a constant slope [L, called the infinite shear viscosity. 

When crazing limits the ductility in tension, large plastic strains may still be possible in compression shear banding (Fig. 23.12). Within each band a finite shear has taken place. As the number of bands increases, the total overall strain accumulates. 

Fig. 2.4. Within the elastic range it is possible to relate uniaxial strain data obtained under shock loading to isotropic (hydrostatic) loading and shear stress. Such relationships can only be calculated if elastic constants are not changed with the finite amplitude stresses.

Fig. 4.3. Typical normalized piezoelectric current-versus-time responses are compared for x-cut quartz, z-cut lithium niobate, and y-cut lithium niobate. The y-cut response is distorted in time due to propagation of both longitudinal and shear components. In the other crystals, the increases of current in time can be described with finite strain, dielectric constant change, and electromechanical coupling as predicted by theory (after Davison and Graham [79D01]).

Strain gages may be applied to the test unit at all points where high stresses are anticipated, provided that the configuration of the units permits such techniques. The use of finite element analysis, models, brittle lacquer, etc., is recommended to confirm the proper location of strain gages. Three-element strain gages are recommended in critical areas to permit determination of the shear stresses and to eliminate the need for exact orientation of the gages. 

In simple shear flow where vorticity and extensional rate are equal in magnitude (cf. Eq. (79), Sect. 4), the molecular coil rotates in the transverse velocity gradient and interacts successively for a limited time with the elongational and the compressional flow component during each turn. Because of the finite relaxation time (xz) of the chain, it is believed that the macromolecule can no more follow these alternative deformations and remains in a steady deformed state above some critical shear rate (y ) given by [193] (Fig. 65) ... 

Some materials have the characteristics of both solids and liquids. For instance, tooth paste behaves as a solid in the tube, but when the tube is squeezed the paste flows as a plug. The essentia] characteristic of such a material is that it will not flow until a certain critical shear stress, known as the yield stress is exceeded. Thus, it behaves as a solid at low shear stresses and as a fluid at high shear stress. It is a further example of a shear-thinning fluid, with an infinite apparent viscosity at stress values below the yield value, and a falling finite value as the stress is progressively increased beyond this point. 

A fluid with a finite yield. stress is sheared between two concentric cylinders, 50 mm long. The inner cylinder is 30 mm diameter and the gap is 20 mm. The outer cylinder is held stationary while a torque is applied to the inner. The moment required just to produce motion was 0.01 N m. Calculate the force needed to ensure all the fluid is flowing under shear if the plastic viscosity is 0.1 Ns/ni2. 

I.H. Gregory and A.H. Muhr, Stiffness and fracture analysis of bonded rubber blocks in simple shear, in Finite Element Analysis of Elastomers, ed. by D. Boast and V.A. Coveny, Professional Engineering Publications, Bury St. Edmunds, United Kingdom, 1999, pp. 265-274. 

The load on the wheel produces a contact area of finite length a. The distortion caused by the load is ignored in the brush model. This means that the above relation is really a shear relation. The fibers have a large compression stiffness and a small shear stiffness, which in fact is true for rubbers. The large contact length is created by the air inflation chamber of the tire. Solid mbber wheels bulge out. 

The y-velocities are all set to zero the problem is numerically underconstrained otherwise. Figure 2 also shows the finite-element prediction of this velocity profile for two cases a Newtonian fluid (power-law exponent = 1) and a shear-thinning fluid (power-law... 

Ciccotti et aV° have evaluated shear viscosity by applying a shearing force that is periodic in space and has by necessity a finite wave vector ... 

Apparently, the slip length diverges when 0. In practice, the shear strain in Eq. (5) will approach zero in such a case, thus leaving the velocity jump finite. Several experimental results on cf have been reported [6, 7], most of them indicating values between 0.8 and 1.0, compatible with rough walls. 

Rheological determinations are destructive of the structures they measure for this reason they do not portray the actual structure of the dispersion at rest. Accordingly, various methods have been devised for extrapolating to zero the results of measurements at various shear rates. The most useful one has been the conversion of viscosities to fluidities at various shear rates and the extrapolation of the resulting nearly linear relationship to zero shear, as shown in Figure 7. Sometimes a power of the shear rate, D, provides a better distinction between a sol (essentially a liquid) and a gel (essentially a solid), as shown in the figure, but the difference between a finite intercept (sol) and zero fluidity (gel) is largely fictitious because of the dependence of the intercept on the exponent n. 

Introduction of the reptation concept by De Gennes [43] led to further essential progress. Proceeding from the notion of a reptile-like motion of the polymer chains within a tube of fixed obstacles, De Gennes [43-45], Doi [46,47] and Edwards [48] were able to confirm Bueche s 3.4-power-law for polymer melts and concentrated polymer solution. This concept has the disadvantage that it is valid only for homogeneous solutions and no statements about flow behaviour at finite shear rates are analysed. 

In order to evaluate the viscosity of a polymeric liquid at finite rates of deformation, two parameters must be determined, i.e. (i) the critical shear rate y (y=l/7.) at which T) becomes a function of the of deformation, and (ii) the slope in the linear range of the flow curve. 

Viscoelastic properties have been discussed in relation to molar mass, concentration, solvent quality and shear rate. Considering the molecular models presented here, it is possible to describe the flow characteristics of dilute and semi-dilute solutions, as well as in simple shear flow, independent of the molar mass, concentration and thermodynamic quality of the solvent. The derivations can be extended to finite shear, i.e. it is possible to evaluate T) as a function of the shear rate. Furthermore it is now possible to approximate the critical conditions (critical shear rate, critical rate of elongation) at which the onset of mechanical degradation occurs. With these findings it is therefore possible to tune the flow features of a polymeric solution so that it exhibits the desired behaviour under the respective deposit conditions. 

This flow field is somewhat idealized, and cannot be exactly reproduced in practice. For example, near the planar surfaces, shear flow is inevitable, and, of course, the range of % and y is consequently finite, leading to boundary effects in which the extensional flow field is perturbed. Such uniaxial flow is inevitably transient because the surfaces either meet or separate to laboratory scale distances. 

K. B. Migler 2002, (Layered droplet microstructures in sheared emulsions finite-size effects), /. Colloid Interface Sci. 255, 391. 

By loading is meant an application of force through the neighbors to the contact point between the two granules which are in contact with each other. Since not all couples can survive the shearing field in an agglomerating charge, there is a finite probability of coalescence, which is ... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Fig. 13. Shear stress t12 and first normal stress difference N1 during start-up of shear flow at constant rate, y0 = 0.5 s 1, for PDMS near the gel point [71]. The broken line with a slope of one is predicted by the gel equation for finite strain. The critical strain for network rupture is reached at the point at which the shear stress attains its maximum value...

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