Deformation simple shear

Deformation Simple Shear Bulk Compression Simple Extension" Bulk Longitudinal... 

Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as... 

Until now we have restricted ourselves to consideration of simple tensile deformation of the elastomer sample. This deformation is easy to visualize and leads to a manageable mathematical description. This is by no means the only deformation of interest, however. We shall consider only one additional mode of deformation, namely, shear deformation. Figure 3.6 represents an elastomer sample subject to shearing forces. Deformation in the shear mode is the basis... 

A sliding plate rheometer (simple shear) can be used to study the response of polymeric Hquids to extension-like deformations involving larger strains and strain rates than can be employed in most uniaxial extensional measurements (56,200—204). The technique requires knowledge of both shear stress and the first normal stress difference, N- (7), but has considerable potential for characteri2ing extensional behavior under conditions closely related to those in industrial processes. 

In this case, the shear stress is linear in the shear strain. While more physically reasonable, this is not likely to provide a satisfactory representation for the large deformation shear response of many materials either, since most materials may be expected to stiffen with deformation. Note that the hypoelastic equation of grade zero (5.117) is not invariant to the choice of indifferent stress rate, the predicted response for simple shear depending on the choice which is made. 

A number of other indifferent stress rates have been used to obtain solutions to the simple shear problem, each of which provides a different shear stress-shear strain response which has no latitude, apart from the constant Lame coefficient /r, for representing nonlinearities in the response of various materials. These different solutions have prompted a discussion in the literature regarding which indifferent stress rate is the correct one to use for large deformations. 

Since simple shear is a constant volume deformation, the solution does not depend on coefficients of terms involving tr(various values of a are shown in Fig. 5.9. The solution for a grade zero material using Jaumann s stress rate (5.120) corresponds to = Ug = Ug = 0 so that a = -1, while the solution using Truesdell s stress rate (5.122) corresponds to = 0 and Ug = 1 so that a = 0. The shear... 

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

Not only are there two classes of deformation, there are also two modes in which deformation can be produced simple shear and simple tension. The actual action during melting, as in the usual screw plasticator is extremely complex, with all types of shear-tension combinations. Together with engineering design, deformation determines the pumping efficiency of a screw plasticator and... 

The above description refers to a Lagrangian frame of reference in which the movement of the particle is followed along its trajectory. Instead of having a steady flow, it is possible to modulate the flow, for example sinusoidally as a function of time. At sufficiently high frequency, the molecular coil deformation will be dephased from the strain rate and the flow becomes transient even with a stagnant flow geometry. Oscillatory flow birefringence has been measured in simple shear and corresponds to some kind of frequency analysis of the flow... 

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) ... 

When a mbber block of rectangular cross-section, bonded between two rigid parallel plates, is deformed by a displacement of one of the bonded plates in the length direction, the rubber is placed in a state of simple shear (Figure 1.1). To maintain such a deformation throughout the block, compressive and shear stresses would be needed on the end surfaces, as well as on the bonded plates [1,2]. However, the end surfaces are generally stress-free, and therefore the stress system necessary... 

The stresses set up in a long mbber block or tube under simple shear deformations are found to depend on the shapes of the end surfaces, even when the block or tube is quite long. 

The degree of deformation and whether or not a drop breaks is completely determined by Ca, p, the flow type, and the initial drop shape and orientation. If Ca is less than a critical value, Cacri the initially spherical drop is deformed into a stable ellipsoid. If Ca is greater than Cacrit, a stable drop shape does not exist, so the drop will be continually stretched until it breaks. For linear, steady flows, the critical capillary number, Cacrit, is a function of the flow type and p. Figure 14 shows the dependence of CaCTi, on p for flows between elongational flow and simple shear flow. Bentley and Leal (1986) have shown that for flows with vorticity between simple shear flow and planar elongational flow, Caen, lies between the two curves in Fig. 14. The important points to be noted from Fig. 14 are these ... 

For Ca > Cacri, a drop continually stretches until it breaks. If Ca > KCacr , where k is about 2 for simple shear flow and 5 for elongational flow (Janssen, 1993), the drop undergoes affine deformation, i.e., the drop acts as a material element, and it is stretched into an extended cylindrical thread with length L and radius R according to... 

Rumscheidt, F. D., and Mason, S. G., Particle motions in sheared suspensions. XII. Deformation and burst of fluid drops in shear and hyperbolic flow. J. ColloidScL 16,238-261 (1961). Rwei, S. P., Manas-Zloczower, I., and Feke, D. L., Observation of carbon black agglomerate dispersion in simple shear flows. Polym. Eng. ScL 30, 701-706 (1990). 

What is a fluid It isn t a solid, but what is a solid Perhaps it is easier to define these materials in terms of how they respond (i.e., deform or flow) when subjected to an applied force in a specific situation such as the simple shear situation illustrated in Fig. 3-1 (which is virtually identical to Fig. 1-1). We envision the material contained between two infinite parallel plates, the bottom one being fixed and the top one subject to an applied force parallel to the plate, which is free to move in its plane. The material is assumed to adhere to the plates, and its properties can be classified by the way the top plate responds when the force is applied. 

We now use this to calculate the stress in the melt after the retraction has occurred. The deformation is described by the tensor E defined so that an arbitrary vector V in the material is deformed affinely into the vector E.v. For example, in simple shear of shear strain 7, and in uniaxial extension of strain e, the tensor E takes the forms... 

In the two classic viscometric deformations of simple shear and extension, the appropriate components of Q have very different behaviour. For small shear strains, the shear stress depends on the component Q which has the linear asymptotic form 47/15. This prefactor is the origin of tne constant v in the tube potential of Sect. 3.For large strains, however, Qxy 7 and therefore predicts strong shear-thinning. Physically this comes from the entanglement loss on re-... 

R.A. De Bruijn Deformation and Breakup of Drops in Simple Shear Flows. Ph. D. Thesis Eindhoven University of Technology (1989). 

Note 3 The deformation gradient tensor for the simple shear of an elastic solid is... 

Interpretation of the viscoelastic behaviour of a liquid or solid in simple shear or uniaxial deformation such that... 

Note 1 ymay be in simple shear or uniaxial deformation. 

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