Shear, simple

For pure shear we note that, in this mode of plane-strain deformation with no thinning in the 3-direction, a square of sides ao drawn on the 1-2 plane transforms into a rectangle having a long direction of 212 0 and a short direction of li o = Moreover, a square of sides ao drawn on the 1-2 face with edges 

Consider the square ABCD drawn on the 1-2 face of a planar box with edges parallel to the 1, 2 axes obtained by a rotation of the 1, 2 axes by %jA relative to 

In a following operation the sheared parallelogram A B C D is rotated back counter-clockwise by an angle tp/l to give a final parallelogram A B C D so that the corners A B C and D now lie on the 1, 2 principal axes. The new parallelogram A B C D effectively represents a pure shear distortion of the initial square ABCD. 

We note that this process described above demonstrates that a simple shear distortion by a clockwise shear tp followed by a counter-clockwise rotation back by tp/l results in a pure shear distortion in the original axis system and that the work of simple shear distortion must equal the work of pure shear distortion (Love 1944). 

Meanwhile, a simple transformation of coordinate axes from the 1,2 system to the 1, 2 system by a rotation of 7t/4 gives by a Mohr-circle construction the shear stress T acting on the 1, 2 -axis system 

For a system undergoing simple shear, when the velocity gradient v 2 7 0, the relative permittivity tensor is non-diagonal 

However, the tensor can be turned to diagonal form by rotating the coordinate frame round axis 3 by an angle x (the extinction angle), defined by the formula 

The differences between the refractive indices (the extent of double refraction) in the different principal directions can be determined from equation (10.17). For a beam propagated in direction 3, we find that 

This relation as well as relation (10.18) is valid in linear approximation and can be therefore rewritten as 

A little bit more complicated situation appears, if one considers a beam propagating across the flow in direction characterised by the unit vector 

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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... 

In addition to the apparent viscosity two other material parameters can be obtained using simple shear flow viscometry. These are primary and secondary nomial stress coefficients expressed, respectively, as... 

In packed beds of particles possessing small pores, dilute aqueous solutions of hydroly2ed polyacrylamide will sometimes exhibit dilatant behavior iastead of the usual shear thinning behavior seen ia simple shear or Couette flow. In elongational flow, such as flow through porous sandstone, flow resistance can iacrease with flow rate due to iacreases ia elongational viscosity and normal stress differences. The iacrease ia normal stress differences with shear rate is typical of isotropic polymer solutions. Normal stress differences of anisotropic polymers, such as xanthan ia water, are shear rate iadependent (25,26). 

Non-Newtonian Fluids Die Swell and Melt Fracture. Eor many fluids the Newtonian constitutive relation involving only a single, constant viscosity is inappHcable. Either stress depends in a more complex way on strain, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known coUectively as non-Newtonian and are usually subdivided further on the basis of behavior in simple shear flow. 

Fig. 10. Fluid behavior in simple shear flow where A is Bingham B, pseudoplastic C, Newtonian and D, dilatant.

Pseudoplastic fluids are the most commonly encountered non-Newtonian fluids. Examples are polymeric solutions, some polymer melts, and suspensions of paper pulps. In simple shear flow, the constitutive relation for such fluids is... 

The state of stress in a cylinder subjected to an internal pressure has been shown to be equivalent to a simple shear stress, T, which varies across the wall thickness in accordance with equation 5 together with a superimposed uniform (triaxial) tensile stress (6). 

Fig. 1. Laminar flow in simple shear. FjA — rjdV/dX, where F is the force acting on areaM, H the velocity and X the thickness of the layer, and Tj the...

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. 

The force is direcdly proportional to the area of the plate the shear stress is T = F/A. Within the fluid, a linear velocity profile u = Uy/H is estabhshed due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient y = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. 

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. 

Proceeding as before, the solution may be found for the simple shear problem using the hypoelastic equation of grade one (5.123). The differential equations are found to be... 

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... 

Note that the solution predicts that simple shear produces normal stresses. In fact, although simple shear occurs at constant volume, the normal stresses in general give rise to a hydrostatic pressure. Determination of the normal stresses in the case of a hypoelastic equation of grade one could, at least in principle, determine the coefficients a, Ug, and Ug individually. 

Y.L. Bai, Thermo-Plastic Instability in Simple Shear, J. Mech. Phys. Solids 30, 195-207 (1982). 

Simple shear Above materials plus Poly(methyl methacrylate)... 

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... 

Depending on the relative velocity in opposite pairs of the rollers, flow can be either purely rotational (X = — 1), simple shear (X = 0) or hyperbolic straining (X = 1, shown on this figure)... 

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... 

The Giesekus criterion for local flow character, defined as

extensional flow, 0 in simple shear flow and — 1 in solid body rotation [126]. The mapping of J> across the flow domain provides probably the best description of flow field homogeneity current calculations in that direction are being performed in the authors laboratory. 

Fig. 38. Action of entanglements in simple shear degradation. The tension on the qth link from the center is given by fq = f0(l — 4q2/n2), where f0 qsy f(L) is the tension on center link (f(L) = function of chain length) and n = number of links per chain.

Much fewer experiments are available in solution where the few reported data are generally more concerned about the effect of molecular structure than about bond dissociation energy. In simple shear, it is generally agreed that chain flexibility dominantly influences the rate of bond scission, with the most rigid polymers being the easiest to fracture [157]. The results are interpreted in terms of the presence of good and poor sequences in the chain conformation. 

Chain degradation in turbulent flow has been frequently reported in conjunction with drag reduction and in simple shear flow at high Reynolds numbers [187], Using poly(decyl methacrylate) under conditions of turbulent flow in a capillary tube, Muller and Klein observed that the hydrodynamic volume, [r ] M, is the determining factor for the degradation rate in various solvents and at various polymer concentrations [188], The initial MWD of the polymers used in their experiments are, however, too broad (Mw/Iiln = 5 ) to allow for a precise... 

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