Cylinder torsion

The relations discussed above can be applied directly to homogeneous deformations of a body in which the strains are the same everywhere in the sample and the macroscopic deformation of the sample is essentially the same as that of any infinitesimal element of it. In many deformation geometries commonly used for experimental measurements, however, such as torsion between coaxial cylinders, torsion of a cylindrical rod, flow through a tube, flexure, etc., the magnitudes of the strains and rates of strain vary from point to point. Application of the equations of continuity and motion and integration over the sample geometry are then necessary to relate external forces and displacements to the viscoelastic functions. For some cases, if the deformations are small, the geometry is no real complication ... 

Many types of rotational rheometers and viscometers have been developed. The cone and plate, couette (coaxial cylinder), torsional, and disc spindle types are the most common. 

Plots of the bursting pressures of the Ni—Cr—Mo cylinders (EN 25) vs k derived from equations 16 and 17 show that neither equation is in such good agreement with the experimental results as is the curve derived from Manning s theory. Similar conclusions have been reached for cylinders made of other materials which have been tested (16). Manning s analytical procedure may be programmed for computation and, although torsion tests are not as commonly specified as tension tests, they are not difficult or expensive to carry out (20). 

Coaxial (Concentric Cylinder) Viscometer, The eadiest and most common type of rotational viscometer is the coaxial or concentric cylinder instmment. It consists of two cylinders, one within the other (cup and bob), keeping the specimen between them, as shown in Figure 27. The first practical rotational viscometer consisted of a rotating cup with an inner cylinder supported by a torsion wire. In variations of this design the inner cylinder rotates. Instmments of both types ate useful for a variety of apphcations. 

The MacMichael viscometer is probably the most straightforward rotatioaal viscometer. The outer cup rotates and the inner cylinder is suspended from a torsion wire. The drag on the inner cylinder is measured as degree of twist on the wire. Wires of different stiffness are available, and the maximum viscosity is ca 10 mPa-s. The shear rate range is limited, ca 2-12, but with modification, higher shear rates can be attained. The iastmment is best... 

Direct Indicating Viscometer. This is a rotational type instrument powered by an electric motor or by a hand crank. Mud is contained in the annular space between two cylinders. The outer cylinder or rotor sleeve is driven at a constant rotational velocity its rotation in the mud produces a torque on the inner cylinder or bob. A torsion spring restrains the movement. A dial attached to the bob indicates its displacement. Instrument constants have been so adjusted that plastic viscosity, apparent viscosity, and yield point are obtained by using readings from rotor sleeve speeds of 300 and 600 rpm. 

As the name implies, the cup-and-bob viscometer consists of two concentric cylinders, the outer cup and the inner bob, with the test fluid in the annular gap (see Fig. 3-2). One cylinder (preferably the cup) is rotated at a fixed angular velocity ( 2). The force is transmitted to the sample, causing it to deform, and is then transferred by the fluid to the other cylinder (i.e., the bob). This force results in a torque (I) that can be measured by a torsion spring, for example. Thus, the known quantities are the radii of the inner bob (R ) and the outer cup (Ra), the length of surface in contact with the sample (L), and the measured angular velocity ( 2) and torque (I). From these quantities, we must determine the corresponding shear stress and shear rate to find the fluid viscosity. The shear stress is determined by a balance of moments on a cylindrical surface within the sample (at a distance r from the center), and the torsion spring ... 

ISO 11003-1 2001 Adhesives - Determination of shear behaviour of structural adhesives -Part 1 Torsion test method using butt-bonded hollow cylinders ISO 11003-2 2001 Adhesives - Determination of shear behaviour of structural adhesives -Part 2 Tensile test method using thick adherents ISO 13445 2003 Adhesives - Determination of shear strength of adhesive bonds between rigid substrates by the block-shear method... 

ISO 11003-1 2001 Adhesives - Determination of shear behaviour of structural adhesives -Part 1 Torsion test method using butt-bonded hollow cylinders... 

For thin-walled cylinders subject to in-plane (axial and circumferential) loading and axial torsion, Whitney and Halpin [8] have developed an analytic solution for strains. Their analysis is valid in the central region of the cylinder, end support effects are neglected. 

In-Plane Shear Properties. The basic lamina in-plane shear stiffness and strength is characterized using a unidirectional hoop-wound (90°) 0.1 -m nominal internal diameter tube that is loaded in torsion. The test method has been standardized under the ASTM D5448 test method for in-plane shear properties of unidirectional fiber-resin composite cylinders. D5448 provides the specimen and hardware geometry necessary to conduct the test. The lamina in-plane shear curve is typically very nonlinear [51]. The test yields the lamina s in-plane shear strength, t12, in-plane shear strain at failure, y12, and in-plane chord shear modulus, G12. 

For measurements on polymer melts, an apparatus of the concentric cylinder type can be used. The internal cylinder of such an apparatus is preferably suspended between two torsion wires. One of them is fixed with its lower end in the bottom of the unit, the other is twisted sinusoidally with the prescribed angular frequency at its upper end. Phase difference and ratio of amplitudes of the oscillations of the upper wire end and of the internal cylinder are measured. From these measurements the dynamic shear moduli, as defined above, can be deduced, when the inertia of the system is taken into account. Such an apparatus has been developed by Den Otter (26) at this Institute, making use of earlier experiences, as made by several other authors. (See e.g. ref. [27, 28 and 29).]... 

Typical of this class of viscometer is the coaxial or Couette type of instrument described in Volume l, Section 3.7.4. The sample fluid is contained within the annular space between two coaxial cylinders, either of which may be rotated by a motor with the remaining cylinder suspended elastically in such a way that the torsional couple exerted on the latter can be measured. If the outer cylinder of radius r2 rotates with an angular velocity cou and the inner cylinder of radius r, is stationary, and the torque (or viscous drag) per unit length of cylinder exerted on the inner cylinder is T, then, for a Newtonian fluid(49) ... 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

We never in practice achieve the homogeneity of physical properties we should like to assume in the applications of elasticity theory. Thus the solutions, available in the treatises, of such problems as simple extension, torsion, bending, the effect of point loads, pressurised cylinders and spheres and so on are never exactly applicable to real materials... 

With the supposition that the slip layer is thin and the slip velocity is constant, various analyses have been developed in the search for the ideal experimental method to define slip. The Mooney analysis (20) for both tube flow and concentric cylinder flow has been applied to a wide range of materials including polymer solutions (21), filled suspensions (22), semisolid foods (23), fruit purees (24), and ketchups (25). Alternate estimates of slip velocity have been determined experimentally from, parallel plate torsion flow (26), from flow data in channels and inclined planes, and from porous medium geometries (8). 

When the liquid, solution or lyophobic colloidal suspension contains asymmetric particles or when it is too concentrated, other methods must be applied to measure the viscosity. This is for instance the case with clay suspensions. In the past the viscosity of clay suspensions was measured by means of a bucket with a hole in it. The bucket was filled with clay suspension and after the stopper had been removed from the hole, the time required by the volume to drain was measured as a function of e.g. the volume and composition. Later mechanical methods were applied. One of them is based on the principle that a metal cylinder or disc, suspended from a torsion thread, is exposed to a certain resistance when you rotate it in the solution or suspension. Before the measurement the cylinder or disc is turned 360° anti-clockwise and then released. After having revolved over a certain angle, the cylinder or disc will change its direction of rotation. The rotation angle is a measure for the viscosity. 

In the first of these techniques an approximation to uniform rate of shear throughout the sample is achieved by shearing a thin film of the liquid between concentric cylinders. The outer cylinder can be rotated (or oscillated) at a constant rate and the shear stress measured in terms of the deflection of the inner cylinder, which is suspended by a torsion wire (Figure 9.2) or the inner cylinder can be rotated (or oscillated) with the outer cylinder stationary and the resistance offered to the motor measured. 

A graphic example of the consequences of the existence of in stress in simple steady shear flows is demonstrated by the well-known Weissenberg rod-climbing effect (5). As shown in Fig. 3.3, it involves another simple shear flow, the Couette (6) torsional concentric cylinder flow,3 where x = 6, x2 = r, x3 = z. The flow creates a shear rate y12 y, which in Newtonian fluids generates only one stress component 112-Polyisobutelene molecules in solution used in Fig. 3.3(b) become oriented in the 1 direction, giving rise to the shear stress component in addition to the normal stress component in. 

Viscosity and Plasticity—Viscosity and plasticity are closely related. Viscosity may be defined as the force required to move a unit-area of plane surface with unit-speed relative to another parallel plane surface, from which it is separated by a layer of the liquid of unit-thickness. Other definitions have been applied to viscosity, an equivalent one being the ratio of shearing stress to rate of shear. When a mud or slurry is moved in a pipe in more or less plastic condition the viscosity is not the same for all rates of shear, as in the case of ordinary fluids. A material may be called plastic if the apparent viscosity varies with the rate of shear. The physical behavior of muds and slurries is markedly affected by viscosity. However, consistency of muds and slurries is not necessarily the same as viscosity but is dependent upon a number of factors, many of which are not yet clearly understood. The viscosity of a plastic material cannot be measured in the manner used for liquids. The usual instrument consists of a cup in which the plastic material is placed and rotated at constant speed, causing the deflection of a torsional pendulum whose bob is immersed in the liquid. The Stormer viscosimeter, for example, consists of a fixed outer cylinder and an inner cylinder which is revolved by means of a weight or weights. 

The sample, usually in the form of a cylinder, can be subjected to uniaxial compression (the simplest and most common test), uniaxial tension, shear, bending or torsion. In compression, the sample rests on the base-plate and is compressed by a horizontal flat plate attached to the crosshead when... 

FIG. 15.2 Types of simple shear flow. (A) Couette flow between two coaxial cylinders (B) torsional flow between parallel plates (C) torsional flow between a cone and a plate and (D) Poisseuille flow in a cylindrical tube. After Te Nijenhuis (2007). 

In Couette flow and in torsional flow between a cone and a plate the shear rate may be considered constant, provided the slit Ar between the two cylinders and the angle A between cone and plate, respectively, are small. On the other hand, the shear rates in... 

Torsional vibration can also be a problem and results from pressure variations in the cylinders which can produce cyclic torques with harmonics ranging from half speed to 10 or 12 times running speed. 

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