Determination of shear modulus

Figure 5.80 Torsion pendulum for the determination of shear modulus aud damping as functions of temperature at frequencies around 1 Hz. Reprinted, by permission, from N. G. McCrum, C. P. Buckley, and C. B. BucknaU, Principles of Polymer Engineering, 2nd ed., p. 133. Copyright 1997 by Oxford University Press.

For anisotropic materials torsion is discussed in the books by Love, Lekhnitskii175 and Hearmon185. The torque M now depends not upon one elastic constant only, as in the isotropic case, but upon two. This makes the determination of shear modulus by a torsion test a difficult task and requires careful experimentation. Early work on this for polymers was done by Raumann195, by Ladizesky and Ward205 and by Arridge and Folkes165. 

Applications where there is compression will often have a shear component, even if not by design. The following methods detail the determination of shear modulus. 

Rg. 7.12 A torsion pendulum for the determination of shear modulus and damping at frequencies near 1 Hz. The support wire has negligible torsional rigidity. (Reproduced by permission of L. C. E. Struik.)... 

Deformation behaviour Biaxial, parallel thread tests Shear behaviour Determination of shear modulus Relaxation behaviour... 

The continuing search to determine the shear modulus and shear strength consists of a collection of tests. Several tests are discussed because each has faults, as will be seen, and because, to some extent, there is no universal agreement on the best way to measure the shear properties. 

Another test used to determine the shear modulus and shear strength of a composite material is the sandwich cross-beam test due to Shockey and described by Waddoups [2-17]. The composite lamina... 

A test method to evaluate the shear stress capability of a seal material is reported [36], An electrolyte-anode-electrolyte trilayer was glass sealed to two metal interconnect plates as shown in Figure 5.11. Shear testing was done in two different modes, constant loading rate and constant displacement rate, to determine the shear modulus and viscosity. 

ISO 1827 1991 Rubber, vulcanized or thermoplastic - Determination of modulus in shear or adhesion to rigid plates - Quadruple shear method ISO 1922 2001 Rigid cellular plastics - Determination of shear strength ISO 4587 2003 Adhesives - Determination of tensile lap-shear strength of rigid-to-rigid bonded assemblies... 

The above results, which were formulated in the early thirties (TVeloar 1958), explain the main features of the elastic behaviour of polymer networks. Nevertheless, there are notable discrepancies between empirical data and the cited results. Further investigations demonstrated that the values of shear modulus and stress-strain dependence are determined substantially by topological constraints due to the proximity of the chains. The theory was improved by taking into account the discussed issue (Edwards 1967a, 1967b, 1969 Flory 1977 Erman and Flory 1978 Priss 1957, 1980, 1981). More recent developments are summarised in the work of Panyukov and Rabin (1996), where many additional relevant references could be found. 

To determine the shear modulus and the tensor of recoverable strains, we calculate the determinants of the left-hand and right-hand sides of equation (9.74). Taking into account the incompressibility of the polymer liquid, i.e. relation AyAjy = 1, we obtain... 

One can determine the shear modulus of a fiber from a torque per unit area versus twist curve. In practice, a simple apparatus called a torsion penduliun is used more commonly. An experimental setup to measure the shear modulus of small fibers is shown in Fig. 9.8 (Mehta, 1996). The torsional pendulum, placed in a vacuum oven, allows the measurement of shear modulus as a func-... 

Oakenfull, D. G., Parker, N. S., and Tanner, R. I. 1989, Method for determining absolute shear modulus of gels from compression tests. J. Texture Stud. 19 407-417. 

The automation of the torsion pendulum utilizing a desktop computer eliminates the tedious data analysis previously associated with that technique. Any one of four data reduction methods can be used the experimental conditions will determine which is the optimum one to employ. The torsion pendulum technique provides quantitative values of shear modulus and logarithmic decrement and in the torsion braid mode provides a qualitative analysis of materia i, especially in the liquid-to-solid transition region. 

The important, lesson from this example is that we can determine tee shear modulus of a material by deforming it in an experiment that looks nothing like the picture of simple shear in Figure 2 2. This is an illustration of the advantage of the full, consideration of the three-dimensional character of deformation. 

Dynamic loading problems in the offshore environment depend on either estimated or measured values of shear modulus. In practice, in situ determination of shear wave velocity on land has been used as the best approximation to the actual values for laboratory tests on samples (Richart, 1975). The techniques for using these seismic methods and data acquisition techniques to determine shear wave velocity for land-based applications have been well developed. The problem in the marine environment has been to develop methods to determine in situ shear wave velocity measurements both at the seabed surface and at known depths in the sediment column, which can be determined in a cost-effective manner. 

To solve the problem of measuring shear wave velocity in the soil column, a seismic cone penetrometer has been developed. The seismic cone contains a triaxial set of geophones (i.e., detectors) incorporated in a conventional in situ piezocone. It is typically pushed into the soil from the seabed or from the bottom of an advancing borehole. The source is typically a hydraulically driven spring hammer located on the seabed. It is ideally coupled to the sediment surface and preferentially generates horizontally polarized shear waves. It is important to decouple the drill rods and tools from the seismic cone prior to testing because compression wave energy may be transmitted. This allows the full characteristics of the soil in terms of shear modulus to be determined. 

The instrument for determination of shear moduli was a Rheometric Scientific dynamic mechanical thermal analyzer, model DMTA V. Round shear sandwich geometry was used. The instrument was inverted so that the sandwich fixtures and sample were in water. A water-jacketed 1000 mL Pyrex cylinder supplied by Rheometric Scientific allowed control of temperature with a circulating temperature bath. A sinusoidal linear shear was applied by moving a flat plate between two identical disk-shaped samples over a specified range of frequencies. The two identical disk-shaped samples were sandwiched between the moving plate and two 12 mm diameter plates (called studs) fastened to a frame. The dynamic frequency sweeps to obtain the loss shear modulus, G", from 0.16 Hz to 318 Hz were reported in log scale. For our purposes the applied initial static force was 0.05 N. Sample size was 12 mm diameter and 0.7 mm thickness. The sample was equilibrated at 40°C for 12 hours prior to starting the series of measurements. Shear moduli were measured from high to low temperature with an equilibration time of 2 hours at each temperature. The sequence of measurements was 40°C, 30°C, 25°C, and 15°C. 

A more simplified predictive model has been developed by Al-Khateeb et al. (2006), which, for the determination of dynamic modulus, uses only two parameters the voids in the mineral aggregate and the dynamic shear modulus of the binder. As concluded, the model is capable of predicting the dynamic modulus of an asphalt concrete at a broader range of temperatures and loading frequencies than the Hirsch model. It also has the advantage of estimating the dynamic modulus of an asphalt concrete with modified bitumen (Al-Khateeb et al. 2006). The mathematical formulation of the model developed is as follows ... 

It should be noted that the elastic modulus in bending is not necessarily the same as that in tension or compression. It may be determined analytically or by physical test. If it is determined by test it should be ensured that the test span is sufficiently large that deflection due to shear is negligible. The shear rigidity of a section may be defined as the product of shear modulus and the cross section of the shear area. 

The full characterization of a unidirectional composite under an in-plane shear load parallel to the fiber direction will entail the measurement of shear modulus G12, shear strength Fg and ultimate shear strain j/g. These can be determined by ... 

Although they are slightly higher than some of the others, Rosen s (1969a) values at room temperature and 4.2 K are given in table 8.22 as the best values because they provide a more complete set. It should also be noted that Savitskiy and Dashkovskiy (1961) have determined the shear modulus from 293 to about... 

Mayer, B. P. Reimer, J. A. Maxwell, R. S., A Methodology for the Indirect Determination and Spatial Resolution of Shear Modulus of PDMS-SUica Elastomers. Macromolecules 2008,41,1323-1327. 

The data obtained using the test method reviewed should be reported as direct shear strength. These data can only be compared to data determined by the same direct-shear methods. This test cannot be used to develop shear S-S curves or determine a shear modulus, because bending or compression rather than pure shear transfer a considerable portion of the load. The test results depend on the susceptibility of the material to the sharpness of load faces. 

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