Apparent shear modulus

Similarly, the orthotropic macroscopic response of the representative element is 

Finally, substituting Equations 8.23 and 8.24 into Equation 8.26 gives the desired equation relating the macroscopic shear modulus Glt to the constituent shear moduli as a function of fiber volume fraction  

FIGURE 8.12 Variation of G,t with fiber volume fraction. 

5 Summary of Results from Micromechanics Analysis of Lamina Elastic Moduli 

The four main sets of equations relating the macroscopic engineering elastic properties of the lamina to the constituent elastic properties as a function of fiber volume fraction are grouped together below  

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This reduces the apparent shear modulus, Ga, which is given by ... 

If a circular cylinder of orthotropic symmetry is used with axis along 3, the apparent shear modulus to insert in Eq. (2.1) above is G = 2/(844 + S55) so that for this geometry a unique value for S44 or S55 is unobtainable. 

Fig. 2. Apparent shear modulus (from torsion data) of oriented S. B. S copolymer as a function of sample aspect ratio. Curve indicates the general trend of results (From Ref. 12)...

Torsional modulus n. Shear modulus (G) as measured in a test in which the specimen is twisted. In ASTM D 1043, the test specimen is a flat rectangular bar with length about seven times its width of 6.35 mm and thickness 1/3 the width. The apparent shear modulus is given by... 

In our laboratory, we have measured the apparent shear stiffness of balsa wood specimens with D = 112 kg/m by means of the experimental method described on page 37 of Ref. 24, Using three prismatic specimens of balsa wood cut so that the grain directions were inclined 0°, 45, and 90° to the specimen axis and bonding strain gages to their prism faces, we measured E, E, and Ei+5. The apparent shear modulus G. was then computed by equation (5) with 0 = 45°. The result was... 

The second purpose is to test lubricants of different nature (naphthenic base oil, naphthenic base oil plus polymer, silicone oil and grease) in rolling-sliding point contacts. This is done in order to either correlate the lubricant rheological properties (the mean apparent viscosity, the mean apparent shear modulus and the mean limiting shear stress) to existing data obtained independently by Bair and Winer (3), or to find limitations if any, in the use of the film thickness formulae and of the traction model under fully flooded elastohydrodynamic conditions. 

Hutton et al fS llJ have studied many types of lubricants with several techniques ultrasonic technique in longitudinal or in shear mode, shear flow in concentric cylinders. Viscoelastic behaviour in shear but also in compression have been shown and the differences between Qy, (shear modulus at infinite frequency), G (apparent shear modulus deduced from traction curves) and G (mechanical shear modulus) have been clearly pointed out. 

This is the dependence expected for purely electrostatic stabilization of polyelectrolyte layers. Electro-steric interactions involve steric contributions [Eq. (3)] and electrostatic contributions [Eq. (4)]. For high ionic strength as in the interior of the polyelectrolyte shell the measured shear modulus should exhibit a characteristic G (r) e Kr-dependence which is apparent in a semi-logarithmic presentation of the data as in Fig. 13. The measured shear moduli are plotted as a function of the reduced distance r/2Rm which is... 

Experimental load deflection curves (Fig. 3.) illustrate the large difference in crack propagation observed in each case. A difference in stiffness between both bonded specimens is observed and results from either a difference in the bond line quality or from interfacial conditions. For both specimens, adherends were made from the same sample of wood. Both wood substrates contained no apparent defects and had the same longitudinal Young s modulus (14500 MPa). Both also had the same growth characteristics (oven dry specific density, annual growth rings), and as a consequence very close values of transverse and shear modulus adjacent to the bond line. Thus, any difference in stiffness is likely to be due to... 

Figure 29. Storage shear modulus vs. frequency for System 1. The points are experimental, uncorrected for the apparent yield stress, the lines computed from the relaxation spectrum.

There is, however, one basic principle, which derives from the fact that fractal clusters are scale invariant. When a gel is formed, clusters of size Rg make bonds with each other via strands at the periphery of the cluster, and the average number of the strands involved per cluster will not depend on cluster size. Since the apparent surface area of a cluster scales with R2 the number of junctions between clusters per unit area of cross section of the gel will scale with R 2. By using the equation for Rg we arrive at the following equation for the shear modulus of a fractal particle gel ... 

On deformation of the system, the bubbles are deformed, which increases their Laplace pressure p. Moreover, some films between particles are stretched and others are compressed, causing surface tension gradients to form, which also needs energy. Above a certain stress, yielding may occur, which means that bubbles (or drops) start to slip past each other, which generally occurs in planes about parallel to the direction of flow. Calculation of the shear modulus and the yield stress from first principles is virtually impossible because of the intricacy of the problem for a three-dimensional and polydisperse system, but trends can be predicted. One relation is that these parameters are proportional to the average apparent Laplace pressure... 

Fig. 9 The acoustic contrast easily saturates. The figure shows a sketch of the contrast function (integrands in Eqs. 78 and 79) as a function of the polymer volume fraction of an adsorbed polymer film. It is assumed that both the shear modulus Gf and the dielectric constant Sf are roughly proportional to the polymer concentration. However, Gf increases much more strongly than gf. If, for instance, a swollen polymer film contains 50% water, this will not appreciably decrease the apparent acoustic thickness because the modulus of the film is still much larger than the modulus of water and (Gf - Gijq)/Gf remains about unity. This is different in optics because the contrast is roughly proportional to the concentration. As a consequence, the apparent optical film thickness is proportional to the product of concentration and thickness, which is the adsorbed amoimt. In acoustics, the apparent thickness is close to the geometric thickness. Trapped water appears as part of the film in acoustics...

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