Shear modulus calculation

The shear modulus, calculated in this way for parallel arranged elements... 

Williams, G. (1962) Upper limit of the shear modulus calculated from the Rouse theory. 

There was still a variation in break properties however, so that at the highest rates the curves showed an upturn below X=2 (IOO56 strain). Raising the t nperature to kO C removed even this small upturn, since the strain never exceeded lOO t, Figure l l-. The observed increase in Ci is just what would be expected if is proportional to the absolute temperature. The only disturbing feature of the results is that the shear modulus calculated from 2Cq at 20 C is 6.0 X 10 dynes/cm, rather than the values of about 8 x 10 obtained in the stress relaxation experiments. 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

A simplified performance index for stiffness is readily obtained from the essentials of micromechanics theory (see, for example. Chapter 3). The fundamental engineering constants for a unidirectionally reinforced lamina, ., 2, v.,2, and G.,2, are easily analyzed with simple back-of-the-envelope calculations that reveal which engineering constants are dominated by the fiber properties, which by the matrix properties, and which are not dominated by either fiber or matrix properties. Recall that the fiber-direction modulus, is fiber-dominated. Moreover, both the modulus transverse to the fibers, 2, and the shear modulus, G12. are matrix-dominated. Finally, the Poisson s ratio, v.,2, is neither fiber-dominated nor matrix-dominated. Accordingly, if for design purposes the matrix has been selected but the value of 1 is insufficient, then another more-capable fiber system is necessary. Flowever, if 2 and/or G12 are insufficient, then selection of a different fiber system will do no practical good. The actual problem is the matrix systemi The same arguments apply to variations in the relative percentages of fiber and matrix for a fixed material system. 

In many isotropic materials the shear modulus G is high compared to the elastic modulus E, and the shear distortion of a transversely loaded beam is so small that it can be neglected in calculating deflection. In a structural sandwich the core shear modulus G, is usually so much smaller than Ef of the facings that the shear distortion of the core may be large and therefore contribute significantly to the deflection of a transversely loaded beam. The total deflection of a beam is thus composed of two factors the deflection caused by the bending moment alone, and the deflection caused by shear, that is, S = m + Ss, where S = total deflection, Sm = moment deflection, and Ss = shear deflection. 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

The ratio of shear stress to shear strain. A property which determines the rate at which elastomers stiffen as the temperature is lowered. The force required to twist the test piece through 90° is measured at each temperature and the modulus calculated from a formula. 

The viscoelastic behavior of concentrated (20% w/w)aqueous polystryene latex dispersions (particle radius 92nm), in the presence of physically adsorbed poly(vinyl alcohol), has been investigated as a function of surface coverage by the polymer using creep measurements. From the creep curves both the instantaneous shear modulus, G0, and residual viscosity, nQ, were calculated. 

Estimates of the ultimate shear strength r0 can be obtained from molecular mechanics calculations that are applied to perfect polymer crystals, employing accurate force fields for the secondary bonds between the chains. When the crystal structure of the polymer is known, the increase in the energy can be calculated as a function of the shear displacement of a chain. The derivative of this function is the attracting force between the chains. Its maximum value represents the breaking force, and the corresponding displacement allows the calculation of the maximum allowable shear strain. In Sect. 4 we will present a model for the dependence of the strength on time and temperature. In this model a constant shear modulus g is used, thus r0=gyb. 

Figure 2.14 The relative shear modulus, G> = G/GN, of an HEUR gel filled with polyethylmethacrylate particles. GN = 0.4 kPa. Experimental points are shown with the curves calculated for a non-interactive and an interactive filler. The amount of adsorbed polymer, T = 78g/kg offiller, gave a good fit to the experimental data...

Figure 5.11 The shear modulus obtained from experiments compared with the calculated value from Equation (5.39). The system is the same as that used to calculate the pair potential in Figure 5.9...

Figure 5.18 The high frequency shear modulus versus volume fraction for a polystyrene latex for three different electrolyte concentrations. The symbols are the experimental data and the solid lines are calculated fits using a cell model. The radius of the latex particles was 38 nm...

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