Effective shear modulus

If a disklike specimen is sheared between two end plates by rotation of one over the other to obtain the shear modulus, then at any moderate twist angle the strain (and strain rate) vary along the radius, so only an effective shear modulus is obtained. For better results the upper plate is replaced with a cone of very small angle. Figure 4 shows fche cone-and-plate and two other possible test geometries for making shear measurements. 

Under the effect of uniform hydrostatj.c pressure and uniform tension, the effective bulk modulus, K = 1/P, and effective shear modulus, p, are calculated. The result is... 

Basing on this statement, we shall calculate the shear moduli of the charged polymeric composite. According to Ref. 172 the effective shear modulus of such a composite (real part of the modulus) at limiting doping may be estimated as p = 2 106[Pa] at m = 0, which exceeds the shear modulus of the unperturbed polymetric compound . Note that... 

In calculating of the effective shear modulus of the composite, it is assumed that the shear modulus of the particles comprising the doping compound along with its boundary region is p) = 2 106[Pa], p" = 0 [174] so that p is... 

The calculation of the real part of the effective shear modulus p of a composite with fractal structure is illustrated. According to this calculation (Fig. 58) the percolation transition appears after go < 10 4 and at doping concentration p 0.12, i.e. for p > 0.12 in a composite with a continuous and strong skeleton composed of particles of a doping compound connected by a boundary stratum of a polymetric compound. 

As mentioned earlier, fractal structures can exhibit properties different from uniform structures. To illustrate this we compare the effective shear modulus and the damping coefficient of a medium with a composite material corresponding to the Hashin-Strikman formulae (referred to as the Hashin-Strikman composite ). 

In Fig. 60, the ratio of the real part of the effective shear modulus to the shear modulus of the elastic phase is shown as a function of the concentration, p, of... 

Figure 62. Ratio of the imaginary to the real part of the effective shear modulus (a, b) in the fractal composite and (c, d) in the Hashin-Strikman composite the other details are the same as in Fig. 60.

According to the calculations (Figs. 60 and 61) the peaks in the dependence of the real part of the effective shear modulus p of the composite of fractal structure are grouped near x 0.2 and 0.8 < x < 1. 

The nature of the peaks and the way in which they are formed depending on the effective shear modulus p, the frequency co, and the concentration p can be understood by considering a single insert with negative shear modulus Pj = p (p = —xK ) immersed in a medium (stabilizing matrix) of positive shear modulus p2 = p2. 

As the model studied constitutes a self-similar, chaotic system of clusters of various sizes, on the jth scale level, each cluster will have its own resonance parameters—that is, its own characteristic frequency, which produces the system of peaks (characteristic frequencies) in the dependence of the effective shear modulus p on the parameters of the composite. 

We also have the effective shear modulus relation, = G, and the effective parameter, 0, which includes the effective fluid inertia in the relative motion parameter, p, i.e.. 

These ideas provide some insight, though we assume that the particles are freely dispersed, but have themselves elastic properties, which is different from the matrix. Examples for such filler particles are elastic microgels. The general result for the effective shear modulus Eq. (36.1) can be written as ... 

Penzien J. and Didriksson T., Effective Shear Modulus of Honeycomb Cellular Structure , AAIA J, March 1964, pp531-5. 

The main defonnation characteristics of objects include effective bulk modulus of the material k and the effective shear modulus X. If necessary, transition is possible from this values to Young s modulus E and Poisson s ratio V aceording to (7). Backward calculations are possible too. 

It is known from previously conducted studies [30, 31] and the effect of the Hall-Petch, the effective bulk modulus and effective shear modulus of nano-objects is not a linear function of their size. Generalized dependence of these quantities can be written in the form of expressions. 

Of special interest here to bcc screw dislocation motion is the effective shear modulus along the < 111 > slip direction,... 

The last approach is a non-linear extension of conventional theory, which is called bifurcation-type model. It leads to an effective shear modulus which we will call also C5 in the following for convenience, and which follows the same qualitative behavior as the one predicted by version 2 of soft elasticity (Menzel et al. 2009b). 

---