Reduced equilibrium shear modulus

Figure 5.11 Dependence of the reduced equilibrium shear modulus, Ge/wg// 7" on the molar ratio of [OH]/[NCO] groups, ah, for poly(oxypropylene)triol (Niax LG 56)-4,4 -diisocyanatodiphenylmethane system (—-) limits of the Flory-Erman junction fluctuation rubber elasticity theory. The dependence has been reconstructed from data of ref. [78]...

Fig. 182. Reduced equilibrium shear modulus, as calculated from Eq. (30), plotted vs reduced concentration for crosslink functionalities f = 4,6 and 1( the broken lines are the asymptotes for high concentrations...

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

Clark and Ross-Murphy [1,487] calculated the concentration dependence of the equilibrium shear modulus of gels by making use of Hermans treatment [488], in which the a ociation of primary chains is described by a monomer-dimer equilibrium. The crosslinks ate treated as dimer interactions and, hence, their functionality is 4. From that consideration they were aUe to calculate the equilibrium shear modulus as a function of the reduced concentration, Le. c/Po, where c is the minimal concentration needed for network formation. 

Hence, the reduced modulus is proportional to (c/c ) whereas the equilibrium shear modulus G is proportional to c° Results of calculations are shown in Fig. 182, where G , is plotted vs c, on double logarithmic scales for... 

To reduce the number of unknown parameters of the model, it was assumed [203] that the dependence (Jmm( h) can be determined from existing experimental data [200,212]. Fig. 8.13 compares the shear modulus for two limiting states of the membrane matrix equilibrium (rj) and nonequilibrium (77 ). The model proposed qualitatively agrees with few reliable experimental data on the effect of dissolved hydrogen on the mechanical modules. It correctly predicts the decrease in both Young s modulus and sheer modulus during the saturation of the metal with hydrogen in a wide... 

The Epons 828,1001,1002,1004, and 1007 fully cured with stoichiometeric amounts of DDS are examples of well-characterized networks. Therefore, mechanical measurements on them offer insight into the viscoelastic properties of rubber networks. The shear creep compliance J t) of these Epons were measured above their glass temperatures [11, 12, 14]. From the statistical theory of rubber elasticity [1-5, 29-33] the equilibrium modulus Ge is proportional to the product Tp, where p is the density at temperature T, and hence the equilibrium compliance is proportional to (Tpy Thus J t) is expected to be proportional to and J(t)Tp is the quantity which should be compared at different temperatures. Actually the reduced creep compliance... 

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