Fractional linear solid model

0) =1 Hz.. 100 Hz frequency range y =0.0001,0.0002... 10 strain range Strain and frequency dependency of the intrinsic time scale fc =51000 constant x =l sec characteristic time 

Intrinsic time scale vs. strain at fixed frequency 

The fractional exponent a gives a nonlinear variation of the intrinsic time scale with frequency (at constant strain). 

2 Modeling the Dynamic Strain Softening Effect Elastic modulus  

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Figure 4.12 Cell survival fractions SF(D) as a function of absorbed radiation dose D in Gy (top panel). The bottom panel is the so-called reactivity R(D) given by product of the reciprocal dose D-1 and the negative natural logarithm of SF(D), as the ordinate versus D as the abscissa. Experiment (symbols) the mean clonogenic surviving fractions SF(D) (top panel) and R(D) = — (1/D) ln(SF) (bottom panel) for the Chinese hamster cells grown in culture and irradiated by 50 kV X-ray [73]. Theories solid curve - PLQ (Pads Linear Quadratic) model and dotted curve - LQ model (the straight line a + /SD on the bottom panel).

Modeling fat crystal networks started with Van den TempeTs work in 1961, when he proposed the linear chain model (Van den Temple 1961). It was postulated that solid fat particles form linear chains that are held together by two types of bonds— irreversible primary bonds (stronger bonds) and reversible secondary bonds (weaker bonds). In this early model, the shear modulus (G) is predicted to be directly proportional to the volume fraction O of solids and to particle diameter D according to the equation ... 

Adsorption of Mo to Mn oxyhydroxides produces an isotopic fractionation that appears to follow that of a closed-system equilibrium model as a function of the fraction of Mo adsorbed (Fig. 8). Barling and Anbar (2004) observed that the 5 TVlo values for aqueous Mo (largely the [MoOJ species) were linearly correlated with the fraction (/) of Mo adsorbed (Fig. 8), following the form of Equation (14) above. The 5 Mo-f relations are best explained by a MOaq-Mn oxyhydroxide fractionation of +1.8%o for Mo/ Mo, and this was confirmed through isotopic analysis of three solution-solid pairs (Fig. 8). The data clearly do not lie... 

The findings about the isotope effects thus indicate that in liquid LiCl and solid Li2S04 the groups of lithium ions moving together comprise about two members. The model just described involves that Uv, and u depend linearly on the molar fraction say of Li, a fact that has not yet been checked experimentally. 

A paper by Prandtl [18] on the kinetic theory of solid bodies, which was published in 1928, one year prior to Tomlinson s paper [17], never achieved the recognition in the tribology community that it deserves. PrandtI s model is similar to the Tomlinson model and likewise focused on elastic hysteresis effects within the bulk. Nevertheless, Prandtl did emphasize the relevance of his work to dry friction between solid bodies. In particular, he formulated the condition that can be considered the Holy Grail of dry, elastic friction If the elastic coupling of the mass points is chosen such that at every instance of time a fraction of the mass points possesses several stable equilibrium positions, then the system shows hysteresis. In the context of friction, hysteresis translates to finite static friction or to a finite kinetic friction that does not vanish in the limit of small sliding velocities. Note that the dissipative term that is introduced ad hoc in Eq. (19) does vanish linearly with small velocities. 

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