Moduli and damping factor

The storage component is calculated thus (E has been used, but replace with G for shear measurements) 

Sometimes the tan 8 is distinguished as tan 8e or tan Sq. Generally they have a similar magnitude, so comparison is usually straightforward. An approximate relationship between these is given below [7]  

Normally tan 8 will be considerably smaller than tan 8 and the approximation that 

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Seed, H. and Idriss, I. (1970) Soil moduli and damping factors for dynamic response analysis , Report EERC 70, College of Engineering, University of California, Berkeley. 

Seed HB, Wong RT, Idriss IM, Toldmatsu K (1986) Moduli and damping factors for dynamic analyses of cohesionless soils. J Geotech Eng ASCE 112(11) 1016-1032... 

It was of general interest to investigate the effectiveness of the damping materials in terms of any synergisms that may arise because of special mixing modes. The bandwidth "constant K indeed is not a constant and depends on the chemistry and other factors of the system, see above. However K is only useful for loss moduli-temperature behavior. No similar theory for tan 6 temperature exists, and... 

Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental fimction. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time rai e. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a cert2un characteristic time. It is interesting to note that this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. 

Most real systems, as distinct from idealized assemblies of point masses linked by two-body forces, show deviations from the simple behaviour discussed in the previous sections. One of these types of more complex behaviour is associated with the internal friction of the material which leads to the damping of free vibrations, even in the most perfectly isolated systems. We have seen that the most convenient way of determining elastic moduli is by examining the vibration frequencies of carefully shaped samples and, in the same way, internal friction or mechanical relaxation can be studied by examining the logarithmic decrement of these same vibrations when the external excitation is removed. The logarithmic decrement 8 is so defined that an oscillation of angular frequency (o is damped by a factor exp (- It is thus n times the loss... 

E and G are the in-phase components and E" and G" are the out-of-phase components of their respective complex moduli. A parameter termed the damping coefficient or loss factor is defined as... 

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