Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 136

where G is a complex modulus.

The dissipated energy during a loading cycle is equal to:

2 2

[4.6]

' W = ʌ C ȦȖ ʌ Im (G ) Ȗ

m m

where Im (G * ) refers to the imaginary part of G * .

As C is constant, we note that the dissipated energy depends on the stress
frequency f = Z/2S.

If we refer to Figure 4.7, without going into too much detail, it is possible to
define a damping term for a material that is frequency-independent, a standard
expression of the energy 'W dissipated during a cycle. The standardization is made
owing to the elastic energy W accumulated during a cycle:

1

W = G J

2

[4.7]

2

Therefore, we can define either a damping ratio E or a loss coefficient K,
independent of the stress frequency:

1W '

KE = 2 = [4.8]

2 S W