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− τ
c
and γ
c
: shear stress and strain, respectively, at the last change of loading
direction;
−
G
MAX
: initial tangent shear modulus;
−
n
= 1 at first loading and n = 2 afterwards;
−
r
, α: material constants;
− τ
y
: maximum stress corresponding to Coulomb's criterion;
− τ
y
=
G
MAX
.γ
y
.
Figure 10.11.
Shear modulus and damping ratio versus shear strain
During one cycle, the elastic energy
W = 1/2.
τ
C
.
γ
C
is restored, whereas the
plastic energy is dissipated during the material shearing. The damping ratio
D
measures the dissipated fraction of the energy:
W
4π
Δ
D =
By integrating Ramberg-Osgood's expression, the coefficient of the damping
ratio
D
permits us to measure the parameter
r
and the ratio
G/G
MAX
:
2r-1
G
D =
1-
π
r + 1
G
max
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