Evaluating Seismic Stability of Embankment Dams - Multiscale Geomechanics: From Soil to Engineering Projects - page 364

10.6.4. Shearstrainlaw

During an earthquake, the soil is shaken by cyclic loads that induce strains of
various amplitudes, from values below 10 -5 corresponding to reversible strains to
significant irreversible strains. The cyclic stress−strain relationship is non-linear; the
shear modulus decreases and the soil dissipates a fraction of the seismic energy. In
order to reproduce this non-reversible behavior, a non-linear constitutive law,
Ramberg-Osgood's law, has been introduced [RAM 43].

In the elastic domain, the shear modulus G MAX and the bulk modulus K MAX are
made dependent on the effective mean stress, following Hertz' law:

n

⎛⎞
⎜ ⎝⎠

p '

G = G

MAX 1

p

1

n

⎛ ⎜ ⎝⎠

p

K = K

MAX 1

p

1

with:

− G 1 : shear modulus for p' = 1MPa;

− K 1 : bulk modulus for p' = 1MPa;
− n : non-linearity exponent (n = 0.5);

− p ': mean effective stress (MPa);

− p 1 : reference stress, p 1 = 1MPa.

Beyond the elastic domain, the hysteretic non-linear law represents the decrease
of the moduli and the increase of the hysteretic damping due to energy dissipation
during shearing and to grain rearrangement using Ramberg-Osgood's expression
(see Figure 10.11):

τ − τ 1

C

= G .

MAX

γ − γ ⎡

r −1

⎤

⎛ − ⎞

τ τ

C

⎜
⎝

⎟
⎠

⎢
⎣

C

⎥
⎦

1 + α

⎢ ⎥

n . τ

Y

with:
− τ: shear stress;