Geoscience Reference
In-Depth Information
256
Multiscale Geomechanics
non-linearity is equal to 1/3 . Hardin [HAR 78] generalized this theory to the case of
non-isotropic loadings, studying the shear wave propagation velocity. Having observed
a linear variation between these velocities and the void ratio ( e ), he proposed the
following form for the elasticity modulus:
E = A (B − e) 2
1+e
0 ) n
[9.9]
where A and B are two constants to be calibrated experimentally for each material. σ 0
is the normal stress in the direction in which the measurement was made. The modulus
is thus independent of the normal stresses in orthogonal directions. Tatsuoka [TAT 01]
completed this result and generalized it to anisotropic materials:
σ v
σ h
n
E v
E h = (E v ) 0
[9.10]
(E h ) 0
where the indices v and h denote the horizontal and vertical directions. (E v ) 0 and
(E h ) 0 are the vertical and horizontal Young's moduli obtained for σ h = σ v . He also
found that the Poisson's ratio increases with value of σ v
σ h . Experience has shown that
the coefficient n of relationship [9.8] is rather close to 0.5 for sands and can reach
higher values for clays [BIA 94]. For a non-linear elastic law to be thermodynamically
admissible, it must not introduce any dissipation. Therefore, it should derive from a
potential (specific free energy). The relationship between stress increment and strain
increment can be written as
tr σ
3
1
3K(p ,q)
1
2G(p ,q)
1
2G(p ,q)
e =
· I +
[9.11]
where the coefficients could depend on p and q . In order to have a reversible law:
∂ε rs
∂σ kl = ∂ε kl
∀(k,l)
∀(r, s)
[9.12]
∂σ rs
this condition can be formulated as
p
K 2 ∂K
3G 2 ∂G
q
rs δ kl − σ kl δ rs )=0 ∀(k,l),∀(r, s)
∂q
[9.13]
∂p
For an isotropic stress field ( r = s and k = l ), the above condition is satisfied.
But for any stress field, it is necessary that the first term on the left vanishes. The
expression, which is not thermodynamically admissible, introduces dissipation even in
the elastic domain. This error may be quantified [PIC 91] and we notice that its value
is very low in current applications. However, in cases where the model is subjected to
a large number of loading/unloading cycles, the accumulation of this error can lead to
very inaccurate results. From a practical point of view, only the dependency of Young's
modulus with the mean stress is regularly considered. This non-linearity results in a
 
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