Geoscience Reference
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The ratio C / C c = c/b will usually be in the range 0.025 - 0.10, the higher values
pertaining to more compressible soils, such as peat. The total strain by the loading
step
' is supposed to be a superposition of natural strains, and using (5.7), (5.8)
and (5.10), it leads to a compact formula in terms of specific stress increments. 26
D -
A = (
B -
A )
(
C -
B )
(
D -
C )
=
a ln(
' B /
' A )
b ln (
' C /
' B )
c ln(t D /t C )
' A ) a (
' B ) b ( t D /t C ) c ]
= ln[(
' B /
' C /
' A ) -a (
' B ) -b ( t D /t C ) -c
1
l = v D /v A = (
' B /
' C /
(6.2)
During strain development in branch BC, the strain rate is considered a constant.
During creep (branch CD) strain continues with attenuating strain rate. The creep
time-frame of t C = t i and t D = t i + t , where t is the time from C to D, is not related to
the moment of loading, see Fig 6.1b. In fact t i is unknown. The creep time is
referred to as intrinsic time , according to which creep proceeds logarithmically at
constant stress. In any creep process, after some time, the difference of intrinsic
time and loading time t becomes insignificant, i.e. t D = t . Historically, the creep
effect is determined empirically, with t C = 1 day and t D = 10 4 days
27 years, an
appropriate period when considering the maintenance period (life-cycle) of a
geotechnical structure (road, dam, foundation).
An alternative way to consider creep is by associating it to the strain rate itself.
The strain rate is expressed by
,t = c/t which gives t = c/
,t . This leads to ln( t D /t C )
= ln(
C,t /
D,t ) . So, expressing the right-hand graph of Fig 6.1 not in terms of ln( t ),
but by ln(
,t ) avoids the problem of intrinsic time and time frame difference.
Although formula (6.2) is expressed in terms of linear strain l , the formula is
fully based on the concept of natural strain. It therefore properly reflects convective
and large-strain aspects. The formulation in terms of linear strain makes it
convenient for practical use, as it can be directly related to measurements in terms
of linear strain.
Formula (6.2) corresponds to a non-linear Maxwell material. A Maxwell
material is a visco-elastic material having both elastic and viscous properties. It is
named for James Clerk Maxwell who proposed the model in 1867. The Maxwell
model can be represented by a purely viscous damper and a purely elastic spring
connected in series. In this configuration, under an applied axial stress, the total
stress and the total strain can be defined as follows: = S = D and = S
D ,
where the subscript D indicates the stress/strain in the damper and the subscript S
indicates the stress/strain in the spring. Taking the derivative of total strain with
respect to time, we obtain:
26 The 2 nd line of (6.2) states = ln( '/ ' A ) a for '< ' B and = ln[( ' B / ' A ) a ( '/ ' B ) b ] for
' B < '< ' C .
The 3 rd of (6.2) states 1
' A ) a for
' A ) a (
' B ) b for
l = (
'/
'<
' B and 1
l = (
' B /
'/
' B < '< ' C
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