The 8th Victor de Mello lecture: role played by viscosity on the undrained behaviour of normally consolidated clays

Martins, Ian Schumann Marques

doi:10.28927/SR.2023.006123

Ian Schumann Marques Martins ^##Corresponding author. E-mail address: ian@coc.ufrj.br

Universidade Federal do Rio de Janeiro, Departamento de Engenharia Civil, Rio de Janeiro, RJ, Brasil.

https://orcid.org/0000-0002-3642-7693

#Corresponding author. E-mail address: ian@coc.ufrj.br

Declaration of interest

The author has no conflicts of interest to declare. The author has observed and affirmed the contents of the paper and there is no financial interest to report.

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Figure 1
Example of strain rate effect on consolidated-undrained tests [after Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 2
Example of effective stress path (ESP) and total stress path (TSP) during stress relaxation [after Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 3
Pore pressure increase after closing drainage after isotropic consolidation (Thomasi, 2000Thomasi, L. (2000). On the existence of a viscous component on the normal effective stress [Master’s dissertation, Federal University of Rio de Janeiro]. Federal University of Rio de Janeiro’s repository (in Portuguese).).

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Figure 4
Sarapuí II Clay specimen after a

\bar{CIUCL}

test with lubricated ends: (a) Inside the triaxial chamber

(ε_{a} = 17 %)

; (b) Outside the triaxial chamber with the rubber membrane; (c) Without the rubber membrane and with the lateral filter paper; (d) Without the filter paper.

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Figure 5
Strain rate effect on the undrained strength of Sarapuí II Clay.

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Figure 6
Values of

S_{u}

as a function of

{\dot{ε}}_{a}

for Sarapuí II Clay measured in a

\bar{CIUCL}

test.

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Figure 7
Influence of strain rate

({\dot{ε}}_{a})

on state boundary surface [adapted from Leroueil et al. cited by Jamiolkowski et al. (1991)Jamiolkowski, M., Leroueil, S., & Lo Presti D.C.F. (1991). Theme lecture: design parameters from theory to practice. In Coastal Development Institute of Technology (Ed.), Procceedings of the International Conference on Geotechnical Engineering for Coastal Development (Vol. 91, pp. 877-917). Yokohama: Geo-Coast.].

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Figure 8
Mohr-Coulomb envelope for soils and rocks.

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Figure 9
Failure modes: (a) By separation in a tensile test (see the piece of chalk after failure by separation at the right side of ); (b) By shear during an unconfined compression test.

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Figura 10
(a) Curved strength envelope showing true cohesion

c^{'}

; (b) Strength envelope of a residual soil showing true cohesion

c^{'}

[adapted from Rodriguez (2005)Rodriguez, T.T. (2005). Pourpose of getechnical classification for brazilian colluvium [Doctoral thesis, Federal University of Rio de Janeiro]. Federal University of Rio de Janeiro’s repository (in Portuguese).]; (c) Sketch of a Brazilian test carried out on a submerged specimen of residual soil.

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Figure 11
Weathering/litification as a loss/gain in true cohesion of rocks/soils.

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Figure 12
Determination of

c_{e}

and

ϕ_{e}^{'}

from drained direct shear tests.

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Figure 13
Definition of the equivalent stress

σ_{e}^{'}

.

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Figure 14
Relationship between normalized shear stress on failure plane at failure

(τ_{f f} / σ_{e}^{'})

and effective stress on failure plane at failure

(σ_{f f}^{'} / σ_{e}^{'})

.

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Figure 15
Comparison between Hvorslev (1937)Hvorslev, M.J. (1937). Über die festigkeitseigenschaften gestörter bindiger böden. Ingeniörvidenskabelige Skrifter. strength envelope for an overconsolidated clay and Mohr-Coulomb strength envelope for the same clay in the normally consolidated condition.

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Figure 16
Action of the adsorbed water layer during the relative displacement between two neighbour particles. (a) Clay particles are separated by a thin water layer of high viscosity or (b) Clay particles are in direct touch. (c) Sliding resistance between clay particles is made up of frictionplus viscosity. After relative displacement, clay particles can be separated by a water viscous layer again. [adapted from Terzaghi & Frölich (1936), pTerzaghi, K., & Frölich, O.K. (1936). Théorie du tassement des couches argileuses. Dunod..19].

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Figure 17
Effect of speed of shear on the compressive strength of clay (Taylor, 1948Taylor, D.W. (1948). Fundamentals of soil mechanics. John Wiley & Sons.).

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Figure 18
Newton’s law of viscosity.

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Figure 19
Illustration of adsorbed water and types of contact between clay particles (Terzaghi, 1941Terzaghi, K. (1941). Undisturbed clay samples and undisturbed clays. Journal of the Boston Society of Civil Engineers, 28(3), 45-65.).

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Figure 20
Forces acting on a plane P – P passing through a plastic soil mass.

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Figure 21
Hypothetical variation of the coefficient of viscosity

μ

of the adsorbed water along the plane P – P within the contact zones between clay particles.

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Figure 22
(a) State of stress in a

\bar{CIUCL}

test during the undrained shear stage; (b) Corresponding state of strain.

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Figure 23
Mohr’s circle of strain during the undrained shear stage of a

\bar{CIUCL}

test.

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Figure 24
Normal linear strains and shear strains of elements ABCD and EFGH.

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Figure 25
Definition of distortion

= γ_{45 °} = 2 ε_{s 45 °}

.

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Figure 26
The viscosity ellipse or Taylor’s ellipse.

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Figure 27
The friction ellipse or Coulomb’s ellipse.

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Figure 28
Failure criterion for a normally consolidated clay.

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Figure 29
“Viscosity jump”- instantaneous mobilization of viscous resistance in

t^{'} \times ε_{t}

and

Δ u \times ε_{t}

plots (Martins, 1992Martins, I.S.M. (1992). Fundamentals for a model of saturated clay behaviour [Doctoral thesis, Federal University of Rio de Janeiro]. Federal University of Rio de Janeiro’s repository (in Portuguese).).

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Figure 30
“Viscosity jump”- instantaneous mobilization of viscous resistance in

s^{'} \times t^{'}

and

s \times t

plots (Martins, 1992Martins, I.S.M. (1992). Fundamentals for a model of saturated clay behaviour [Doctoral thesis, Federal University of Rio de Janeiro]. Federal University of Rio de Janeiro’s repository (in Portuguese).).

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Figure 31
The effect of duration of test on undrained strength [adapted from Bishop & Henkel (1962)Bishop, A.W., & Henkel, D.J. (1962). The measurement of soil properties in the triaxial test. London: Edward Arnold Ltd..]: (a) Mohr circles at failure for a consolidated-undrained test on a normally consolidated clay; (b) Variation in measured strength with time to failure.

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Figure 32
Curves

t^{'} \times ε_{t}

and

Δ u \times ε_{t}

for a set of ideal

\bar{CIUCL}

tests on a normally consolidated clay.

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Figure 33
Paths on the

s^{'} \times t^{'}

and

s^{'} \times e

planes followed by ideal

\bar{CIUCL}

tests carried out on normally consolidated clay specimens under a given shear strain rate

{\dot{ε}}_{t} = constant

.

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Figure 34
Effective stress paths in

\bar{CIUCL}

tests and their respective “viscosity jumps” [adapted from Fonseca (2000)Fonseca, A.P. (2000). Compressibility and shear strength of a gully soil from Ouro Preto-MG [Master’s dissertation, Federal University of Rio de Janeiro]. Federal University of Rio de Janeiro’s repository (in Portuguese).].

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Figure 35
Effective stress paths in

\bar{CIUCL}

tests and their respective “viscosity jumps” [adapted from Lira (1988)Lira, E.N.S. (1988). Automatic data acquisition system for triaxial tests [Master’s dissertation, Federal University of Rio de Janeiro]. Federal University of Rio de Janeiro’s repository (in Portuguese).].

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Figure 36
Normalized curves

(t^{'} / p_{e}^{'}) \times ε_{t}

and

(Δ u / p_{e}^{'}) \times ε_{t}

and normalized ESPs

(s^{'} / p_{e}^{'}) \times (t^{'} / p_{e}^{'})

for a fixed strain rate

{\dot{ε}}_{t}

.

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Figure 37
Definition of angles

α^{'}, β^{'}, ϕ^{'}

and

ϕ_{e}^{'}

.

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Figure 38
Homothetic (same eccentricity) friction ellipses at failure.

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Figure 39
Friction ellipses at failure with different eccentricities, both tangent to the same failure envelope of slope

\tan ϕ_{e}^{'}

, resulting from

\bar{CIUCL}

tests with different distortion rates

\dot{γ}

.

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Figure 40

\bar{CIUCL}

test with different strain rates and stress relaxation stages.

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Figure 41
Effective stress paths for

\bar{CIUCL}

tests starting out from the same

p_{e}^{'}

but corresponding to different strain rates

{\dot{ε}}_{t 1}

and

{\dot{ε}}_{t 2}

with

{\dot{ε}}_{t 1} < {\dot{ε}}_{t 2}

.

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Figure 42
Normalized effective stress paths

(s^{'} / p_{e}^{'})

×

(t^{'} / p_{e}^{'})

for

\bar{CIUCL}

tests corresponding to different strain rates

{\dot{ε}}_{t 1}

and

{\dot{ε}}_{t 2}

with

{\dot{ε}}_{t 1} < {\dot{ε}}_{t 2}

.

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Figure 43

\bar{CIUCL}

tests carried out with a fixed

p_{e}^{'}

and

{\dot{ε}}_{t} \neq 0

and

{\dot{ε}}_{t} = 0

(basic curves).

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Figure 44
Relationship between coordinates of a point Y on the ESP for a fixed

{\dot{ε}}_{t} \neq 0

and coordinates of a corresponding point X on the bESP (associated with

{\dot{ε}}_{t} = 0

).

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Figure 45
A point Y on the ESP of a

\bar{C I U C L}

test carried out with

{\dot{ε}}_{t} \neq 0

and its corresponding point X on the bESP (associated with

{\dot{ε}}_{t} = 0

).

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Figure 46
General normalization taking into account different values of

p_{e}^{'}

and

{\dot{ε}}_{t}

.

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Figure 47
Normalized effective stress paths

s^{'} / p_{e}^{'}

×

t^{'} / p_{e}^{'}

for different strain rates

({\dot{ε}}_{t})

.

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Figure 48
Uniqueness of

t a n ϕ_{e m o b}^{'} \times ε_{t}

relationship for each and every value of

p_{e}^{'}

and

{\dot{ε}}_{t}

.

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Figure 49
Virgin isotropic compression line – VICL – San Francisco Bay Mud (Lacerda, 1976Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.).

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Figure 50
Curves

t^{'} / p_{e}^{'} \times ε_{t}

and

Δ u / p_{e}^{'} \times ε_{t}

for

\bar{CIUCL}

tests on San Francisco Bay Mud carried out with

{\dot{ε}}_{t} = 0.09 % / min .

[data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 51
Normalized ESPs

(s^{'} / p_{e}^{'}) \times (t^{'} / p_{e}^{'})

for tests with a fixed strain rate

{\dot{ε}}_{t}

.

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Figure 52
Virgin isotropic compression line (VICL) and critical state line (CSL) for San Francisco Bay Mud corresponding to a strain rate

{\dot{ε}}_{t} ≅ 0.10 % / min .

.

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Figure 53
Critical state line for San Francisco Bay Mud corresponding to a strain rate

{\dot{ε}}_{t} ≅ 0.10 % / min .

[data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 54
Curves

t^{'} / p_{e}^{'} \times ε_{t}

and

Δ u / p_{e}^{'} \times ε_{t}

for undrained shear phases of stress relaxation tests on San Francisco Bay Mud carried out with different values of

{\dot{ε}}_{t}

.

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Figure 55

t^{'} / p_{e}^{'} \times ε_{t}

and

Δ u / p_{e}^{'} \times ε_{t}

for

\bar{CIUCL}

and stress relaxation tests on San Francisco Bay Mud for several different

{\dot{ε}}_{t}

values [data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 56
Normalized effective stress paths (ESPs) for undrained creep tests on normally consolidated specimens of San Francisco Bay Mud.

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Figure 57
Normalized ESPs for 3 different

{\dot{ε}}_{t}

values determined from undrained creep tests.

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Figure 58
Normalized effective stress paths

(s^{'} / p_{e}^{'})

×

(t^{'} / p_{e}^{'})

for all tests with different strain rates

({\dot{ε}}_{t})

.

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Figure 59

(t^{'} / p_{e}^{'})

×

ε_{t}

and

(Δ u / p_{e}^{'})

×

ε_{t}

curves for

\bar{CIUCL}

, stress relaxation and undrained creep tests for different strain rates

{\dot{ε}}_{t}

.

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Figure 62
Curves

t a n ϕ_{e m o b}^{'} \times ε_{t}

and

ϕ_{e m o b}^{'} \times ε_{t}

for normally consolidated San Francisco Bay Mud [data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 60
Normalized basic curve

t_{b}^{'} / p_{e}^{'} \times ε_{t}

for normally consolidated specimens of San Francisco Bay Mud [data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 61
bESPn – normalized basic effective stress path

({\dot{ε}}_{t} = 0)

for normally consolidated San Francisco Bay Mud [data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 63
Effective stress paths during undrained creep tests.

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Figure 64
Mohr’s circles of effective stress, friction elipses and mobilized states of friction during undrained creep ESP HNDP. (a) At point H – no friction mobilized (b) Mobilized state of friction at point N (c) Mobilized state of friction at point D (d) At point P – friction fully mobilized.

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Figure 65
Normalized ESPs corresponding to different

{\dot{ε}}_{t}

values and the normalized basic effective stress path (bESPn).

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Figure 66

ε_{t} \times {\dot{ε}}_{t}

plots for undrained creep tests on normally consolidated San Francisco bay Mud specimens [data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Figure 67
Effective stress paths for

\bar{CIUCL}

tests with stress relaxation stages.

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Figure 68
Normalized effective stress paths during stress relaxation stages for normally consolidated specimens of San Francisco Bay Mud [data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Table 1
Characterization tests results of San Francisco Bay Mud samples (Lacerda, 1976Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.).

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Table 2
Lacerda’s tests (Lacerda, 1976Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.) analyzed in this article.

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Table 3
Summary of undrained creep tests of (Lacerda, 1976Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.).

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Table 4
Validation checking of the generalized complementary principle 1 for San Francisco Bay Mud [data from Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley.].

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Table 5
Undrained creep tests on normally consolidated San Francisco Bay Mud carried out by Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley., analyzed in this article.

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Table 6
Summary of data from stress relaxation tests carried out by Lacerda (1976)Lacerda, W.A. (1976). Stress-relaxation and creep effects on soil deformation [PhD thesis dissertation]. University of California at Berkeley. on normally consolidated specimens of San Francisco Bay Mud.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(78)

(79)

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

(90)

(91)

(92)

(93)

(94)

(95)

(96)

(97)

(98)

(99)

(100)

(101)

(102)

(56)

(103)

Natural water content w (%)	Liquid limit w_L (%)	Plastic limit w_P ₍%)	Plasticity index I_P (%)	Specific gravity G	Clay fraction % < 2μm	Activity
88 to 93	88 to 90	35 to 44	45 to 55	2.75	60	0.83

Test	Description	Specific volume after isotropic consolidation $v$	Isotropic compression stress $p_{e}^{'}$ (kPa)	Shear strain rate ${\dot{ε}}_{t}$ $(% / min .)$
FP-13	$\bar{CIUCL}$	3.00	118	0.09
FP-23	$\bar{CIUCL}$	2.81	158	0.09
FP-32	$\bar{CIUCL}$	2.98	98.0	0.09
FP-42	$\bar{CIUCL}$	2.92	137	0.09
SR-I-5	stress relax.	3.11	78.4	1.15
SR-I-8	stress relax.	not informed	78.4	5.5 × 10^-4
SR-I-9	stress relax.	not informed	314	0.10
CR-I-1	undrained creep	3.15	78.4	variable
CR-I-2	undrained creep	3.02	78.4	variable
CR-71-1	undrained creep	2.78	196	variable
CR-I-ST-2	undr. step creep	2.60	314	variable

Point	Test	$ε_{t} (%)$	${\dot{ε}}_{t} (% / min .)$	$t^{'} / p_{e}^{'}$	$s^{'} / p_{e}^{'}$	$Δ u / p_{e}^{'}$
a	CR–I–1	0.29	0.09	0.234	0.934	0.300
b	CR–I–1	0.45	1.0 × 10^-2	0.234	0.874	0.360
c	CR–I–1	0.65	1.0 × 10^-3	0.234	0.824	0.410
d	CR–I–1	0.75	6.5 × 10^-4	0.234	0.804	0.430
e	CR–I–1	0.82	3.4 × 10^-4	0.234	0.784	0.450
f	CR–71–1	0.60	0.075	0.281	0.871	0.410
g	CR–71–1	0.97	~1.0 × 10^-2	0.281	0.811	0.470
h	CR–I–ST–2	0.06	1.0 × 10^-2	0.102	1.02	0.080
i	CR–I–ST–2	0.08	1.0 × 10^-3	0.102	1.00	0.100
j	CR–I–ST–2	0.12	5.5 × 10^-4	0.102	0.972	0.130
k	CR–I–ST–2	0.14	8.4 × 10^-2	0.172	1.01	0.160
l	CR–I–ST–2	0.19	1.0 × 10^-2	0.172	0.952	0.220
m	CR–I–ST–2	0.27	1.4 × 10^-3	0.172	0.892	0.270
n	CR–I–ST–2	0.30	3.5 × 10^-1	0.242	0.962	0.280
p	CR–I–ST–2	0.40	5.2 × 10^-2	0.242	0.892	0.350
q	CR–I–ST–2	0.50	1.3 × 10^-2	0.242	0.862	0.380
r	CR–I–ST–2	0.55	9.4 × 10^-3	0.242	0.852	0.390
s	CR–I–ST–2	1.00	9.5 × 10^-2	0.312	0.832	0.48
t	CR–I–ST–2	1.70	1.7 × 10^-2	0.312	0.752	0.560
u	CR–I–ST–2	2.10	1.2 × 10^-1	0.347	0.747	0.600
v	CR–I–ST–2	2.21	9.5 × 10^-2	0.347	0.742	0.610
x	CR–I–ST–2	3.70	2.7 × 10^-1	0.365	0.695	0.670
y	CR–I–ST–2	3.70	~1.0 × 10^-1	0.365	0.695	0.670
z	CR–I–ST–2	6.83	~1.0 × 10^-1	0.365	0.695	0.670

Test	$ε_{t} (%)$	${\dot{ε}}_{t} (% / min .)$	$t^{'} / p_{e}^{'}$	$s^{'} / p_{e}^{'}$	$Δ u / p_{e}^{'}$	$C_{η}$	$t a n ϕ_{e m o b}^{'}$
FP–23	0.13	~1.0 × 10^-1	0.123	0.953	0.17	0.09	0.035
FP–32	0.13	~1.0 × 10^-1	0.18	0.95	0.23	0.09	0.096
FP–42	0.13	~1.0 × 10^-1	0.17	0.960	0.21	0.09	0.075
CR–I–ST–2	0.14	8.4 × 10^-2	0.172	1.01	0.16	0.09	0.082
FP–32	0.29	~1.0 × 10^-1	0.241	0.920	0.32	0.09	0.170
FP–42	0.29	~1.0 × 10^-1	0.231	0.931	0.30	0.09	0.156
CR–I–1	0.29	0.09	0.234	0.934	0.300	0.09	0.159
FP–23	0.35	~1.0 × 10^-1	0.217	0.917	0.30	0.09	0.143
FP–32	0.36	~1.0 × 10^-1	0.264	0.914	0.35	0.09	0.199
FP–42	0.35	~1.0 × 10^-1	0.25	0.91	0.34	0.09	0.183
SR–I–8	0.33	5.5 × 10^-4	0.193	0.866	0.33	0.035	0.187
FP–23	0.45	~1.0 × 10^-1	0.25	0.89	0.36	0.09	0.187
FP–32	0.46	~1.0 × 10^-1	0.28	0.90	0.38	0.09	0.222
CR–I–1	0.45	1.0 × 10^-2	0.234	0.874	0.36	0.06	0.207
CR–I–ST–2	0.55	1.0 × 10^-2	0.242	0.852	0.39	0.06	0.223
FP–23	0.58	~1.0 × 10^-1	0.265	0.885	0.38	0.09	0.207
CR–71–1	0.60	0.075	0.281	0.871	0.41	~0.09	0.232
SR–I–8	0.68	5.5 × 10-4	0.244	0.794	0.45	0.04	0.270
FP–13	0.69	~1.0 × 10^-1	0.29	0.88	0.41	0.09	0.241
FP–23	0.70	~1.0 × 10^-1	0.28	0.87	0.41	0.09	0.231
FP–32	0.70	~1.0 × 10^-1	0.315	0.875	0.44	0.09	0.276
FP–42	0.71	~1.0 × 10^-1	0.304	0.854	0.45	0.09	0.268
CR–I–1	0.75	6.5 × 10^-4	0.234	0.804	0.43	0.04	0.252
SR–I–8	0.75	5.5 × 10^-4	0.247	0.784	0.46	0.035	0.285
FP–23	0.75	~1.0 × 10^-1	0.285	0.86	0.43	0.09	0.240
FP–42	0.75	~1.0 × 10^-1	0.304	0.851	0.46	0.09	0.269
FP–13	1.0	~1.0 × 10^-1	0.319	0.833	0.48	0.09	0.298
FP–23	1.0	~1.0 × 10^-1	0.325	0.833	0.47	0.09	0.306
FP–32	1.0	~1.0 × 10^-1	0.334	0.846	0.49	0.09	0.314
FP–42	1.0	~1.0 × 10^-1	0.322	0.825	0.50	0.09	0.305
SR–I–9	1.0	1.0 × 10^-1	0.317	0.863	0.45	0.09	0.283
CR–71–1	0.97	~1.0 × 10^-2	0.281	0.811	0.47	0.06	0.290
CR–I–ST–2	1.00	~1.0 × 10^-1	0.312	0.832	0.48	0.09	0.288
FP–13	1.15	~1.0 × 10^-1	0.328	0.82	0.51	0.09	0.317
FP–23	1.16	~1.0 × 10^-1	0.312	0.812	0.50	0.09	0.296
FP–32	1.18	~1.0 × 10^-1	0.345	0.835	0.51	0.09	0.335
FP–42	1.20	~1.0 × 10^-1	0.33	0.800	0.53	0.09	0.329
FP–13	1.38	~1.0 × 10^-1	0.34	0.80	0.54	0.09	0.345
FP–23	1.40	~1.0 × 10^-1	0.324	0.794	0.53	0.09	0.323
FP–32	1.42	~1.0 × 10^-1	0.356	0.816	0.54	0.09	0.362
FP–42	1.44	~1.0 × 10^-1	0.339	0.784	0.56	0.09	0.352
FP–13	1.62	~1.0 × 10^-1	0.347	0.79	0.56	0.09	0.362
FP–23	1.62	~1.0 × 10^-1	0.332	0.782	0.55	0.09	0.342
FP–32	1.64	~1.0 × 10^-1	0.364	0.809	0.56	0.09	0.379
FP–13	1.83	~1.0 × 10^-1	0.354	0.774	0.58	0.09	0.384
FP–23	1.86	~1.0 × 10^-1	0.336	0.766	0.57	0.09	0.357
FP–42	1.80	~1.0 × 10^-1	0.345	0.755	0.59	0.09	0.380
FP–42	2.05	~1.0 × 10^-1	0.345	0.735	0.61	0.09	0.393
FP–13	2.08	~1.0 × 10^-1	0.357	0.757	0.60	0.09	0.400
FP–23	2.12	~1.0 × 10^-1	0.34	0.74	0.60	0.09	0.380
FP–32	2.10	~1.0 × 10^-1	0.372	0.772	0.60	0.09	0.417
CR–I–ST–2	2.21	~1.0 × 10^-1	0.347	0.742	0.61	0.09	0.392
FP–32	2.24	~1.0 × 10^-1	0.375	0.770	0.61	0.09	0.424
FP–13	2.30	~1.0 × 10^-1	0.36	0.74	0.62	0.09	0.418
FP–23	2.35	~1.0 × 10^-1	0.345	0.735	0.61	0.09	0.393
FP–32	2.35	~1.0 × 10^-1	0.375	0.765	0.61	0.09	0.427
FP–42	2.42	~1.0 × 10^-1	0.349	0.719	0.63	0.09	0.412
FP–13	2.51	~1.0 × 10^-1	0.36	0.73	0.63	0.09	0.425
FP–23	2.55	~1.0 × 10^-1	0.347	0.717	0.63	0.09	0.410
FP–32	2.84	~1.0 × 10^-1	0.375	0.750	0.63	0.09	0.439
FP–23	2.90	~1.0 × 10^-1	0.35	0.710	0.64	0.09	0.421
FP–13	2.98	~1.0 × 10^-1	0.365	0.715	0.65	0.09	0.447

Test	Stage	$p_{e}^{'}$ (kPa)	$t_{u c}^{'}$ (kPa)	$t_{u c}^{'} / p_{e}^{'}$
CR–I–1	unique	78.4	18.4	0.234
CR–I–2	unique	78.4	24.5	0.313
SR–I–3	unique	78.4	15.3	0.195
CR–I–5	unique	78.4	27.6	0.351

Test	Stress relaxation stage	$s^{'} / p_{e}^{'}$ initial value	$t^{'} / p_{e}^{'}$ initial value	$s^{'} / p_{e}^{'}$ final value	$t^{'} / p_{e}^{'}$ final value	Stage duration (minutes)
SR–I–5 $p_{e}^{'} = 78.4$ (kPa)	1	1.00	0.278	0.850	0.075	3070
	2	0.888	0.463	0.700	0.213	1320
	3	0.725	0.388	0.581	0.244	2700
	4	0.568	0.356	0.534	0.234	8370
SR–I–8 $p_{e}^{'} = 78.4$ (kPa)	1	0.781	0.244	0.751	0.151	4530
	2	0.830	0.400	0.741	0.201	1660
	3	0.819	0.444	0.612	0.237	365
	4	0.745	0.425	0.528	0.208	1000
SR–I–9 $p_{e}^{'} = 314$ (kPa)	1	0.979	0.323	0.831	0.128	1250
	2	0.844	0.313	0.775	0.173	1250
	3	0.758	0.303	0.712	0.196	1280
	4	0.764	0.347	0.671	0.249	100

ε_{V} = - \frac{δ h}{h_{0}} - \frac{2 δ r}{r_{0}} - \frac{2 δ r δ h}{r_{0} h_{0}} - \frac{δ r^{2}}{r_{0}^{2}} - \frac{δ r^{2} δ h}{r_{0}^{2} h_{0}} = ε_{a} + 2 ε_{r} + ε_{r} [2 ε_{a} + ε_{r} (1 + ε_{a})]

ε_{l 45 °} = \frac{\bar{E E'} c o s 45 ° - \bar{F F'} c o s 45 °}{\bar{E F}} = \frac{\frac{δ_{1}}{2} \frac{\sqrt{2}}{2} - \frac{δ_{3}}{2} \frac{\sqrt{2}}{2}}{\frac{l \sqrt{2}}{2}} = \frac{\frac{δ_{1}}{2} - \frac{δ_{3}}{2}}{l} = \frac{δ_{1}}{4 l} = \frac{ε_{1} + ε_{3}}{2}

ε_{l 45 °} = \frac{ε_{1} + ε_{3}}{2} + \frac{ε_{1} - ε_{3}}{2} c o s 90 ° = \frac{ε_{1} + ε_{3}}{2} = \frac{\frac{δ_{1}}{l} - \frac{δ_{3}}{l}}{2} = \frac{1}{2} (\frac{δ_{1}}{l} - \frac{δ_{1}}{2 l}) = \frac{δ_{1}}{4 l} = \frac{ε_{1}}{4}

ε_{s 45 °} = \frac{\bar{E E'} s i n 45 ° - (- \bar{F F^{'}} s i n 45 °)}{\bar{E F}} = \frac{\frac{δ_{1}}{2} \frac{\sqrt{2}}{2} - (- \frac{δ_{3}}{2} \frac{\sqrt{2}}{2})}{\frac{l \sqrt{2}}{2}} = \frac{\frac{δ_{1}}{2} + \frac{δ_{3}}{2}}{l} = \frac{3}{4} ε_{1}

\begin{array}{l} \frac{\partial (\frac{τ_{ϕ α}}{σ_{α}^{'}})}{\partial α} = \frac{\partial}{\partial α} \{\frac{[\frac{(σ_{1}^{'} - σ_{3}^{'})}{2} - V] s i n 2 α}{\frac{(σ_{1}^{'} + σ_{3}^{'})}{2} + \frac{(σ_{1}^{'} - σ_{3}^{'})}{2} c o s 2 α}\} = \\ \frac{\partial}{\partial α} [\frac{(t^{'} - V) s i n 2 α}{s^{'} + t^{'} c o s 2 α}] = 0 \end{array}

\begin{array}{l} \frac{\partial (\frac{τ_{ϕ α}}{σ_{α}^{'}})}{\partial α} = \frac{2 (t^{'} - V) \cos 2 α (s^{'} + t^{'} \cos 2 α)}{{(s^{'} + t^{'} \cos 2 α)}^{2}} + \\ \frac{2 t^{'} (t^{'} - V) \sin^{2} 2 α}{{(s^{'} + t^{'} \cos 2 α)}^{2}} = 0 \end{array}

\begin{array}{l} t a n ϕ_{e m o b}^{'} (Y) = \frac{t^{'} (Y) - V}{\sqrt{{(s^{'} (Y))}^{2} - {(t^{'} (Y))}^{2}}} = \\ t a n ϕ_{e m o b}^{'} (X) = \frac{t_{b}^{'} (X)}{\sqrt{{(s_{b}^{'} (X))}^{2} - {(t_{b}^{'} (X))}^{2}}} \end{array}

t a n ϕ_{e m o b}^{'} (G) = \frac{\frac{t^{'} (G)}{p_{e 2}^{'}} - \frac{V_{12}}{p_{e 2}^{'}}}{\sqrt{{(\frac{s^{'} (G)}{p_{e 2}^{'}})}^{2} - {(\frac{t^{'} (G)}{p_{e 2}^{'}})}^{2}}}

t a n ϕ_{e m o b}^{'} (G) = \frac{\frac{t^{'} (C)}{p_{e 1}^{'}} - \frac{V_{11}}{p_{e 1}^{'}}}{\sqrt{{(\frac{s^{'} (C)}{p_{e 1}^{'}})}^{2} - {(\frac{t^{'} (C)}{p_{e 1}^{'}})}^{2}}}

\begin{array}{l} t a n ϕ_{e m o b}^{'} (X) = t a n ϕ_{e m o b}^{'} (C) = t a n ϕ_{e m o b}^{'} (G) = \\ t a n ϕ_{e m o b}^{'} (J) = \frac{t_{b}^{'} (X)}{\sqrt{s_{b}^{'}^{2} (X) - t_{b}^{'}^{2} (X)}} \end{array}

\frac{t_{b}^{'} (X)}{p_{e 1}^{'}} = t a n^{2} ϕ_{e m o b}^{'} (J) [1 + c s c ϕ_{e m o b}^{'} (J)] [\frac{s^{'} (J)}{p_{e 2}^{'}} - \frac{t^{'} (J)}{p_{e 2}^{'}}]

\begin{array}{l} t a n ϕ_{e}^{'} = \frac{(t_{f}^{'} - V)}{\sqrt{s_{f}^{'}^{2} - t_{f}^{'}^{2}}} = \frac{(t_{f}^{'} / p_{e}^{'} - V / p_{e}^{'})}{\sqrt{{(s_{f}^{'} / p_{e}^{'})}^{2} - {(t_{f}^{'} / p_{e}^{'})}^{2}}} = \\ \frac{(t_{f}^{'} / p_{e}^{'} - C_{η} ({\dot{ε}}_{t}))}{\sqrt{{(s_{f}^{'} / p_{e}^{'})}^{2} - {(t_{f}^{'} / p_{e}^{'})}^{2}}} \end{array}

\begin{array}{l} t a n ϕ_{e}^{'} = \frac{(t_{f}^{'} / p_{e}^{'} - C_{η} ({\dot{ε}}_{t}))}{\sqrt{{(s_{f}^{'} / p_{e}^{'})}^{2} - {(t_{f}^{'} / p_{e}^{'})}^{2}}} = \\ \frac{0.36 - 0.09}{\sqrt{{0.70}^{2} - {0.36}^{2}}} = 0.45 \to ϕ_{e}^{'} ≅ 24 ° \end{array}

t_{b}^{'} / p_{e}^{'} = \frac{{(t^{'} / p_{e}^{'} - V / p_{e}^{'})}^{2}}{(s^{'} / p_{e}^{'} + t^{'} / p_{e}^{'})} [1 + \frac{\sqrt{{(s^{'} / p_{e}^{'})}^{2} + {(V / p_{e}^{'})}^{2} - 2 V t^{'} / p_{e}^{'}^{2}}}{(t^{'} / p_{e}^{'} - V / p_{e}^{'})}]

t_{b}^{'} / p_{e}^{'} = \frac{{(t^{'} / p_{e}^{'} - C_{η} ({\dot{ε}}_{t}))}^{2}}{(s^{'} / p_{e}^{'} + t^{'} / p_{e}^{'})} \{1 + \frac{\sqrt{{(s^{'} / p_{e}^{'})}^{2} + {[C_{η} ({\dot{ε}}_{t})]}^{2} - 2 (t^{'} / p_{e}^{'}) C_{η} ({\dot{ε}}_{t})}}{(t^{'} / p_{e}^{'} - C_{η} ({\dot{ε}}_{t}))}\}

\frac{s_{b}^{'}}{p_{e}^{'}} = (\frac{s^{'}}{p_{e}^{'}} - \frac{t^{'}}{p_{e}^{'}}) + \frac{{(\frac{t^{'}}{p_{e}^{'}} - \frac{V}{p_{e}^{'}})}^{2}}{(\frac{s^{'}}{p_{e}^{'}} + \frac{t^{'}}{p_{e}^{'}})} [1 + \frac{\sqrt{{(\frac{s^{'}}{p_{e}^{'}})}^{2} + {(\frac{V}{p_{e}^{'}})}^{2} - \frac{2 V t^{'}}{p_{e}^{'}^{2}}}}{(\frac{t^{'}}{p_{e}^{'}} - \frac{V}{p_{e}^{'}})}]

\frac{s_{b}^{'}}{p_{e}^{'}} = (\frac{s^{'}}{p_{e}^{'}} - \frac{t^{'}}{p_{e}^{'}}) + \frac{{(\frac{t^{'}}{p_{e}^{'}} - C_{η} ({\dot{ε}}_{t}))}^{2}}{(\frac{s^{'}}{p_{e}^{'}} + \frac{t^{'}}{p_{e}^{'}})} [1 + \frac{\sqrt{\frac{s^{'}^{2}}{p_{e}^{'}^{2}} + {[C_{η} ({\dot{ε}}_{t})]}^{2} - 2 \frac{t^{'}}{p_{e}^{'}} C_{η} ({\dot{ε}}_{t})}}{(\frac{t^{'}}{p_{e}^{'}} - C_{η} ({\dot{ε}}_{t}))}]

t a n ϕ_{e m o b}^{'} = \frac{(\frac{t^{'}}{p_{e}^{'}} - \frac{V}{p_{e}^{'}})}{\sqrt{{(\frac{s^{'}}{p_{e}^{'}})}^{2} - {(\frac{t^{'}}{p_{e}^{'}})}^{2}}} = \frac{\frac{t^{'}}{p_{e}^{'}} - C_{η} ({\dot{ε}}_{t})}{\sqrt{{(\frac{s^{'}}{p_{e}^{'}})}^{2} - {(\frac{t^{'}}{p_{e}^{'}})}^{2}}}

[1] #Corresponding author. E-mail address: ian@coc.ufrj.br

Brasil

Brasil

The 8th Victor de Mello lecture: role played by viscosity on the undrained behaviour of normally consolidated clays

Abstract