124

Figure 4.1: Typical monotonic stress-strain behavior assumed in Tresca model.

Figure 4.2: Tresca failure criterion in a 3-D stress space.

Figure 4.3: Isotropic model: evolution of the yield surface in 2-D (left) and 3-D (right)

principal stress space.

Loading

Unloading

Stress

Strain

125

Rankine

Drucker-Prager

Figure 4.4: Two ways of modeling crack in finite element analysis (after Pankaj, 1990).

Figure 4.5: Yield surfaces for CDP model in the deviatoric plane, corresponding to different

values of (modified from Abaqus 6.11, 2011).

Figure 4.6: Yield surface for CDP model in plane stress (after Abaqus 6.11, 2011).

(a) Discrete crack (b) Smeared crack

126

Figure 4.7: Definition of cracking strain ̃ used in tension data (after Abaqus 6.11, 2011).

Figure 4.8: Definition of compressive inelastic (or crushing) strain ̃ used in compression

data (after Abaqus 6.11, 2011).

127

Figure 4.9: Drained triaxial compression test for cement-treated Toyoura sand by Namikawa

(2006).

Figure 4.10: Loading and boundary conditions simulated in finite element models.

(a) Drained triaxial

compression test (b)Direct tension test

(Full

specimen

model)

(Quarter

specimen

model)

0

500

1000

1500

2000

-0.3 -0.1 0.1 0.3 0.5D

ev

iato

r str

ess (kP

a)

z (%)x (%)y (%)

128

-0.4

-0.3

-0.2

-0.1

0

-0.020-0.015-0.010-0.0050.000

Ten

sile

str

ess

(MP

a)

strain z (%)

Figure 4.11: Calibrated stress-strain curves for three constitutive models and laboratory

measurement for drained triaxial compression test of cement-treated Toyoura sand.

Figure 4.12: Direct tension test for cement-treated Toyoura sand by Koseki et al. (2005).

0

500

1000

1500

2000

-0.3 -0.1 0.1 0.3 0.5

Dev

iato

ric

stre

ss (

kP

a)

axial strain z (%)

LabFEM-TrescaFEM-IsotropicFEM-CDP

lateral strain y (%)

z

129

Figure 4.13: Classical and tension truncated for Tresca criteria (after Antão et al., 2007).

Figure 4.14: Projection of Rankine surface in deviatoric plane.

Figure 4.15: Post-failure stress-fracture energy curve simulated by CDP model (after Abaqus

6.11, 2011).

(a) Classical Tresca (b)Tension truncated Tresca

admissible stress state imitated truncation

1 3

- 1

2

- 3

- 2

1

2

3

cr

c

130

Figure 4.16: Calibrated tensile stress-strain curves for three constitutive models and

laboratory measurement for direct tension test of cement-treated Toyoura sand.

Figure 4.17: Three-point bending notched beam test for cement-treated Toyoura sand by

Namikawa (2006).

-0.5

-0.4

-0.3

-0.2

-0.1

0

-0.025-0.020-0.015-0.010-0.0050.000

Ten

sile

str

ess

(MP

a)

strain y (%)

Lab

FEM-Tresca

FEM-Isotropic

FEM-CDP

Beam width = 40mm

Specimen length = 180mm

(a) (b)

Load F

131

Figure 4.18: Load-deflection curves of three-point bending notched beam test results for

cement-treated Toyoura sand by Namikawa (2006).

Figure 4.19: Tension-softening relation for cement-treated Toyoura sand by Namikawa

(2006).

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Lo

ad,

F (

N)

Beam Deflection, v (mm)

Bab-1

Bab-2

Bab-3

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15

t(M

Pa)

Opening crack displacement, c (mm)

Bab-1 Bab-2

Bab-3

132

Figure 4.20: Finite element meshes for simulating the three-point bending notched beam test:

(a) coarser mesh with 5,353 elements; and (b) finer mesh with 9,866 elements.

Figure 4.21: Load-deflection curves for two different mesh sizes in simulating the three-point

bending notched beam test for cement-treated Toyoura sand.

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6

Lo

ad,

F (

N)

Beam Deflection, v (mm)

5353 elements

9866 elements

Load, F

(a)

Load, F

(b)

133

Figure 4.22: FEM input tension-softening relation in Case T2 for cement-treated Toyoura

sand.

Figure 4.23: Calibrated load-deflection curves and laboratory measurement for three-point

bending notched beam test of cement-treated Toyoura sand.

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6

Lo

ad,

F (

N)

Beam Deflection, v (mm)

Bab-1

Bab-2

Bab-3

Case T1

Case T2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2

t(M

Pa)

Opening crack displacement, c (mm)

Bab-1

Bab-2

Bab-3

FEM input

134

0

200

400

600

800

1000

0 1 2 3 4 5

Axia

l st

ress

(kP

a)

Axial strain (%)

14Days

Aw=30%

Figure 4.24: Tensile damage process happened around the notch tip observed in FE analysis

Case T2.

Figure 4.25: Schematic experiment setup and stress-strain curves for UCT of cement-treated

Singapore marine clay.

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5

Lo

ad,

F (

N)

Beam Deflection, v (mm)

Case T2

a b

c

d

(a) (b)

(c) (d)

sample

load cell

Displacement

transducer

loading rate

1 mm/min

135

Figure 4.26: FEM model and stress-strain curves for UCT calibration of cement-treated

Singapore marine clay.

Figure 4.27: Schematic diagram of three-point bending notched beam test B1 of cement-

treated Singapore marine clay (unit: mm).

0

200

400

600

800

1000

0 1 2 3 4 5

Axia

l st

ress

(kP

a)

Axial strain (%)

Lab

FEM-CDP

(Single element

model)

200

25 25 50

25

2.5

Load, F

136

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3

Lo

ad,

F (

N)

Beam deflection, v (mm)

Test B1 (14Days

Aw=30%)

Figure 4.28: Load-deflection curve of three-point bending notched beam test B1 for cement-

treated Singapore marine clay.

Figure 4.29: Finite element mesh for simulating the three-point bending notched beam Test

B1.

Load, F

137

Figure 4.30: Calibrated load-deflection curves and laboratory measurement for three-point

bending notched beam Test B1 for cement-treated Singapore marine clay.

Figure 4.31: Post-test damage observed in numerical (Top) and laboratory (Bottom) for three-

point bending notched beam Test B1.

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Loa

d,

F (

N)

Beam Deflection, v (mm)

Lab

E=212qu

E=400qu

E=800qu

crack

specimen

notch