Idealized and Real Stress-Strain Behavior of Soils; Basic Concepts of Mohr-Coulomb Model; Mohr-Coulomb Model Modeling Friction Angle; Influence of Intermediate Principal Stress on Friction Angle; Drained Simple Shear Test; How to Understand ; Drained Triaxial Test; Constitutive modelling; stress strain behaviour;

1. CL- Constitutive Modelling of Frictional Material
Samirsinh P Parmar
Spp.cl@ddu.ac.in
Asst. Professor of Civil Engineering Department
Dharmsinh Desai University, Nadiad, Gujarat, Bharat.
Mohr-Coulomb Model
M.Tech Geotechnical Engineering

2. Contents
• Idealized and Real Stress-Strain Behavior of Soils
• Basic Concepts of Mohr-Coulomb Model
• Mohr-Coulomb Model Modeling
• Friction Angle
• Influence of Intermediate Principal Stress on
Friction Angle
• Drained Simple Shear Test
• How to Understand 
• Drained Triaxial Test
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 2

3. Pre-requisite:
Real Soil stress vs strain
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 3

4. Idealized and Real Stress-Strain
Behavior of Soils
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 4

5. Basic Concepts of Mohr-Coulomb Model
Bilinear approximation of triaxial test
Basic law:
for
reversible elastic strain
irreversible plastic strain f = yield function

i  i
e
 i
p

i  x,y,etc.

i
e

i
p
 
i
p

 0

f  0

f  f (x,y,z,xy,yz,zx)
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 5

6. Mohr-Coulomb Soil Modeling
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 6

7. Mohr-Coulomb Soil Modeling
Flow rule for plastic strain:
This means: , , etc.

i
p

 
g
i
x
p

 
g
x
y
p

 
g
y
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 7

8. Mohr-Coulomb Soil Modeling
a multiplier that determines the magnitude of plastic strains
determines the direction of plastic strains
Classical associated plasticity: g = f
General non-associated plasticity: g  f
M-C model: f  r – s sin – c cos : yield function
g  r – s sin – c cos : plastic potential function

 

g
i
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 8

9. Mohr-Coulomb Soil Modeling
Nonlinear Failure Envelope Representation
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 9

10. Mohr-Coulomb Soil Modeling
Friction Angle Definitions
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 10

11. Mohr-Coulomb Soil Modeling
Strength Envelopes for a Range of Soil Types
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 11

12. Mohr-Coulomb Soil Modeling
Versus Relative Density and Unit Weight
tc
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 12

13. Friction Angle
Dilatancy Angle Relationships
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 13

14. 1 = Correction for particle shape
1 = -6o
For high sphericity and subrounded shape
1 = +2o
For low sphericity and angular shape
2 = Correction for particle size (effective size, d10)
2 = -11o
For d10 > 2.0 mm (gravel)
2 = -9o
For 2.0 > d10 >0.6 mm (coarse sand)
2 = -4o
For 0.6 > d10 >0.2 mm (medium sand)
2 = 0 For 0.2 > d10 >0.06 mm (fine sand)
3 = Correction for graduation (uniformity coefficient, Cu)
3 = -2o
For Cu > 2.0 (well-graded)
3 = -1o
For Cu = 2.0 (medium graded)
3 = 0 For Cu < 2.0 (poorly graded)
4 = Correction for relative density (Dr)
4 = -1o
For 0 < Dr <0.5 (loose)
4 = 0 For 0.5 < Dr < 0.75 (intermediate)
4 = +4o
For 0.75 < Dr < 1.00 (dense)
5 = Correction for type of mineral
5 = 0 For quartz
5 = +4o
For feldspar, calcite, chlorite
5 = +6o
For muscovite mica

tc  36o
 1  2  3  4  5
in which:
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 14

15. 
cv
max  cv  0.8
for NC Clays vs. PI
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 15

16. Influence of
Intermediate
Principal
Stress on
Friction
Angle
Introduction:
Models and soil
mechanics
Special conditions Name of test Diagram
c
b
a 

 
 True triaxial
r
c
b 

 
 Cylindrical compression
The “triaxial” test
0

b
 Plane strain or biaxial
0

b
 Plane stress
0


 r
c
b 

 One-dimensional compression
The oedometer test
0


 r
c
b 

 Uniaxial compression or
Unconfined compression
0


 c
b
a 

 Isotropic compression
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 16

17. Influence of Intermediate Principal Stress on
Friction Angle
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 17

18. Drained Simple Shear Test
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 18

19. How to Understand 
 =  + i or  =  - i
i = inter particle angle of friction
quartz sand:    – 30o
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 19

20. Drained Triaxial Test

v  x  y  z  2x  y
tan 
2sin
1 sin
(identical to biaxial test)
Computational Geotechnics Mohr-Coulomb Model
Prof. S.P.Parmar, DoCL, DDU, Nadiad. 20

21. Prof. S.P.Parmar, DoCL, DDU, Nadiad. 21