This document discusses various models used in geotechnical engineering to describe soil behavior, including elastic, plasticity, and constitutive models. It provides an overview of linear and non-linear elastic models, and describes the equations that relate stresses to strains for different material conditions, such as Hooke's law and elastic models based on stress invariants. The document also discusses plasticity theory and concepts such as perfectly plastic and strain hardening responses, as well as plasticity theorems for lower and upper bound analysis.

1. MB March 10, 2002
University of Arizona
Department of Civil Engineering & Engineering Mechanics
CE544 – Special Topics in Geotechnical Engineering
FIELD INSTRUMENTATION AND MONITORING
Instructor: Muniram Budhu
Date: Apeil 14, 2003

2. MB March 10, 2002
Model
 Two requirements for a good model
 1: It must accurately describe a large class of
observations with few arbitrarily elements.
 2: It must make definite predictions about the
results of future observations.
(extracted from Stephen Hawking ‘Á brief history of time,’ Bantam Books)

3. MB March 10, 2002
Types of Models (CE544)
 ELASTIC
 LINEAR
 NON-LINEAR
 PLASTICITY
 NON-LINEAR
 CAM-CLAY FAMILY
 Elasto-plastic

4. MB March 10, 2002
NON-LINEARITY
 GEOMETRIC – change of shape, size,
etc.
 MATERIAL – change of properties
 CAUSES: stress state. History of loading,
change in stiffness, physical conditions, in
situ stress, water content, voids ratio

5. MB March 10, 2002
ELASTIC MODELS
 LINEAR – magnitude of response
proportional to excitation
 Non-LINEAR – magnitude of response not
proportional to excitation
stress
strain
Linear
Non-linear

6. MB March 10, 2002
CONSTITUTIVE LAWS
 SET OF EQUATIONS THAT RELATE
STRESSES TO STRAINS.
 F(stress, stress rate, strain, strain rate) = 0
 Homogeneity of time.

7. MB March 10, 2002
CONSTITUTIVE EQUATIONS
ij ijkl kl
C
  
Stiffness matrix
ij ijkl kl
D
  
Compliance matrix

8. MB March 10, 2002
ELASTIC MODELS
Elastic materials: State of stress is a
function of the current state of
deformation; no history effects
 Cauchy – stress is a function of strain
(infinitesimal strain, first order)
 Green – based on strain energy function
(Hyper-elastic)

9. MB March 10, 2002
Hooke’s law – simple case
 Simple one dimensional case:
E = Young’s modulus (Elastic modulus)
E
  

10. MB March 10, 2002
Hooke’s law – General state
 
 
 
x x
y y
z z
xy xy
yz yz
zx zx
1 0 0 0
1 0 0 0
1 0 0 0
1
0 0 0 2 1 0 0
E
0 0 0 0 2 1 0
0 0 0 0 0 2 1
 
 
   
 
 
   
 
 
   
  
 
   
  
   
 
 
 
   
 
 
 
   
 
 
 
   
 
   

11. MB March 10, 2002
Shear stresses and strains
  zx
zx zx
2 1
E G
  
   
 
E
G
2 1

 
G is the shear modulus. Only G or E and u are
required to solve linear elastic problems

12. MB March 10, 2002
Typical values of E and G
Soil Type Description E*
(MPa) G (MPa)
Clay Soft
Medium
Stiff
1 to 15
15 to 30
30 to 100
0.4 to 5
5 to 11
11 to 38
Sand Loose
Medium
Dense
10 to 20
20 to 40
40 to 80
4 to 8
8 to 16
16 to 32
*These are average secant elastic moduli for drained condition

13. MB March 10, 2002
Principal stresses
1 1
2 2
3 3
1
1
1
E 1
 
   
 
 
   
    
 
   
 
 
 
   
 
   
  
1 1
2 2
3 3
1
E
1
1 1 2 1
 
   
   
 
   
      
 
   
       
 
 
   
 
   

14. MB March 10, 2002
AXISMMETRIC CONDITION
   
1 1
3 3
1 1 2
1
E
 
 
 

 
  
 
 
      
1 1
3 3
E 1 2
1
1 1 2
 
  
 

 

 
 
   

15. MB March 10, 2002
PLANE STRAIN CONDITION
   
1 1
3 3
1 1
1
E
 
 
  
 

 
  
 
 
      
1 1
3 3
E 1
1
1 1 2
 
  
 

 
  
 
 
   

16. MB March 10, 2002
Hooke’s law using stress
invariants
p
K
1
e
p




p is mean stress, K is bulk modulus; the prime denotes effective
)
2
1
(
3
E
p
K e
p 







q
G
3
1
e
q 

 
v
1
2
E
G
G






SPECIAL CASE : 1/2; K 0

  

17. MB March 10, 2002
Constitutive elastic model – stress
invariants

















 






 
e
q
e
p
G
3
0
0
K
q
p
Decoupling - Mean effective stress causes
volumetric strain; deviatoric stress (shear
stress) causes deviatoric strain

18. MB March 10, 2002
Lame’s constant
G(E 3G)
K 2G /3
3G E
3KE
G
9K E

   




19. MB March 10, 2002
Poisson’s ratio
 



 2
1
'
K
3
E
also
 



 1
G
2
E
Then
 
 
1
1
G
2
2
1
K
3







K
6
G
2
G
2
K
3







20. MB March 10, 2002
Green’s elastic model
The work done by external forces in altering
the configuration of a body from its natural
state is equal to the sum of the kinetic
energy and the strain energy
i i ij i i
v s
ij i i ij i ,j
s v
i ,j ij ij
w Fu dv n u ds
using Gauss's divergence theorem
n u ds ( u) dv
( u) w
    
    
    
 
 
Strain tensor - symmetrical
Strain tensor – skew symmetrical

21. MB March 10, 2002
ANISOTROPIC ELASTICITY
 Anisotropic materials have different elastic
parameters in different directions.
 Structural anisotropy or transverse anisotropy –
manner in which soil is deposited.
 Stress induced anisotropy – differences in normal
stresses in different directions.

22. MB March 10, 2002
Transverse anisotropy
- most prevalent in soils
     
rz
z r
z z
rr
r r
zr
z r
2
1
E E
1
E E
 
 
 
 
 

 
 

 
 
 

23. MB March 10, 2002
ELASTICITY AND PLASTICITY
Theory of elasticity: uniqueness – behavior
of the material expressed by a set of
equations
Theory of plasticity: discontinuity in stress-
strain relationship (involves discontinuities
and inequalities); deals with initial stress
problems, state of structure at collapse, at
post-yield.

24. MB March 10, 2002
THEORY OF PLASTICITY
 TO ADEQUATELY DESCRIBE THE
PLASTIC DEFORMATION OF SOILS
 TO USE RELATIONSHIPS DEVELOPED
TO PREDICT FAILURE LOADS AND
SETTLEMENT.

25. MB March 10, 2002
PLASTIC RESPONSES
stress
stress
stress
strain strain strain
Rigid – perfectly
plastic
Elastic- perfectly
plastic
Elastic- plastic
strain hardening

26. MB March 10, 2002
FULL PLASTIC STATE
(COLLAPSE)
 Guess a plastic collapse mechanism
 For small deformation of this mechanism,
integrate the work consumed in plastic
deformation over the whole body
 Equate this to the work supplied to find the
collapse load
(ref: Calladine, C. R. “Engineering plasticity”, Pergamon Press, London)

27. MB March 10, 2002
PLASTICITY THEOREMS
 LOWER BOUND –IF ANY STRESS DISTRIBUTION
THROUGOUT THE STRUCTURE CAN BE FOUND WHICH IS
EVERYWHERE IN EQUILIBRIUM INTERNALLY AND BALANCES
CERTAIN EXTERNAL LOADS AND AT THE SAME TIME DOES NOT
VIOLATE THE YIELD CONDITION, THESE LOADS WILL BE CARRIED
SAFELY BY THE STRUCTURE.
 UPPER BOUND –IF AN ESTIMATE OF THE PLASTIC
COLLAPSE LOAD OF A BODY IS MADE BY EQUATING INTERNAL RATE
OF DISSIPATION OF ENERGY TO THE RATE AT WHICH EXTERNAL
FORCES DO WORK IN ANY POSTULATED MECHANISM OF
DEFORMATION OF THE BODY, THE ESTIMATE WILL BE EITHER HIGH,
OR CORRECT.