Description of Cam-Clay and Modified-Cam-Clay Critical State Strength Models Introduction The first critical state models for describing the behaviour of soft soils such as clay, the Cam-Clay (CC) and Modified Cam-Clay (MCC) were formulated by researchers at Cambridge University. Both models describe three important aspects of soil behaviour:

(i) Strength (ii) Compression or dilatancy (the volume change that occurs with shearing) (iii) Critical state at which soil elements can experience unlimited distortion without any changes in stress or

volume. A large proportion of the volume occupied by a soil mass consists of voids that may be filled by fluids (primarily air and water). As a result, deformations in soil are accompanied by significant, and often non-reversible, volume changes. A major advantage of cap plasticity models, a class to which the CC and MCC formulations belong, is their ability to model volume changes more realistically. The primary assumptions of the CC and MCC models are described next. In critical state mechanics, the state of a soil sample is characterized by three parameters:

• Effective mean stress 'p • Deviatoric (shear stress) q , and • Specific volume ν .

Under general stress conditions, the mean stress, 'p , and the devaitoric stress, q , can be calculated in terms of principal stresses '

1σ , '2σ and '

3σ as

( )' ' ' '1 2 3

13

p σ σ σ= + +

( ) ( ) ( )2 2 2' ' ' ' ' '1 2 2 3 3 1

12

q σ σ σ σ σ σ= − + − + −

The specific volume is defined as 1 eν = + , where e is the void ratio. Virgin Consolidation Line and Swelling Lines The models assume that when a soft soil sample is slowly compressed under isotropic stress conditions ( )' ' ' '

1 2 3 pσ σ σ= = = , and under perfectly drained conditions, the relationship between specific volume, ν , and 'ln p consists of a straight virgin consolidation line (also known as the normal compression line) and a set of straight swelling lines (see Figure 1). Swelling lines are also called unloading-reloading lines. The loading and unloading behaviour of the CC and MCC models is best described with an example. When a soil element is first loaded to isotropic stress '

bp , on the plane of ln pν ′− , it moves down the virgin consolidation line from point a to point b. If the sample is unloaded the specific volume–mean stress behaviour moves up the swelling line bc to the point c. If the sample is now reloaded to a stress '

dp , it will first move down the swelling line for stress values up to 'bp .

Once 'bp is exceeded, the sample will again move down the virgin consolidation line to the point d. If the sample is

then unloaded to a stress value of 'ap , this time it will move up the swelling line de .

The virgin consolidation line in Figure 1 is defined by the equation

'lnv N pλ= − , while the equation for a swelling line has the form

'lnsv v pκ= − . The values λ , κ and N are characteristic properties of a particular soil. λ is the slope of the normal compression (virgin consolidation) line on ln 'v p− plane, while κ is the slope of swelling line. N is known as the specific

volume of normal compression line at unit pressure, and is dependent on the units of measurement. As can be seen on Figure 1, sv differs for each swelling line, and depends on the loading history of a soil. If the current state of a soil is on the virgin consolidation (normal compression) line the soil is described as being normally consolidated. If the soil is unloaded such as is described by the line bc , it becomes overconsolidated. In general, soil does not exist outside the virgin consolidation line; when it does that state is unstable. The Critical State Line Sustained shearing of a soil sample eventually leads to a state in which further shearing can occur without any changes in stress or volume. This means that at this condition, known as the critical state, the soil distorts at constant state of stress with no volume change. This state is called the Critical State and characterized by the Critical State Line (CSL). In p q′ − plane the CSL is a straight line passing through the origin with the slope equal to M , one of the characteristic of the material (see Fig. 3). The location of this line relative to the normal compression line is shown on Figure 2. As seen in the picture, the CSL is parallel to the virgin consolidation line in ln 'v p− space. The parameter Γ is the specific volume of the CSL at unit pressure. Like N , the value of Γ depends on measurement units. There is a relationship between the parameter N of the normal compression line and Γ . For the Cam-Clay model the two parameters are related by the equation

( )N λ κΓ = − − , while for the Modified Cam-Clay model the relationship is

( ) ln 2N λ κΓ = − − . Due to this relationship between N and Γ , only one of them needs to be specified when describing a Cam-Clay or Modified Cam-Cam material. Yield Functions The yield functions of the CC and MCC models aree determined from the following equations: Cam-Clay

''

'ln 0o

pq Mpp

+ =

Modified Cam-Clay

'22

'2 '1 0opq Mp p

+ − =

In 'p q− space, the CC yield surface is a logarithmic curve while the MCC yield surface plots as an elliptical curve (Figure 3). The parameter '

op (known as the yield stress or pre-consolidation pressure) controls the size of the yield surface. The parameter M is the slope of the CSL in 'p q− space. A key characteristic of the CSL is that it intersects the yield curve at the point at which the maximum value of q is attained. In three-dimensional space 'v p q− − the yield surface defined by the CC or MCC formulation is known as the State Boundary Surface. The State Boundary Surface for the Modified Cam-Clay model is shown in Figure 4.

Elastic Material Constants for Cam-Clay and Modified Cam-Clay In geotechnical engineering, the elastic material constants commonly used to relate stresses to strains are Young’s modulus, E , shear modulus, G , Poisson’s ratio, µ , and bulk modulus, K . Only two of these parameters must be specified in an analysis. In soil modelling, the more fundamental elastic parameters of shear modulus, G , and bulk modulus, K , are preferred. This is because they allow the effects of volume change and distortion to be decoupled. For Cam-Clay and Modified Cam-Clay soils, the bulk modulus is not constant. It depends on mean stress, 'p , specific volume, ν , and the slope of the swelling line, κ , and is calculated as

vpKκ

=

Cam-Clay and Modified Cam-Clay formulations require specification of either shear modulus G or Poisson’s ratio µ . When G is given as a constant then µ is no longer a constant, and is calculated from the equation

3 22 6

K GG K

µ −=

+.

When µ is given as a constant then G is determined using the relationship

3(1 2 )2(1 )

G Kµµ

−=

+

The Overconsolidation Ratio The current state of a soil can be described by its stress state (mean effective stress 'p ), specific volume, ν , and yield stress, op (also known as preconsolidation pressure is a measure of the highest stress level the soil has ever experienced). The ratio of preconsolidation pressure to current mean effective stress is known as the overconsolidation ratio (OCR), i.e.

'op

OCRp

= .

The in-situ distribution of preconsolidation pressure for a Cam-Clay or Modified Cam-Clay material can be generated using the OCR. An OCR value of 1 represents a normal consolidation state; a state in which the maximum stress level previously experienced by a material is not larger than the current stress level. OCR > 1 describes an overconsolidated state indicating that the maximum stress level experienced by the material is larger than the present stress level. Hardening and Softening Behaviour If yielding occurs to the right of the point at which the CSL intersects a yield surface, hardening behaviour, accompanied by compression, is exhibited. This side of the yield surface is known as the wet or subcritical side. Figure 5a illustrates the soil behaviour on the wet side for the case of simple shearing. When a sample is sheared, it behaves elastically until it hits the initial yield surface. From then on the yield surface begins to grow/expand and exhibits hardening behaviour (yielding and plastic strain is accompanied by an increase in yield stress). The figure shows two intermediate growth stages of the yield surface. At the point C, the sample reaches critical state at which it will continue to distort without any accompanying changes in shear stress or volume. Figure 5b portrays the stress-strain hardening behaviour that occurs for the sample loaded on the wet side. If yielding occurs to the left of the intersection of the CSL and yield surface (called the dry or supercritical side), the soil material exhibits softening behaviour, which is accompanied by dilatancy (increase in volume). In softening regimen the yield stress curve decreases after the stress state touches the initial envelope. To depict the reduction in yield stress curve, the loading line in Figure 6a doubles back. The yield curve and sustained load move downwards

until the sample comes to the critical state. The softening stress-strain curve for dry side loading is shown on Figure 6b. Specification of Initial States for Cam-Clay and Modified Cam-Clay Models To compute models involving Cam-Clay or Modified Cam-Clay materials, non-trivial initial effective stresses must be specified. Phase2 allows specification of gravity in-situ stresses or a constant stress field.

Next, the initial state of consolidation (initial yield surfaces for all stress states) must be specified. This can be done in one of two ways:

(a) Specification of the OCR, or (b) Specification of a pre-consolidation stress, op

If a current stress state completely lies within a specified yield surface, the soil will initially respond elastically to loading. This implies that it is overconsolidated. If, however, the initial stress state is located on the yield surface, the soil will respond elasto-plastically upon loading, indicating that it is normally consolidated.

Since initial stress states that lie outside yield surfaces have no physical meaning for Cam-Clay and Modified Cam-Clay models, Phase2 will not allow them. If such stress states are specified, Phase2 will change the pre-consolidation stress to a value (see below) that makes the soil normally consolidated. If in a model an initial mean effective pressure is negative, Phase2 warn about the occurrence and will not compute the model.

Specification of Initial Consolidation State for Gravity Loading Case I: When OCR is specified

(a) Determine vertical stress distribution, 'vσ (b) Determine horizontal stresses, 'hσ , using oK (c) For each element, calculate mean effective stress, 'p , and deviatoric stress, q (d) Calculate pre-consolidation pressure, *

op , such that ( )*', , 0oF p q p =

For Cam-Clay: * ''q

Mpop p e=

For Modified Cam-Clay: 2

*2 '

'oqp p

M p= +

(e) Set initial element pre-consolidation pressure, op , equal to *

o op p OCR= ⋅ (f) Set initial element specific volume, initv , to be consistent with op and the specified parameters M, λ , and κ

'ln lninit oo

pv N pp

λ κ

= − −

Case II: When pre-consolidation pressure, op , is initially specified

(a) Determine vertical stress distribution, 'vσ (b) Determine horizontal stresses, 'hσ , using oK (c) For each element, calculate mean effective stress, 'p , and deviatoric stress, q (d) Calculate pre-consolidation pressure *

op such that ( )*', , 0oF p q p =

For Cam-Clay: * ''q

Mpop p e=

For Modified Cam-Clay: 2

*2 '

'oqp p

M p= +

(e) Set initial element pre-consolidation pressure, op , equal to

( )*,max ,o o o specifiedp p p=

(f) Set initial element specific volume, initv , to be consistent with op and the specified parameters M, λ , and κ

'ln lninit oo

pv N pp

λ κ

= − −

Specification of Initial Consolidation State for Constant In-situ Stress Field The specification of initial states for a constant stress field is the same as for gravity loading except that initial effective vertical and horizontal stresses do not have to be determined. Summary of Input Parameters for Cam-Clay and Modified Cam-Clay Materials Specification of Cam-Clay and Modified Cam-Clay models requires five material parameters. These parameters are outlined below. 1. λ – the slope of the normal compression (virgin consolidation) line and critical state line (CSL) in ln 'v p−

space

2. κ – the slope of a swelling (reloading-unloading) line in ln 'v p− space

3. M – the slope of the CSL in 'q p− space

N – the specific volume of the normal compression line at unit pressure 4. or Γ – the specific volume of the CSL at unit pressure

µ – Poisson’s ratio

5. or G – shear modulus.

The initial state of consolidation of such materials must also be specified. This is accomplished by indicating

OCR – the overconsolidation ratio: the ratio of the previous maximum mean stress to the current mean stress or op – the preconsolidation pressure. Remark on the application of Cam-Clay and Modified Cam-Clay models in FE analyses: The Cam-Clay and Modified Cam-Clay models may allow for unrealistically large ratios of shear stress over mean stress when the stress state is above the critical state line. Furthermore, these models predict a softening behaviour for the state of stress on the dry side of the yield surface. Without special considerations, the softening behaviour leads to mesh dependency of a finite element analysis. The use of these two models in simulations of practical applications including general boundary valued problems is not recommended.

Figure 1. Behaviour of soil sample under isotropic compression.

Figure 2. Location of CSL relative to virgin compression line.

v

ln p

Virgin consolidation line (Normal compression line) N

λ

κ

1

1

vs1

vs2

Swelling lines

a

c

e b

d

1 pb pd

ln

v

N

Γ

Virgin compression li

CSL

Figure 3. Cam-Clay and Modified Cam-Clay yield surfaces (in ' 'p q− ) space. The parameter M is the slope of the CSL.

p

q

Critical State Line (CSL)

Modified Cam-Clay (MCC) yield curve

Cam-Clay (CC) yield curve

M

1

po

Figure 4. The State Boundary Surface for the Modified Cam-Clay model.

p

p

q

v

State boundary surface

CSL

Figure 5a. Evolution of the yield curve on the wet side of Modified Cam-Clay under simple shearing.

Figure 5b. Hardening stress-strain response on wet side of Modified Cam-Clay material under simple shearing.

shear strain

q

p

qCSL

Initial yield surface

Yield surface at critical state

Line of loading

C

wet or subcritical (hardening behaviour)

Figure 6a. Evolution of the yield curve on the dry side of Modified Cam-Clay under simple shearing.

Figure 6b. Softening stress-strain response on dry side of Modified Cam-Clay material under simple shearing.

p

qCSL

Initial yield surface

Yield surface at critical state

Line of loading

C

Dry or supercritical (softening behaviour)

shear strain

q

Modified Cam-Clay model: Implicit integration of the constitutive equations The numerical integration method for the modified Cam-Clay model is presented here. The formulation is almost the same as what is presented in an article by Borja (1991). Nonlinear elastic behaviour In the formulation of Cam-Clay model the bulk modulus of the material is dependent on the mean stress, 'p , specific volume, v , and the slope of the swelling line, κ ,

vpKκ

′=

Thus, the relationship between the rate of mean stress, 'p , and the rate of elastic volumetric strain, e

vε , can be written as

e ev v

vpp Kε εκ

′′ = =

Integrating the above equation over a finite time increment (step n to 1n + ), assuming that the change in specific volume is insignificant, results in the following incremental equation

1 exp enn n v

vp p ε

κ+ ′ ′= ∆

Assuming an average bulk modulus from step n to 1n + one can write

1e

n n vp p p K ε+′ ′ ′∆ = − = ∆ Based on the above equations, the average bulk modulus over step n to 1n + can be obtained as follow

exp 1en nve

v

p vK ε

κε′ = ∆ − ∆

In case the shear modulus is not constant, i.e. the poison’s ration is constant, the shear modulus can be calculated as

(3 6 )2(1 )

G rK Kµµ

−= =

+

Integration of elasto-plastic constitutive equations The mechanical behaviour of a wide range of elasto-plastic materials can be characterized by means of a set of constitutive relations in the general form of

( ), 0F κ =σ ; ( ) .ijQ constσ = e p= +ε ε ε

e e=σ D ε

p Qλ ∂=

∂ε

σ

pQκκ λ ∂ ∂

=∂ ∂ε σ

In the equations above, F is the yield function, Q is the plastic potential, σ and ε are the stress and strain

tensors/vectors, eD is the elastic constitutive tensor/matrix, λ is the plastic multiplier and κ is the hardening parameter. The superscript e and p stand for elastic and plastic. The above equations are yield and plastic potential functions, the additivity postulate that allows division of the increment of total strain into the elastic and plastic parts, the generalized Hooke’s law, flow rule and the hardening rule, respectively. In the modified Cam-Clay model the constitutive equations are much simpler to deal with if they are expressed in terms of stress/strain invariants. The set of constitutive equations, considering an implicit integration scheme, are presented below

( )2

11 1 1 02 1

0nn n n n

qF p p p

M+

+ + + +′ ′ = + − = Yield function and plastic potential

1

1 3

en n v

en n q

p p K

q q G

ε

ε+

+

′ ′ = + ∆

= + ∆ Hooke’s law

e p

v v ve p

q q q

ε ε ε

ε ε ε

∆ = ∆ + ∆

∆ = ∆ + ∆ The additivity postulate

1

1

pv

n

pq

n

Fdp

Fdq

ε λ

ε λ

+

+

∂∆ = ′∂

∂ ∆ = ∂

The flow rule

( ) ( )0 01exp

pn v

n np pν ελ κ+

∆= −

Hardening/softening rule

Form the definition of the yield function one can obtain:

( )1 0 11

12

1

2

2

n nn

n

n

F p pp

qFq M

+ ++

+

+

∂ ′= − ′∂

∂= ∂

The set of constitutive equations presented above can be simplified in a nonlinear set of equations with 4 equations and 4 unknowns ( ( )1 1 0 1

, , ,n n np q p dλ+ + +′ ) 4 independent unknowns

( )( )( )

( )

( ) ( ) ( )( )

1 1 1 0 1

12 1 2

21

3 1 1 02 1

4 0 0 1 01 1

exp 2 0

23 0

0

exp 2 0

nn n v n n

nn n q

nn n n

nnn n n

g p p d p p

qg q q G d

Mq

g p p pM

g p p d p p

νε λ

κ

ε λ

νλ

λ κ

+ + +

++

++ + +

++ +

′ ′ ′= − ∆ − − = = − + ∆ − = ′ ′ = + − =

′= − − = −

4 independent nonlinear equations

The integration/solution algorithm for a material point starts from an initial state of stress and hardening parameters ( ( )0, , ,n n nnp q p ν′ ) with the introduction of the increment of strains ( vε∆ , qε∆ ). The solution algorithm is based on a Newton iterative technique and follows these steps, sequentially: 1- Initializing the unknown variables:

1n np p+′ ′= , 1n nq q+ = , ( ) ( )0 01n np p+

= , 0dλ = 2- Calculate the ig functions, and check if they are all close enough to zero. If yes terminate the process, if not go to step 3 to modify the current values of unknowns. 3- Update the unknowns by solving the linear system of equations presented below, and then go to step2. The term

,i jg represents the partial derivative of the function ig with respect to the j th variable.

( )

11,1 1,2 1,3 1,4 1

12,1 2,2 2,3 2,4 2

03,1 3,2 3,3 3,4 31

4,1 4,2 4,3 4,4 41

n

n

n

n

pg g g g gqg g g g gpg g g g g

g g g g gdp

+

+

+

+

∆ ∆ = ∆ ∆

After finding the updated state of stress and state variables, i.e. ( )1 1 0 1

, , ,n n np q p ν+ + +′ , the transformation to general

state of stress, σ , is straightforward.

Cam-Clay model: Explicit integration of the constitutive equations The numerical integration method for the Cam-Clay model is presented here. The formulation is based on the classical plasticity as the integrations process takes advantage of an explicit scheme. Nonlinear elastic behaviour In the formulation of Cam-Clay model the bulk modulus of the material is dependent on the mean stress, 'p , specific volume, v , and the slope of the swelling line, κ ,

vpKκ

′=

Considering an explicit scheme over a finite time increment (step n to 1n + ), the bulk modulus will be calculated based on the state of the material at step n .

n nn

v pK

κ′

=

In case the shear modulus is not constant, i.e. the poison’s ration is constant, the shear modulus can be calculated as

(3 6 )2(1 )n n nG rK Kµ

µ−

= =+

Integration of elasto-plastic constitutive equations The mechanical behaviour of a wide range of elasto-plastic materials can be characterized by means of a set of constitutive relations in the general form

( ), 0F κ =σ ; ( ) .ijQ constσ = e p= +ε ε ε

e e=σ D ε

p Qλ ∂=

∂ε

σ

pQκκ λ ∂ ∂

=∂ ∂ε σ

In the equations above, F is the yield function, Q is the plastic potential, σ and ε are the stress and strain

tensors/vectors, eD is the elastic constitutive tensor/matrix, λ is the plastic multiplier and κ is the hardening parameter. The superscript e and p stand for elastic and plastic. The above equations are yield and plastic potential functions, the additivity postulate that allows division of the increment of total strain into the elastic and plastic parts, the generalized Hooke’s law, flow rule and the hardening rule, respectively. Like the Modified Cam Clay model, in Cam-Clay model the constitutive equations are much simpler to deal with if they are expressed in terms of stress/strain invariants. The set of constitutive equations, considering an explicit integration scheme, are presented below

0

ln 0pF q Mpp

′′= + =

; 0dF = Yield function and plastic potential; The consistency condition

1

e en n n+ = + ∆ = + ∆σ σ σ σ D ε Hooke’s law

e p∆ = ∆ + ∆ε ε ε The additivity postulate

p

n

Fdλ ∂ ∆ = ∂ ε

σ The flow rule

( ) ( )0 01exp

pn v

n np pν ελ κ+

∆= −

Hardening/softening rule

The derivative of the yield function with respect to the stress tensor/vector can be calculated as

F F p F qp q

∂ ∂ ∂ ∂ ∂= +

′∂ ∂ ∂ ∂ ∂σ σ σ

Form the definition of the yield function one can obtain:

1

n

nn

n

qF Mp p

Fq

∂= − ′ ′∂

∂= ∂

The set of constitutive equations presented above can be simplified in one nonlinear equation with one independent variable, i.e. dλ , which can be calculated as

0

0

e

ep

v

F

dpF F F F

p p

λ

ε

∂∆

∂=∂∂ ∂ ∂ ∂

−′∂ ∂ ∂ ∂∂

Dεσ

Dσ σ