CHAPTER 3. LINEAR AND NONLINEAR CONTINUA Contents · Chapter 3. Linear and Nonlinear Continua Page 44 CHAPTER 3. LINEAR AND NONLINEAR CONTINUA Contents 3.1 Linear elastostatics 45

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Chapter 3. Linear and Nonlinear Continua Page 44

CHAPTER 3. LINEAR AND NONLINEAR CONTINUA

Contents

3.1 Linear elastostatics 45

3.1.1 Invariants 49

3.2 Basic elasto-plastic models 50

3.2.1 Mohr-Coulomb and Tresca yield criteria 52

3.2.2 Drucker-Prager and Misès criteria 54

3.2.3 Dilatancy and flow rule 55

3.2.4 ZSOIL data 57

3.3 Finite elements 60

3.4 Newton-Raphson procedure 62

3.4.1 Convergence 65

3.5 Geotechnical aspects 66

3.5.1 Initial state 66

3.5.2 Locking in quasi incompressible media 68

3.5.3 Spurious pressure oscillations in consolidation flow 69

3.6 References 70



3.1 Linear elastostatics

Equilibrium of a 2D infinitesimal element of dimensions 1 2*dx dx can be represented

graphically as shown in Fig 3.1. Equilibrium in direction of x1 then writes:

11 121 2 2 1 1 1 2

1 2

0dx dx dx dx f dx dxx x

σ τ∂ ∂+ + =

∂ ∂

which can be simplified into:

11 121

1 2

0fx x

σ τ∂ ∂+ + =

∂ ∂

and generalized to 1, 2 or 3 directions as:

0; for i=1 to 2 and sum over j=1 to 2, in the 2D caseiji

j

fx

σ∂+ =

∂

We rewrite this expression as:

, 0 with sum on repeated indicesij j ifσ + = (1 to 2 in the 2D case)

The corresponding multi-dimensional boundary value problem can be stated as, in

differential form:

, 0ij j i

i i u

ij j i u

f on

u u on

n on andσ σ

σ

σ σ

+ = Ω = Γ = Γ Γ = Γ + Γ

The first equation expresses that equilibrium must be satisfied on domain Ω, the second

defines imposed displacements on part of the boundary, and the third imposed surface

tractions on the rest of the boundary ( jn are components of the local normal) (Fig. 3.1).

Fig. 3.1 Elastostatic equilibrium



If the medium is elastic, then Hooke’s law applies which can be stated as:

( )ij ijkl klC sum on repeated indicesσ ε=

For isotropic elasticity, this can be simplified to:

2ij kk ij ijσ λε δ µε= +

where λ and µ are Lamé’s constants, and ijδ is Kronecker’s symbol

( )1 ,0if i j if i j= ≠ .

Alternatively, a volumetric-deviatoric split of the stress tensor is possible, then:

- 11 22 33kk kkKσ σ σ σ ε= + + = , is the volumetric stress, function of the volumetric

strain, and K is the bulk modulus

- 2ij ijs eµ= , is the deviatoric stress, function of the deviatoric strain, and

hence:

- ijijkk

ij s+

= δσσ3

The same split applied to strain leads to:

- ijijkk

ij e+

= δεε3



Expanding Hooke’s law for the isotropic 2 and 3-dimensional cases, we obtain, in matrix

form:

_______________________________________________________________

Table 3.1: Hooke’s law, 3D isotropic elasticity

1/ / /

/ 1/ / 0

/ / 1/

1/

0 1/

1/

2

2 0

2

0

x x

x y

x z

yz yz

zx zx

xy xy

x

y

z

yz

zx

xy

E E E

E E E

E E E

G

G

G

G

G

G

G

G

G

ε συ υε συ υε συ υγ τγ τγ τ

σ λ λ λσ λ λ λσ λ λ λτττ

− − − − − − =

+ + + =

0 2

x

x

x

yz

zx

xy

xy xyempty spaces in matrices and

εεεγγγ

γ ε

≡ =

_______________________________________________________________

_______________________________________________________________

Table 3.2: Hooke’s law, Plane strain

2

2

0

(1 ) / (1 ) / 0

(1 ) / (1 ) /

0 1/

2 0

2

0

( )

z xz yz

x x

y y

xy xy

x x

y y

xy xy

z x y

E E

E E

G

G

G

G

ε γ γ

ε υ υ υ σε υ υ υ σγ τ

σ λ λ εσ λ λ ετ γ

σ υ σ σ

= = =

− − + = − + −

+ = +

= +

_______________________________________________________________



_______________________________________________________________

Table 3.3: Hooke’s law, Plane stress (only in ZSOIL custom version)

2 2

2 2

0

1/ / 0

/ 1/

0 1/

/(1 ) /(1 ) 0

/(1 ) /(1 )

0

( )(1 )

z xz yz

x x

y y

xy xy

x x

y y

xy xy

z x y

E E

E E

G

E E

E E

G

σ τ τ

ε υ σε υ σγ τ

σ υ υ υ εσ υ υ υ ετ γ

υε ε ευ

= = =

− = −

− − = − −

= − +−

_______________________________________________________________

Only two constants are needed for isotropic elasticity, but different pairs can be used

(see Table 3.4).

_______________________________________________________________

Table 3.4 : Elastic constants

_______________________________________________________________

Remark:

- Soils are sometimes almost incompressible. Incompressible elasticity

corresponds to ν = 0.5. This value of ν will induce numerical problems and must

be approximated, by ν = 0.49999, but even this value will lead to mesh locking

unless appropriate elements are used. A special section is dedicated to these

questions.



3.1.1 Invariants

Stress and strain components depend on the orientation of coordinates but stress and

strain invariants remain unaffected by a change of the coordinates. Invariants play an

essential role in nonlinear analysis because yield and failure criteria must be expressed

in terms of invariants in order to avoid dependence on the coordinate system.

Different invariants or linear combinations of invariants can be defined, but we will use

only a few to start with.

_______________________________________________________________

Table 3.5: Stress and strain invariants in 3D and in plane strain

_______________________________________________________________



3.2 Basic elasto-plastic models

Linear elastostatics are not sufficient for the simulation of soil behavior.

ε

E

σ

σy

EepH’

softening

hardening

eε∆pε∆

σ∆perfect plastisticity

ε

E

σ

σy

EepH’

softening

hardening

eε∆pε∆


ε

E

σ

σy

EepH’

softening

hardening

eε∆pε∆


ε

E

σ

σy

E

σ

σy

EepH’

softening

hardening

eε∆pε∆


∆εp

plasticity

Fig.3.2 Uniaxial elastoplasticity

Permanent deformation, hysteretic cyclic behavior, strain hardening and strain

increments which are not coaxial with stress increments are typical of soil behavior.

Incremental plasticity supports such type of behavior, which makes it an appropriate

constitutive theory for soils.

The 1-dimensional case is examined in Fig. 3.2. The figure illustrates three typical plastic

behaviors: perfectly plastic behavior, hardening, and softening. All three are

supported by ZSOIL, but not with all models. The first two can be used without special

precautions, but softening induces a dependence on the discretization (mesh size) which

must be accounted for.

Loading up to σy, the yield stress, is elastic and unloading from this point will leave no

permanent deformation. Applying a positive stress increment ∆σ from the yield point σy

leads to a permanent deformation pε∆ after unloading.

Incremental plasticity assumes split of strain into additive elastic and plastic strain

components, such that e p

∆ε = ∆ε + ∆ε .

The corresponding incremental constitutive law can then be written in different forms, as

should be obvious from Fig. 3.2:

; 1 : epin D Eσ ε∆ = ∆ep ep e p∆σ = D ∆ε = D (∆ε + ∆ε )

; 1 : 'e e pin D E Hσ ε ε∆ = ∆ = ∆p p∆σ = D ∆ε = D ∆ε



Elastoplasticity requires the definition of three essential ingredients:

1) A yield criterion ( ) 0F =σ , a surface in stress space which limits the domain of

elastic behavior. Given an arbitrary stress state σσσσ we have:

In the 1D case the yield criterion is simply a limiting stress σy, where index y stands for

yield.

2) A hardening law, which governs the evolution of the size of F under increasing

plastic strain, then, ( ) 0F =σ becomes:

( , ) 0F h =σ

where h is a hardening parameter, a function of plastic strain invariants.

In the 1D case the yield criterion is simply ( , ) ( ', )pyF h Hσ ε=σ .

3) A flow rule. The plastic strain increment direction will be governed by:

( ( ) / )d Gλ= ∂ ∂p∆ε σ σ

where ( )G σ defines the plastic potential, ( ) /G∂ ∂σ σ defines the direction of plastic

flow, normal to ( )G σ , dλ is the plastic multiplier which defines the amplitude

of plastic flow. When ( ) ( )F G≡σ σ we have associative plasticity, otherwise non-

associative plasticity.

Remark:

- In the 1D case, plastic flow will be coaxial with elastic flow as there is no other

option.

The consistency condition ( , ) 0F h•

=σ completes the formulation and will be used to

define the amplitude of plastic flow via dλ . It expresses that the stress point remains

on the yield surface during plastic flow. This point is essential for theory and

implementation but not for applications, see ZSOIL manuals for more details.



3.2.1 Mohr-Coulomb and Tresca yield criteria

Mohr-Coulomb is the most frequently used yield criterion in soil mechanics, it

expresses that the shear stress τ should not exceed a given limit, function of the

effective normal stress nσ , on the physical failure surface:

tannCτ σ φ≤ +

where C the cohesion, and φ the effective friction angle are material constants.

Remark:

- Sign convention: underlined values are positive in compression.

A more convenient form, for numerical implementation, can be written in terms of stress

invariants, in plane strain:

2 sin 2 cos 0d sF Cσ σ φ φ≡ − − =

and in 3D:

sin sinsin (cos ) cos 0

3 3

qmF C

σ φ θσ φ θ φ≡ − + − − =

Fig. 3.3 Mohr-Coulomb criterion in 3D stress space

In a three-dimensional principal stress space, Mohr-Coulomb criterion corresponds to a

cone with hexagonal cross-section (Fig. 3.3). This is a multi-surface criterion, in fact six

surfaces are present, and this criterion is therefore slightly more difficult to manage

numerically than smooth criteria.



Remarks:

- Tresca criterion corresponds to Mohr-Coulomb criterion with 0φ = , the

corresponding criterion is a cylinder with regular hexagonal cross-section.

- Notice that the shape of the cross-section of Mohr-Coulomb criterion moves from

a regular hexagonal for 0φ = , to nearly triangular for large friction angles (see

Fig. 3.3).

- The criterion is composed of three pairs of planes associated with the min and

max principal stresses. The traces of failure planes associated with (σ1, σ2), in the

deviatoric cross section are illustrated in red in Fig. 3.3.

Mohr-Coulomb is an elastic-perfectly plastic model in ZSOIL, there is therefore no

hardening law associated with it.

The plastic flow direction is associated - normal to the yield surface F(σ,φ) in the

deviatoric direction and characterized by an angle of dilatancy ψ which assumes the

existence of a plastic potential G(σ,ψ), analogue to the yield criterion F(σ,φ), in

meridional planes, see Fig. 3.5. The value of the angle of dilatancy is usually extracted

from an experiment. Dilatancy will be discussed in Section 3.2.3.



3.2.2 Drucker-Prager and Misès criteria

Drucker-Prager and Misès criteria are slightly more convenient for numerical

implementation than Mohr-Coulomb.

The criterion proposed by Drucker and Prager as an approximation to Mohr Coulomb can

be written

2 sin 2 cos 0q mF Cσ σ φ φ≡ − − = ; or alternatively :

1 2 0F a I J kφ≡ + − =

In 3D principal stress space, Drucker-Prager criterion corresponds to a cone with

circular cross-section and Misès criterion to a cylinder (Fig. 3.4). Both are single-surface

criteria.

Remark: Misès criterion corresponds to 0φ ψ= = .

When a Drucker-Prager criterion is used to approximate a Mohr-Coulomb criterion it

appears that matching both criteria cannot be unique. Adjustment is possible at external

edges, which correspond to uniaxial compression, internal edges, corresponding to

uniaxial tension, matching elastic domains for a particular stress state or plane strain

collapse. A table of size adjustments available in ZSOIL is given below and illustrated in

Fig. 3.5. Plane strain states (σ1 σ2) are in planes which intersect deviatoric planes as

shown in Fig. 3.5.

Fig. 3.4 Drucker-Prager and Misès criteria



Fig. 3.5 Size adjustments of Drucker-Prager with respect to Mohr-Coulomb, in

deviatoric cross section

The flow rule, which defines the direction of ∆εp, for Drucker-Prager criterion is

introduced again by assuming the existence of a plastic potential G whose normal, in

meridional planes can be characterized by a dilatancy angle ψ(for MC criterion), and aψ

for DP criterion, Fig. 3.6. Incompressible flow is coaxial with the normal to I1 axis

( 1 2 3σ σ σ= = ).

Fig. 3.6 Direction of plastic flow, with respect to yield criterion and plastic potential

3.2.3 Dilatancy and flow rule

Except for incompressible flow, shearing causes a volume increase during plastic flow,

this is called dilatancy. Considering Mohr-Coulomb criterion, with associative flow

(ψ φ= ), and the definition of plastic flow, we derive:



which corresponds to a dilatant behavior.

Usually, associated plasticity overestimates dilatancy observed in experiments.

If an experiment is available from which p

vdε can be evaluated, then ψ can be calculated

from the above formula. In the absence of available experiments take:

Remark

- Dilatant and incompressible flow both lead to mesh locking phenomena

unless appropriate elements are used. ZSOIL handles the choice of elements

automatically (Fig. 3.7, toggle Advanced version option in the menu if needed),

depending on the constitutive choices of the user. But the interested user who

wants to investigate locking phenomena can release the default options.

Fig. 3.7 Selection of elements



3.2.4 ZSOIL data

Mohr-Coulomb model requires 2 elastic constants E, ν and 3 plasticity parameters,

cohesion C, friction angle φ, and dilatancy angle ψ. Dilatancy is “0” by default

(incompressible case), it can be specified differently by activation of option advanced.

Material specification also requires the unit weight γ; and a tension cut-off is available

for no-tension materials. A dilatancy cut-off is also available, see Appendices.

Fig. 3.8.1 Material data for Mohr-Coulomb elasto-plastic material

If we use Drucker-Prager as a smooth approximation to Mohr-Coulomb, then cohesion C

and friction angle φ are the natural material data. Plasticity theory requires in addition

the plastic flow direction given by the dilatancy angle ψ and a size-adjustment is

necessary because matching both criteria is not unique, as seen earlier.

The advantage of using Drucker-Prager for soil analysis rather than Mohr-Coulomb is

only in saving some computing time, it should therefore be avoided unless computer

time is an issue, like in multiple parametric analyses, or if the user is knowledgeable

about size adjustments; results are very sensitive to this parameter. Watch that the ratio

of the internal radius and the external one of possible DP approximations is 0.62 for a

friction angle of 45o, see Table 3.6.

The proper choice for size adjustment is however easy in some cases like adjustment at



external edges, which corresponds to uniaxial compression, internal edges,

corresponding to uniaxial tension, or matching plane strain collapse between MC and DP

criteria.

Default options in ZSOIL are plane strain collapse, for plane strain analysis, which

speaks for itself, and intermediate, between external and internal edges for

axisymmetry which will be further discussed when we analyse the ultimate load of

axisymmetric footings.

Matching parameters for the various options are given in Table 3.6.

Fig. 3.8.2 Input screens for Drucker-Prager material



_______________________________________________________________________

Table 3.6: Size adjustments for Drucker–Prager criterion used as approximation

to Mohr-Coulomb

SIZE ADJUSTMENTSD-P vs M-C

))sin3(3/()cos6));sin3(3/(sin2 φφφφφ −=−= Cka

3-dimensional,external apices

3-dimensional,internal apices

))sin3(3/()cos6));sin3(3/(sin2 φφφφφ +=+= Cka

Plane strain failure with (default)

)cos;3/sin φφφ Cka ==

0=ψ

Axisymmetry intermediate adj. (default)

)sin9/(cos36);sin9/(sin32 22 φφφφφ −=−= Cka

NB: Rint/Rext~=0.62 at 45o

SIZE ADJUSTMENTSD-P vs M-C

))sin3(3/()cos6));sin3(3/(sin2 φφφφφ −=−= Cka

3-dimensional,external apices

3-dimensional,internal apices

))sin3(3/()cos6));sin3(3/(sin2 φφφφφ +=+= Cka

Plane strain failure with (default)

)cos;3/sin φφφ Cka ==

0=ψ

Axisymmetry intermediate adj. (default)

)sin9/(cos36);sin9/(sin32 22 φφφφφ −=−= Cka

NB: Rint/Rext~=0.62 at 45o

_______________________________________________________________________



3.3 Finite elements

Finite differences would be a straightforward way to discretize boundary-value problems,

but finite elements lead to more robust numerical methods and are nowadays the most

commonly used technique, and the one used exclusively in ZSOIL, for space

discretization; finite differences are used in time.

In order to formulate our finite element method we first build a weak (integral) form of

the equilibrium equation. If equilibrium is true as stated by:

, 0 ( sum on repeated indices)ij j ifσ + = ,

then the integral form

( , ) 0ij j i if w dσΩ

+ Ω =∫

in which iw is a reasonable arbitrary weighting function, is also true and integration by

parts yields:

( ) ( ) ( )i ij i i i iw d w f d w dσ

σ σΩ Ω Γ

Ω+ = Ω + Γ∫ ∫ ∫

Stress is related to strain by Hooke’s law, strain to displacements by the small strain

kinematic relation , ,0.5( )ij i j j iu uε = +

- remember that ( , /i j i ju u x= ∂ ∂ ) - and u is

approximated on quadrilateral subdomains, called elements, by:

4

1

hi aa ia

u N d=

=∑

where the iad are nodal displacement values, and aN are (often linear) interpolation

functions, see Fig. 3.9.

Similarly w is approximated by the same interpolation:

4

1

hi aa ai

w N c=

=∑

where the aic are nodal values of our arbitrary w test functions.

Invoking arbitrariness of w (think of it as of virtual displacements), we can then deduce

the matrix form of equilibrium.



1 2

3

4

(dx1,dy1)(dx2,dy2)

(…)

(…) 4

1 1 1 2 1 3 4 4

1 1

1

1

1 2

1 44

4

.....

0 .. 0..

0 0 ..

hi a iaa

hx x x x x

hy y

x

yhxhy

x

y

v N d

v N d N d N d N d

v N d

d

dv N N

v N Nd

d

=

= + + +

=

= =

∑

hv

Ω

Fig. 3.9 Interpolation of displacement field within an element

The first integral will generate the stiffness term, a square matrix of coefficients

multiplied by the vector of unknown nodal displacements Kd, the second the body force

term Ff, a vector of given nodal forces and the third the surface traction term Fσσσσ, also a

vector of given nodal forces, i.e. :

with

∫

∫

f σ

T

Ω

T

Ω

Kd = F + F

Kd = B σdΩ

K = B DBdΩ

and :

ε = Bd

σ = Dε = DBd

D was given above for plane strain, plane stress and 3D, matrix forms details can be

found in text books on finite elements and ZSOIL manuals.

K will be symmetric for elasticity and associative plasticity; it will have a symmetric

profile for non-associative plasticity and large displacements.

Displacement boundary conditions will be enforced later directly into the matrix form, as

illustrated earlier for trusses.



3.4 Newton-Raphson procedure

We have seen earlier that an elastoplastic constitutive law will induce a nonlinear

equilibrium relationship between force and displacements. So that the linear relation

Kd = F no longer holds, and is replaced by the nonlinear one:

N(d) = F

Where d is the vector of nodal displacements, F the vector of nodal forces, and N(d)

represents a nonlinear matrix form, function of d.

Load F will be defined by a load time-history, so that:

1

1

( )

( 1)n

n

t and

t n t+

+ = + ∆n+1F = F ,

where t∆ is the time increment, and n is the time increment (real or fictitious) counter.

At each time value, a nonlinear problem must be solved, iteratively; with i as iteration

counter, we write:

11 1

in n++ +N(d ) = F

This expression can be linearized using a Taylor expansion:

exp

11 1

11 1

( ) ( ) ( ) ....

( ) ( )

linear part of the ansion

i in n

i in n

++ +

++ +

= + ∂ ∂ +

≅ + T

N d N d N/ d ∆d

N d N d K ∆d

64444744448

with TK , the tangent stiffness.

From here on we derive the algorithm:

1 1

1 1

,

1, .

in n

in n

for each n

i i iterate as needed until Tol

+ +

+ +

−

= + − <

T

i+1 in+1 n+1

K ∆d = F N(d )

d = d + ∆d

F N(d )

The solution procedure is illustrated in figure 3.10. The application could correspond to a

loaded footing, for example.



Fig. 3.10 Newton-Raphson procedure

To better understand the procedure, let’s assume that equilibrium has been reached at tn

(Fig. 3.10, at “1.”) and that an increase of external load ∆F (Fig. 3.10, at “2.”) is

applied. The dashed part of curve N(d) is of course unknown at this point but we only

need a local tangent to proceed. This situation is illustrated in (Fig. 3.10, at “3.”), which

indicates that application of ∆F will lead to an increment of displacement ∆d, which leads

to a first estimate for dn+1 after one iteration. The procedure is then repeated with the

out-of balance force defined at (Fig. 3.10, at “4.”), till convergence to N(d) is reached

within a prescribed tolerance (Fig. 3.10, at “6.”).

As obvious from Fig. 3.10 there is no need to take the true tangent stiffness at each step

and iteration, which corresponds to the full Newton-Raphson scheme; alternative

schemes are possible: Initial or Constant Stiffness or Modified Newton-Raphson

(see Fig. 3.11).



NEWTON- RAPHSON & al.ITERATIVES SCHEMES

i: iterationn: step

1.Full NR, update KT at each step & iteration, till .TOLFi <∆

ndd

1F∆Fn

Fn+1

1

1+nd

•TonK 1+

1F∆2F∆

ndd

Fn

Fn+1iF∆

2.Constant stiffness,use KTo

till .TOLFi <∆

3.Modified NR, update KT

opportunistically, each step e.g.,till

KTo

.TOLFi <∆

4. BFGS, “optimal”secant scheme

NEWTON- RAPHSON & al.ITERATIVES SCHEMES

i: iterationn: step

1.Full NR, update KT at each step & iteration, till .TOLFi <∆

ndd

1F∆Fn

Fn+1

1

1+nd

•TonK 1+

1F∆2F∆

ndd

1F∆Fn

Fn+1

1

1+nd

•TonK 1+

ndd

1F∆Fn

Fn+1

1

1+nd

•TonK 1+

1F∆2F∆

ndd

Fn

Fn+1iF∆

ndd

Fn

Fn+1

ndd

Fn

Fn+1iF∆

2.Constant stiffness,use KTo

till .TOLFi <∆

3.Modified NR, update KT

opportunistically, each step e.g.,till

KTo

.TOLFi <∆

4. BFGS, “optimal”secant scheme

Fig. 3.11 Alternative iterative strategies

Fig. 3.12 Input of algorithmic parameters for Newton–Raphson and other algorithms



Algorithmic choices are made in ZSOIL under CONTROL/Control. Options described in

Fig. 3.12 are available. Full Newton-Raphson is the default option. Initial stiffness

and BFGS, a quasi Newton method are usually activated when a numerical instability is

detected, in order to pass a point where the tangent stiffness is critical. Modified

Newton-Raphson is an opportunistic alternative, which can be used, e.g., when

nonlinearity is moderate and reforming the stiffness too costly; stiffness is then reformed

from time to time, after some iterations, or after some time steps.

3.4.1 Convergence

Newton-Raphson iterations are repeated until convergence criteria are satisfied. The first

convergence criterion is the Euler norm of the right-hand-side of the linear system

solved at step n, which must be less than some preset tolerance (Fig. 3.12). By default

the tolerance is set to 1% of the initial out-of balance at step (n+1):

1 1i

n n Tolerance+ +− = ≤∑NDOFs i 2

n+1 n+1 kk=1F N(d ) [(F - N(d )]

Convergence of the energy of deformation is also tested and used to limit iterations

when internal energy is stationary, meaning that nothing is happening anyway.

Tolerances are set by default in ZSOIL (Fig. 3.12), but the user can modify default

values if needed.

The maximum number of iteration is automatically updated by ZSOIL, with an absolute

max. used to stop ill-conditioned runs, in particular during batch runs (Fig. 3.12).

Remark:

- Convergence tolerances are set separately for the solid phase, the liquid phase,

structures and interfaces; this is a must.



3.5 Geotechnical aspects

3.5.1 Initial state

Most in-situ analyses start with an existing constructed or at least deformed state, about

which we only know the actual geometry, that gravity was the dominant load over past

history, and that some long-term creep took place.

In order to compute the initial stress state we first compute the stress state

corresponding to gravity loads + all other loads existing at assumed initial time t=0

(existing constructions e.g.). We then apply the computed stress state as an initial stress

state and superpose both computed states. The result will be an undeformed state with

stresses corresponding to gravity; this will be our initial state. The implementation of the

procedure is described step by step in Table 3.7.

_______________________________________________________________________

Table 3.7: Initial state procedure

0

; is the initial stress

then by definition:

( ) , from which

( )

,

;

ext

ext

let

d

let gravity and t loads

yield

= =

∆ ∆ = − Ω

∆ ∆

∫ ∫

∫ ∫

∫

o o

T To

Ω Ω

T To

Ω Ω

ext1 Γ

TΓ

Ω

σ = σ + ∆σ σ

Kd B σdΩ = B σ + ∆σ dΩ F

K d =

compute ∆d due to

B σdΩ F B σ

F = F

K d = B σ gravitydΩ = F

0

0

1 1

2 0 0

2 2

;

superposition finally yields;

s and as results

let with

yields and as results

∆ ∆

−

∆

∆ − ∆ −

∫

∫

Γ Γ

ext Tσ Γ

Ω

Tσ

Ω

Γ Γ

compute ∆d due to gravity induced stress

applied as initial str

d = d σ = σ

F = B σ dΩ = F σ = σ

K d = B ∆σdΩ = F

d = d σ σ

ess

=

1 2+ = + − =1 2 o Γ Γ Γ Γd = ∆d + ∆d = 0; σ =

superpos

σ + ∆σ ∆σ σ σ σ σ

e

define initial stress as recomputed gravity stress

_______________________________________________________________________

The initial state procedure is automated in ZSOIL (Fig. 3.13). The initial state driver,

under CONTROL/Analysis & Drivers applies gravity step by step starting with a user

defined gravity load multiplier of 0.5 e.g., and raising it upto 1 by steps of amplitude 0.1

e.g. The idea is to apply gravity progressively in order to avoid initiating too much

plasticity at once.



Remarks:

- The procedure applies to nonlinear behavior and to 2-phase media.

- All loads associated with a load time function (ASSEMBLY/Load time function)

with nonzero value at t=0 are activated in the initial state procedure (Fig. 3.14).

Fig. 3.13 Initial state driver

Fig. 3.14 Load function with nonzero value at t=0



3.5.2 Locking in quasi incompressible media

Incompressible media are frequently encountered in soil mechanics and this type of

behavior leads to locking. Incompressible elasticity (ν ≅ 0.5) and incompressible plastic

flow are typically inducing such behavior.

1

2

N

1

2

N

Fig. 3.15 Locking mesh

In order to better understand the locking phenomenon let’s look at a simple example

[T.J.R. Hughes, 1987]. Fig. 3.15 shows a small finite element mesh, composed of two

triangular elements with linear interpolation of displacements. Linear interpolation means

constant displacements derivatives within elements and hence constant volume in case

of material incompressibility. With the given boundary conditions this means that, at

node N, element 1 requires a vertical displacement, and element 2 a horizontal

displacement. This will lead to locking and obviously locking will propagate throughout

the mesh if the mesh is extended in both directions with the same stencil. As a matter of

fact, imposing incompressibility adds constraints on the nodal displacements which have

less freedom to satisfy the differential equation of equilibrium.

Remedies to this problem are known: selective underintegration, BBAR elements, EAS

elements, stabilized formulations, or higher order elements. ZSOIL automatically selects

the most appropriate remedy available among BBAR elements, EAS elements, stabilized

formulations, but the user can impose his own preference through the following input

screen, see Fig. 3.16 (toggle Advanced version instead of Basic version in the menu

if needed).



Fig. 3.16 Element options to avoid locking and pressure oscillations

3.5.3 Spurious pressure oscillations in consolidation

A similar phenomenon also occurs in 2-phase media. [Vermeer & Verruijt, 1981]

established the existence of a lower bound to applicable ∆t to avoid pressure oscillations:

2

w

w

1; ( ) /

6where h is the element size, an algorithmic parameter,set to 1

in k is Darcy's coefficient, ZS and the water weightOIL,

v oed

v

ht C E k

Cγ

θθ

γ

•∆ ≥ =⋅

This barrier can however be overcome in ZSOIL with stabilization [Truty &

Zimmermann, 2006]; again activation/deactivation can be done by the user under

CONTROL/finite elements, see Fig. 3.16.



3.6 References

[Truty & Zimmermann, 2006] Stabilized mixed finite element formulations for materially

nonlinear partially saturated two-phase media, in Comput. Methods Appl. Mech. Engrg.

195 (2006), 1517-1546

[T.J.R. Hughes, 1987] The Finite Element Method, Prentice-Hall, 1987.

[Vermeer & Verruijt, 1981] An accuracy condition for consolidation by finite elements.

Int. J. Num. Anal. Meth. Geomech., 5, pp 1-14.

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CHAPTER 3. LINEAR AND NONLINEAR CONTINUA Contents · Chapter 3. Linear and Nonlinear Continua Page 44 CHAPTER 3. LINEAR AND NONLINEAR CONTINUA Contents 3.1 Linear elastostatics 45