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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
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Chapter 3. Linear and Nonlinear Continua Page 45
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
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Chapter 3. Linear and Nonlinear Continua Page 46
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
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Chapter 3. Linear and Nonlinear Continua Page 47
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
= = =
− − + = − + −
+ = +
= +
_______________________________________________________________
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_______________________________________________________________
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.
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Chapter 3. Linear and Nonlinear Continua Page 49
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
_______________________________________________________________
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Chapter 3. Linear and Nonlinear Continua Page 50
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∆
σ∆perfect plastisticity
ε
E
σ
σy
EepH’
softening
hardening
eε∆pε∆
σ∆perfect plastisticity
ε
E
σ
σy
E
σ
σy
EepH’
softening
hardening
eε∆pε∆
σ∆perfect plastisticity
∆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
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Chapter 3. Linear and Nonlinear Continua Page 51
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.
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Chapter 3. Linear and Nonlinear Continua Page 52
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.
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Chapter 3. Linear and Nonlinear Continua Page 53
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.
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Chapter 3. Linear and Nonlinear Continua Page 54
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
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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:
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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
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Chapter 3. Linear and Nonlinear Continua Page 57
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
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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
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_______________________________________________________________________
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
_______________________________________________________________________
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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.
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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.
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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.
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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).
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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
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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.
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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.
Getting started with ZSOIL.PC
Chapter 3. Linear and Nonlinear Continua Page 67
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
Getting started with ZSOIL.PC
Chapter 3. Linear and Nonlinear Continua Page 68
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).
Getting started with ZSOIL.PC
Chapter 3. Linear and Nonlinear Continua Page 69
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.
Getting started with ZSOIL.PC
Chapter 3. Linear and Nonlinear Continua Page 70
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.