2 Equations
2.1 Convection and diffusion of heat
2.1.1 Convection-diffusion equation
r C ( T + bi |
|
) =
k ( |
|
+
|
|
+
|
|
) -
a T + f
|
The primary unknown is the condif_temperature T. Further notation: r
group_condif_density; C group_condif_capacity; x space coordinate; bi
group_condif_flow in i-direction; k group_condif_conductivity;
a group_condif_absorption; f force_element_volume.
Typical applications are heat conduction and heat conduction in a flow.
2.1.2 Convection to environment
qc = ac ( T - Tc )
Here qc is the condif_convection heat flux, ac is the convection
coefficient and Tc is the environmental temperature for convection.
2.1.3 Radiation to environment
qr = ar ( T4 - Tr4 )
Here qr is the condif_radiation heat flux, ar is the radiation
coefficient and Tr is the environmental temperature for radiation.
2.2 Material deformation and flow
r vi = |
|
+
(1-b T) r gi -
d |
|
no sum on i
+ fi
|
Notations:
r materi_density; vi materi_velocity in i-direction;
sij materi_stress matrix;
x space coordinate; b group_materi_expansion_volume;
T (optional) condif_temperature;
gi force_gravity; d group_materi_damping coefficient;
fi force_element_volume.
The equation is given for space coordinates following the material
velocities vi.
TOCHNOG allows you to build your favorite material,
by adding separate contributions to the stresses sij.
In this way you can build solids or fluids or things in between.
The separate contributions will be listed below.
First two typical examples are given.
Nearly incompressible Navier Stokes:
...
materi_velocity
materi_stress
end_initia
...
group_type 0 -materi
group_materi_elasti_compressibility 0 1.0
group_materi_viscosity 0 1.2
Linear solid:
...
materi_velocity
materi_strain_total
materi_stress
end_initia
...
group_type 0 -materi
group_materi_elasti_young 0 1.e10
group_materi_elasti_poisson 0 0.2
group_materi_memory 0 -updated_without_rotation
The -updated Lagrange formulation
Deformations (i.e. the incremental deformation matrix F) refers to the previous time point.
TOCHNOG decomposes the incremental deformation tensor with a polar decomposition into F = R U with
F the incremental deformation matrix, R the incremental rotation matrix and
U the incremental stretch matrix.
The incremental stretch matrix U is used to determine the incremental
strain matrix 0.5(U+UT)-I with I the identity tensor.
The velocities between the two time points are the unknowns to be solved.
Stresses are calculated from adding incremental stresses to the old stresses.
Incremental stresses are caused by the incremental strain matrix and a rigid body rotation
by the incremental rotation matrix of the old stresses.
The -updated_without_rotation Lagrange formulation
Deformations (i.e. the incremental deformation matrix F) refers to the previous time point.
Any rigid body rotation between the two time points are neglected,
so TOCHNOG decomposes the incremental deformation tensor with a polar decomposition into F = U with
F the incremental deformation matrix, and U the incremental stretch matrix.
The velocities between the two time points are the unknowns to be solved.
Stresses are calculated from adding incremental stresses to the old stresses.
Incremental stresses are caused by the incremental strain matrix.
The -total Lagrange formulation
Deformations (i.e. the total deformation matrix F) refers to the time 0.
TOCHNOG decomposes the total deformation tensor with a polar decomposition into F = R U with
F the total deformation matrix, R the total rotation matrix and U the
total stretch matrix.
The total stretch matrix U is used to determine the total
strain matrix 0.5(U+UT)-I with I the identity tensor.
The displacements between the current time and time 0 are the unknowns to be solved.
Stresses are calculated from the total displacements.
The -total_piola Lagrange formulation
Deformations (i.e. the total deformation matrix F) refers to the time 0.
TOCHNOG decomposes the total deformation tensor with a polar decomposition into F = R U with
F the total deformation matrix, R the total rotation matrix and U the
total stretch matrix.
The total stretch matrix U is used to determine the total Green-Lagrange strain
matrix 0.5 ( FT F - I ) = 0.5 ( UT U - I )
Stresses are calculated from the total displacements.
The -total_linear Lagrange formulation
Deformations (i.e. the total deformation matrix F) refers to the time 0.
TOCHNOG neglects any rigid body rotations and uses linear engineering strains
0.5 ( F + FT ) - I.
The displacements between the time current time and time 0 are the unknowns to be solved.
Stresses are calculated from the total displacements.
A remark on the total Lagrange models.
Normally stresses are calculated from the total displacements (and thus the total strains).
The old stresses are not used.
This means that any initial stresses are neglected.
This type of stress calculation for the total Lagrange models
is used whenever materi_strain_total is not initialised.
However, in case materi_strain_total is initialised, the difference
between the total displacements (and thus strains) between two time points
is used to determine incremental stresses, which are added to the stresses at
the previous time point.
And thus, in case materi_strain_total is initialised, the old
stresses are used and any initial stresses are not neglected.
See also group_materi_memory.
2.2.2 Elasticity
The elastic stress rate is
Cijkl eklelas
where Cijkl is the elastic modulus tensor (which is a
doubly symmetric tensor: Cijkl=Cjikl, Cijkl=Cijlk and Cijkl=Cjilk),
and eklelas is the elastic strain rate.
See the plasticity section for a definition of the elastic strain rate.
For an isotropic material
with E group_materi_elasti_young modulus and n group_materi_elasti_poisson ratio
(the remaining non-zero moduli follow from the double symmetry conditions).
For a transverse isotropic material the material has one unique
direction (think of an material with fibers in one direction).
Here we take 'a' as the unique direction; 'b' and 'c' are
the transverse directions. The material is fully defined by Caaaa,
Cbbbb, Caabb, Cabab and Cbcbc and the unique direction
in space (see group_materi_elasti_transverse_isotropy).
The other non-zero moduli follow from Ccccc=Cbbbb,
Cacac=Cabab, Cbbcc=Cbbbb-2Cbcbc and from the
double symmetry conditions.
The nonlinear elasticity polynomials is a strain dependent model.
In this model, the 'young's stiffness' modulus is made dependend of the size
of the strains via a series of polynomials
E = E0 + E1 e1 + E2 e2 + ...
(1)
where
e = ( eij eij )
(2)
with eij the components of the strain matrix.
The parameters E0 etc. need to be specified in the
group_materi_elasti_young_polynomial record.
The power law nonlinear elasticity is a stress dependent model which typically is used
to model the elastic behavior of granular materials.
It can be combined with plastic models, by example with the di Prisco plasticity model for soils,
and with a poisson ratio.
In this model, the 'young's stiffness' modulus is made a function of the average stress state:
E = E0 (p/p0)a (3)
where p is the pressure.
Furthermore, E0 is the reference stiffness at reference pressure p0, and a is a soil dependent
power coefficient.
The parameters E0, p0, and a need to be specified in the group_materi_elasti_young_power record.
The Lade nonlinear elasticity is a stress dependent model which typically is used
to model the elastic behavior of granular materials.
It can be combined with plastic models, by example with the di Prisco plasticity model for soils.
The stress rates are linked to the strain rates by the equation:
where the function W is
where
X = p2 + R* abs( sij sij )
with pressure p=(s11+s22+s33)/3 and deviatoric stresses
sij = sij - p dij.
The model contains three user specified constants B, R, l
which need to be specified in the group_materi_elasti_lade record.
B and l are defined by means of an isotropic unloading test, and R by means of
an unloading-standard-triaxial-compression test.
For example for a loose sand B=1028, R=0.25, l=0.28.
See [7] for the details.
The model cannot be used in combination with a poisson ratio.
2.2.3 Elasto-Plasticity
Plastic strain
In plastic analysis, the materi_strain_elasti rate follows by subtracting from
the materi_strain_total rate the materi_strain_plasti rate
eijelas = eij -
eijplas
where the materi_strain_total rate is
The materi_strain_plasti rate follows from the
condition that the stress cannot exceed the yield surface.
This condition is specified by a yield function fyield(sij)=0.
The direction of the plastic strain rate is specified by the stress gradient
of a flow function ¶ fflow/¶ sij.
If the yield function and flow function are chosen to be the same,
the plasticity is called associative, else it is non-associative.
Von-Mises is typically used for metal plasticity.
Mohr-Coulomb and Drucker-Prager are typically used for soils and other
frictional materials.
The plasticity models can freely be combined;
the combination of the plasticity surfaces defines the
total plasticity surface.
First some stress quantities which are used in most of the plasticity models
are listed.
Equivalent Von-Mises stress:
Mean stress:
Deviatoric stress:
sij = sij - sm dij
Modified CamClay plasticity model
Here we give the equations of the Modified Cam Clay model
(Roscoe and Burland, 1968, summarized e.g. by Wood, 1990, see [12]).
All stresses are effective (geotechnical) stresses, i.e.compression is positive!
Definitions of variables:
p = (s11+s22+s33)/3
q = { |
|
[ (s11-s22)2 +
(s22-s33)2 +
(s33-s11)2 ]
+ 3 ( s122 + s232 + s312 ) }1/2
|
The CamClay yield rule, which is also the flow rule, reads:
f = g = q2 - M2 [ p (p0-p) ] = 0
M is a soil constant and p0 is a history (hidden) variable which
corresponds to the preconsolidation mean pressure.
The hardening function, evolution, of p0 reads:
in which
dvp = d11p+d22p+d33p
and l and k are user specified soil constants.
Further e is the void ratio with the evolution equation:
de = -dv (1+e)
in which
dv = d11+d22+d33
The poisson ratio n reads:
in which the elastic bulk modulus K is given by:
K = (1+e) p / k
and the Young's modulus E:
E = 2.*G*(1+n)
in which G is a user specified soil constant,
By using this n and E the classical isotropic stress-strain law is used to calculate the stresses.
The soil constants M, k, l need to be specified in
group_materi_plasti_camclay.
The soil constant G, need to be specified
in group_materi_elasti_camclay_g.
For an alternative see group_materi_elasti_camclay_poisson.
The history variables e, p0 need to be initialized by
materi_history_variables 2 record
(and given initial values in node_dof records).
Remark 1: An additional parameter N can be often found in textbooks on the Cam Clay model.
We don't include it since it is linked to other model parameters via:
1+e = N - l lnp0 + k ln(p0/p)
Remark 2: If you apply a geometrical linear analysis, see section 8.4, then also the calculation of de
void ratio development is linearized, and so will contain some error as compared to the exact void ratio change.
Hence for very large deformations, say above 10 percent or so, don't use such geometrical linear analysis.
Cap plasticity model
This model accounts for permanent plastic deformations under high pressures for
granular materials.
It is intended to be used in combination with shear plasticity models like
Drucker-Prager, Mohr-Coulomb, etc.
First a deviatoric stress measure t and hydrostatic stress measure p are defined
t = 3 s
p = - sm
See above for s and sm.
The yield rule for the group_materi_plasti_cap model reads:
f = |
(p-pa)2 +
|
é
ê
ê
ê
ê
ê
ë |
|
|
ù
ú
ú
ú
ú
ú
û |
|
|
- R ( c + pa tanf )
|
Here c is the cohesion and f is the friction angle which should be
taken equal to the values in the shear flow rule which you use.
The parameter pa follows from
where the hydrostatic compression yield stress pb is to be defined with an
table of volumetric plastic strains evp versus pb with
evp = e11p + e22p + e33p.
As always, positive strain denote extension whereas negative strains denote compression.
Associative flow is used, so the flow rule is taken equal to the yield rule.
Summarizing the group_materi_plasti_cap model needs as input the cohesion c,
the friction angle f, the parameter a (typically 1. 10-2
up to 5. 10-2), and a table evp versus pb.
Compression limiting plasticity model
This group_materi_plasti_compression model uses a special
definition for the equivalent stress
s =
s12 + s22 + s32
where s1, s2 and s3 are the first, second and third
principal stress respectively.
Each of these is only incorporated if it is a compression stress.
The model now reads
s - sy = 0
This plasticity surface limits the allowed compression stresses.
Tension limiting plasticity model
This group_materi_plasti_tension model uses a special
definition for the equivalent stress
s =
s12 + s22 + s32
where s1, s2 and s3 are the first, second and third
principal stress respectively.
Each of these is only incorporated if it is a tension stress.
The model now reads
s - sy = 0
This plasticity surface limits the allowed tension stresses.
A simple model for concrete can be obtained as follows.
Use group_materi_plasti_tension to limit the tension strength ft.
Use group_materi_plasti_compression to limit the compressive strength fc.
The tension strength could be softened to zero over an effective
plastic strain k of, say, 1 percent.
The compressive strength could be softened to zero over an effective plastic
strain k of, say, 10 percent.
Another possibility for concrete is to combine group_materi_plasti_tension to limit the tension strength ft,
and use the group_materi_plasti_vonmis model
to limit the compressive strength fc.
di Prisco plasticity model
The di Prisco model is an non-associative plastic model for soils,
which can be typically combined with the 'Lade elastic model'.
This di Prisco model is a rather advanced soil model, which is explained in more detail in
[4] and [5].
The yield rule reads:
f = 3 bf (g - 3) ln |
æ
ç
ç
è |
|
|
ö
÷
÷
ø |
- g J3 h* +
|
|
( g - 1 ) J2 h*
|
and the flow rule yields:
g = 9 ( g - 3 ) ln |
æ
ç
ç
è |
|
|
ö
÷
÷
ø |
- g J3 h* +
|
|
( g - 1 ) J2 h*
|
This is an anisotropic model in which the first and second invariant of the
stress rate h* are defined relative to the rotation axes c.
r = sij cij
J3h* = hij* hjk* hki*
J2h* = hij* hij*
where s* follows from
shk* = shk* - r chk
Further rg=1.
The history variables are cij ( rotation axes, 9 values),
b (yield surface form factor), and rc (preconsolidation mean pressure).
The evolution laws for these history variables can be found in the papers listed above.
The history variables cij (9 values), b, rc need to be initialized by
the materi_history_variables 11 record
(and should be given initial values in node_dof records).
In a normally consolidated sand with isotropic initial conditions
cij = dij / 3 , b=0.0001
and rc equals 3 times the means pressure.
The total model, yield rule and flow rule and evolution laws for history variables,
contains a set of soil specific constants.
In group_materi_plasti_diprisco you need to specify these constants.
These constants are explained in more detail in the papers mentioned above, but here
we give a short explanation.
The constants qc, qe, xc and xe are linked to the dilitancy
and the stress state during failure (standard triaxial compression and extension test
in drained conditions).
The constants g, cp, bf and bf0 are defined by means of the experimental
curves ( q-eaxial, evol-eaxial) obtained by performing
a standard compression test in drained conditions.
Moreover, bf, bf0 and tp can also be determined by means of the effective-stress
path obtained by performing a standard triaxial compression test in undrained conditions.
A cohesion C can be also be introduced if required.
Finally bp can determined from an isotropic compression test.
For a loose sand qc=0.253, qe=0.0398, xc=-0.2585,
xe=-0.0394, g=3.7, cp=18., bf=0.5, bf0=1.1, tp=10., and bp=0.0049.
Drucker-Prager plasticity model
The group_materi_plasti_druckprag model reads
3 a sm + s - K = 0
a = |
2 sin( f ) |
|
3 ( 3 - sin(f) ) |
|
K = |
6 c cos( f ) |
|
3 ( 3 - sin(f) ) |
|
Here c is the cohesion, which needs to be specified both for the yield
function and the flow rule; by choosing different values
non-associative plasticity is obtained.
Gurson plasticity model
The group_materi_plasti_gurson model reads
|
|
+
2 q1 f* cosh( q2 |
|
) -
(1 + ( q3 f* ) 2 ) = 0
|
Here f* is the volume fraction of voids.
The rate equation
f* = ( 1 - f*) f* ekkplas
defines the evolution of f* if the start value for f* is specified.
Furthermore, q1, q2 and q3 are model parameters.
HLC plasticity model
The group_materi_plasti_hlc Hau-Liu-Chang model is nearly similar
to the Gurson's equation, and it is reads
|
|
+
f* ( + |
|
) m1 exp( |
|
) - 1 = 0
|
Here f* is the volume fraction of voids.
The rate equation
f* = ( 1 - f*) f* ekkplas
defines the evolution of f* if the start value for f* is specified.
The variable m20 is a function of the porosity:
m20 = ( m21 - m22 f*)
exp( m21 |
|
)
|
Furthermore, sy, m21, m22, m23 and m1 are model parameters.
The parameters can be calculated numerically by simple numerical simulations.
See http://www.tam.nwu.edu/wkl/paper/suhao-paper2.html for further info
about the model.
Von-Mises plasticity model
The group_materi_plasti_vonmis model reads
3 s - sy = 0
where without hardening the yield value is fixed sy = sy0 .
If however the group_materi_plasti_vonmis_nadai hardening law for
Von-Mises plasticity is specified then
sy = sy0 + C ( k_0 + k ) n
where C, k_0 and n are parameters for the hardening law,
and k is the isotropic hardening parameter (see later).
The parameter sy0 is specified by group_materi_plasti_vonmis.
Modified Matsuoka-Nakai model plasticity model
The Matsuoka-Nakai model [8] is a perfectly plastic model
thus the fixed yield surface represents the failure surface as well.
The model is based on experimental results with soils and can be formulated
in terms of three stress invariants
where
I1 |
= |
tr(sij) = s11+s22+s33 =
s1 + s2 + s3 = 3 sm |
I2 |
= |
|
|
( |
tr ( sikskj ) - I12
|
) |
= -s1 s2 - s2 s3 - s3 s1 |
|
I3 |
= |
det(sij) = s1 s2 s3 |
s1, s2 and s3 are the principal stresses
(all stresses are effective; compressive stresses are negative).
The parameter f is equal to the angle of internal friction in axisymmetric
(triaxial) compression [11].
For axisymmetric stress states the Matsuoka-Nakai model corresponds to the
Mohr-Coulomb model. Nevertheless, the Matsuoka-Nakai model is described by
a smooth surface in the stress space and thus it is more
suitable from the computational aspect.
When the cohesion c is considered in the model, the yield condition is
formulated for a modified stress [9]
sij = sij-s0dij
with
s0 = c cotf .
Notice that straightforward usage of the function f above also as flow rule g would
lead to a loss of the information on f in the derivative of g, since the f only appears
in a constant.
For that reason we apply as flow rule g the Drucker-Prager function (see elsewehere in this
theoretical part).
A separate fflow can be specified which enters this Drucker-Prager g.
Since we use the Drucker-Prager function as flow rule g,
the present plasticity model has been named Modified Matsuoka-Nakai.
Mohr-Coulomb plasticity model
The group_materi_plasti_mohrcoul model reads
0.5 ( s1 - s3 ) + 0.5 ( s1 + s3 ) sin( f )
- c cos( f ) = 0
Here c is the cohesion, s1 is the maximal principal stress
and s3 is the minimal principal stress.
The angle f needs to be specified for
both the yield condition and the flow rule; by choosing different
values, non-associative plasticity is obtained.
For a numerically more stable solution, consider
using Matsuoka-Nakai plasticity in stead of Mohr-Coulomb.
Mohr-Coulomb softening plasticity model
The group_materi_plasti_mohrcoul_softening model is the same as the standard Mohr-Coulomb model.
Now, however, the parameters c and f (both for the yield rule and
for the flow rule) are softened on the the effective plastic strain k.
By example, for the cohesion a linear variation is taken between the
initial value c0 at k=0,
up to c1 at a specified critical value of k,
and constant c1 for larger values of k.
The same is done for f for the yield rule and for the flow rule.
Isotropic Hardening
The size of the plastic strains rate is measured
by the materi_plasti_kappa parameter
k = 0.5 eijplas eijplas
This parameter can be used for isotropic hardening.
Use the dependency_diagram for this.
Kinematic Hardening
The materi_plasti_rho matrix rij, governs the kinematic
hardening in the plasticity models.
It is used in the yield rule and flow rule to get a new origin by using the
argument sij - rij:
fyield = fyield(sij - rij)
fflow = fflow(sij - rij)
where the rate of the matrix rij is taken to be
rij = a eijplas
where a is a user specified factor
(see group_materi_plasti_kinematic_hardening).
Plastic heat generation
The plastic energy loss can be partially turned into heat rate per unit
volume q:
q = h sij eijplas
where h is a user specified parameter (between 0 and 1)
specifying which part of the plastic energy loss is turned into heat
(see group_materi_plasti_heatgeneration).
2.2.4 Hypo-Plasticity
In hypoplasticity a direct relation is used between strain rates and stress rates.
Specifically:
sij = Lijkl eij +
Nij ekl ekl
Here the part with Lijkl gives a linear relation between strain rates and
stress rates and the part with Nij gives a nonlinear relation.
The constitutive tensors Lijkl and Nij are functions of the effective
stress tensor sij and void ratio e.
The effective stress tensor sij follows from the total stress tensor
sij minus any pore pressures (see groundflow).
Rigid body rotations (objectivity) are treated elsewhere (see the section on memory).
Basic law, Wolffersdorff
The law proposed by Wolffersdorff [11] is used.
|
Lijkl = fs |
|
|
( |
F2 Iijkl + a2 sijskl |
) |
|
|
Nij = fs fd |
|
|
( |
sij + sij* |
) |
|
|
|
with sij = sij / (smndmn) ,
sij* = sij - |
|
dij ,
Iijkl = dik djl , |
|
|
|
|
|
|
|
|
|
tany=3sij*sij* ,
cos3q=-6 |
|
. |
|
For sij*=0 is F=1.
The scalar factors
fs and fd take into account the influence of mean pressure and
density:
fs |
= |
|
|
æ
ç
ç
è |
|
ö
÷
÷
ø |
|
|
|
æ
ç
ç
è |
- |
|
|
ö
÷
÷
ø |
|
|
é
ê
ê
ê
ê
ê
ë |
3+a2-a3
|
æ
ç
ç
è |
|
|
ö
÷
÷
ø |
|
|
ù
ú
ú
ú
ú
ú
û |
|
, |
|
fd |
= |
|
Three characteristic void ratios -- ei (during isotropic compression at the
minimum density), ec (critical void ratio) and ed (maximum density) --
decrease with mean stress:
|
|
= |
|
=
|
|
= exp |
é
ê
ê
ê
ê
ê
ë |
-
|
æ
ç
ç
è |
- |
|
|
ö
÷
÷
ø |
|
|
ù
ú
ú
ú
ú
ú
û |
The range of admissible void ratios is limited by ei and ed.
The model parameters can be found in Tab. 1.
They correspond to Hochstetten sand from the vicinity of Karlsruhe, Germany
[11].
j [°] |
hs [MPa] |
n |
ec0 |
ed0 |
ei0 |
a |
b |
33 |
1000 |
0.25 |
0.95 |
0.55 |
1.05 |
0.25 |
1.0 |
Table 1:
Basic hypoplastic parameters of Hochstetten sand.
The basic law parameters should be specified in
group_materi_plasti_hypo_wolffersdorff.
Cohesion
A simplistic appraoch to include cohesion is used here.
Instead of feeding the real effective stress state sij into the hypoplastic
law, an alternative effective stress state sijc is used.
Cohesion is modelled by subtracting in each of the normal stress
components a value c representing cohesion: s11c = s11 - c,
s22c = s22 - c and s33c = s33 - c.
The shear stresses are not altered: s12c = s12, etc.
The cohesion value should be specified in
group_materi_plasti_hypo_cohesion.
Intergranular strains
In order to take into account the recent deformation history, an additional
tensorial state variable Sij1 is introduced.
Denoting the normalized magnitude of Sij
(R is a material parameter) and the direction of Sij
(Sij=0 for Sij=0),
the evolution equation for the intergranular strain tensor reads:
Sij = |
ì
í
î |
( Iijkl-rbrSij Skl)ekl |
for |
Sijeij > 0 |
eij |
for |
Sijeij £ 0 |
|
|
,
|
where Sij is the objective rate of intergranular strain.
Rigid body rotations are treated elsewhere (see the section on memory).
From the evolution equation (2.2.4) it follows that r must remain
between 0 and 1.
The general stress-strain relation is now written as
The fourth order tensor
Mijkl represents the incremental stiffness and is calculated from the
hypoplastic tensors
Lijkl and Nij
which may be modified by scalar multipliers mT and mR,
depending on r and on the product Sijeij:
Mijkl |
= |
[ rc mT + (1-rc)mR ] Lijkl + |
|
+ |
ì
í
î |
rc (1-mT) Lijmn Smn Skl
+ rc Nij Skl |
for |
Sijeij > 0 |
rc (mR-mT) Lijmn Smn Skl |
for |
Sijeij £ 0 |
|
|
|
c is an additional material parameter.
An example intergranular parameters can be found in Tab. 2.
R |
mR |
mT |
br |
c |
1· 10-4 |
5.0 |
2.0 |
0.50 |
6.0 |
Table 2:
Example of Intergranular hypoplastic parameters.
The intergranular parameters should be specified in
group_materi_plasti_hypo_intergranularstrain.
Also you need to include materi_strain_intergranular in the initialisation part.
Pressure dependent initial void ratio
You can correct the initial void ratio e0, as specified in the initial value
for the history variable in the node_dof records, for the initial pressure
to obtain a corrected initial void ratio e.
|
|
= exp |
é
ê
ê
ê
ê
ê
ë |
-
|
æ
ç
ç
è |
- |
|
|
ö
÷
÷
ø |
|
|
ù
ú
ú
ú
ú
ú
û |
See the basic law description for the parameters hs and n.
The sij denotes the effective stress tensor (total stresses minus any
groundflow pressure).
This pressure dependent initial void ratio correction can be activated
by group_materi_plasti_hypo_pressuredependentvoidratio.
After the initial void ratio has been established, the development of the void
ratio is governed by volumetric compression or extension of the granular
skeleton.
In the presence of materi_damage d,
the materi_stress follows:
sijdamaged = (1-d) sijundamaged
For the damage, the group_materi_damage_mazars model is available:
d = dt ab + dc (1 - a )b
where
dt = 1. - (1-at) |
|
-
at e -bt(eeq - e0)
|
and
dc = 1. - (1-ac) |
|
-
ac e -bt(eeq - e0)
|
Here eeq contains the positive principal strains.
The parameter a is given by the ratio eeq/e,
where e contains the total strains (both negative and positive).
The parameter e0 is the strain threshold for damage;
other material parameters are b , at , bt , ac , bc.
Typically for concrete:
1.e-4 < e0 < 3.e-4 ;
b = 1. ;
1 < at < 1.5 ;
500 < bt < 2000 ;
0.7 < ac < 1.2 ;
e4 < bc < 5 e4
You can combine damage freely with plasticity models
or other material behavior.
2.2.6 Average stress (hydrostatic compressibility)
An extra average stress contribution on each of s11, s22
and s33 is
where co is the group_materi_elasti_compressibility, which should not be 0.
This pressure term can e.g. be used to model
nearly incompressible fluids.
The compressibility contribution should be combined
with a contribution for the deviatoric stresses (e.g. group_materi_viscosity).
2.2.7 Thermal stresses
Temperature rates cause fictitious thermal strain rates
- a T dij
where dij=1 if i=j
else dij=0
where a is the group_materi_expansion_linear coefficient and T is
the condif_temperature.
These fictitious thermal strain rates in turn lead to stress rates.
2.2.8 Hyper elasticity
Hyper elasticity is used to model rubbers.
It should be combined with a total lagrange formulation for the memory
of the material (so use -total or -total_piola for
group_materi_memory).
The stresses follow from a strain energy function (with Cij components of the matrix C, and
where F is the deformation tensor and U is the stretch tensor following
from the polar decomposition of the deformation tensor)
C = FT F = UT U
Deviatoric contributions
To obtain a purely deviatoric function, the following strain measures
are used (with I1, I2 and I3 the first, second and third
invariant of the elastic strain matrix C respectively)
The group_materi_hyper_besseling function reads ( with K1, K2
and a user defined constants)
W = K1 ( J1 - 3 ) a+ K2 ( J2 - 3 )
The group_materi_hyper_mooney_blatz_ko function reads (with G and b user
defined constants)
W = G * 0.5 * ( I1 - 3.0 + (2.0/b) ( J-b - 1. ) );
This Blatz-Ko hyperelastic material hardens in compression, and softens slightly in
tension; it models a foamlike rubber.
The group_materi_hyper_mooney_rivlin function reads (with K1 and K2 user
defined constants)
W = K1 ( J1 - 3 ) + K2 ( J2 - 3 )
The group_materi_hyper_neohookean function reads (with K1 a user defined constant)
W = K1 ( J1 - 3 )
The group_materi_hyper_reducedpolynomial function reads (with Ki user defined constants)
W = Ki ( J1 - 3 )i
where a summation over i = 1,2,... is applied.
Volumetric contributions
We define J = I3 .
Now a volumetric part can be added to the strain energy.
The group_materi_hyper_volumetric_linear contribution reads:
The group_materi_hyper_volumetric_murnaghan contribution reads:
The group_materi_hyper_volumetric_polynomial contribution reads:
for i=0,1,....
The group_materi_hyper_volumetric_simotaylor contribution reads:
The group_materi_hyper_volumetric_ogden contribution reads:
As an example, you can combine the group_materi_hyper_mooney_rivlin
energy function with the group_materi_hyper_volumetric_linear so that
the total strain energy function becomes:
W = K1 ( J1 - 3 ) + K2 ( J2 - 3 ) + |
|
(J-1)2
|
Here the initial shear modulus and bulk modulus are included as:
initial shear modulus = 2 ( K1 + K2 )
and
initial bulk modulus = K
respectively.
2.2.9 Viscoelasticity
Viscoelasticity is modeled with n parallel group_materi_maxwell_chains.
Each of the chains contains a spring with stiffness Em in line with a dash pot
with relaxation time tm (m indicates the m-th maxwell chain).
The viscoelastic stress rate is given by (with Cijklm is the elastic
tensor modulus of the m-th maxwell chain (depending on Em and the
poisson ratio))
2.2.10 Viscoplasticity
Viscoplasticity is a model for rate-dependent plasticity.
Rate dependent plasticity is important for (high-speed)
transient plasticity calculations. It should be used in combination
with a plasticity law. Viscoplasticity influences the stresses
via the plastic strains.
The group_materi_plasti_visco_exponential model reads
where g and a are material fluidity constants
and p is the pressure.
This model was first developed for visco-plastic soil behaviour.
The group_materi_plasti_visco_power model reads
where h (fluidity constant), fref (reference value for
yield function, e.g. the yield stress for Von-Mises plasticity)
and p (power) are user specified parameters.
The viscous contribution to the total stress is
2 n Dij
where
and divergence is neglected since we only model slightly compressible flows.
Viscous heat generation
The viscous energy loss is turned into heat rate per unit volume q:
q = 2 n Dij Dij
See group_materi_viscosity_heatgeneration.
2.3 Contact analysis
2.3.1 Penalty formulation
In contact analysis, normal forces Fn follow from the condition that
bodies cannot penetrate each other.
Since we use a penalty formulation, the normal force is given by
Fn = l un
where u is the penetration and l is called
the contact_penalty_velocity because its generates
forces on the velocity unknowns.
You can also impose groundflow_pressure and condif_temperature contact conditions
by respectively the penalty factors contact_penalty_pressure and
contact_penalty_temperature.
2.3.2 Friction and frictional heat generation
This normal force leads to a friction force Ff which equals
Ff = n Fn
where n is the friction coefficient (see contact_friction.
The friction force causes heat generation rate Q:
Q = h Ff vf
where vf is the slip velocity, and the factor h is a user
specified factor which determines which part of the frictional energy
loss is transformed into heat (h is between 0 and 1;
see contact_heatgeneration).
2.4 Ground water flow
2.4.1 Storage equation
The hydraulic pressure head h follows from the storage equation:
C h =
( k1p |
|
+
k2p |
|
+
k3p |
|
) +
|
|
+ f
|
Primary unknown is the hydraulic pressure head groundflow_pressure. Further notation:
C group_groundflow_capacity;
kip group_groundflow_permeability in i-direction;
xi space coordinate;
vi material velocity (if present);
f force_element_volume (hydraulic pressure source).
The equation is given for space coordinates following material
velocities vi (if present).
Groundflow velocities
The groundflow velocities, after initializing groundflow_velocity, follow from:
Total groundwater pressure
The total groundwater pressure is by example needed to calculate the total
stresses in soils (total soil stress = effective soil stress + total
groundwater pressure).
The total groundwater pressure follows from:
ptotal = h - r g z
where g is the gravitational acceleration,
and r is the groundflow_density
(Please notice that z typically is a negative number).
Static groundwater pressure
The static pressure due to gravity is:
pstatic = r g D z
where the D z is the distance to the groundwater level, the phreatic level.
The phreatic level needs to be specified with the groundflow_phreaticlevel record.
If that groundflow_phreaticlevel record is not specied, the static
pressure part is not used, so that the static pressure becomes zero.
Dynamic groundwater pressure
The dynamic groundwater pressure follows from
pdynamic = ptotal - pstatic
Boundary conditions
If the groundwater velocity is 0 normal to an edge (say at the interface
with a rock layer it is zero), then you should prescribe nothing on that edge
(Tochnog will then take care of that boundary condition for you).
At the phreatic level where the groundflow meets free air the hydraulic pressure head
should become r g z.
You can either set this yourself by using bounda_unknown combined with
bounda_time or else demand that Tochnog automatically does it for you
by activating the option groundflow_phreaticlevel_bounda.
At edges where you have some other hydraulic head you need to
specify that head yourself with bounda_unknown and bounda_time records.
If gravity is not of importance, e.g. in biomechanics where the storage
equation is used to model fluid transport in soft tissues,
the hydraulic pressure head h is equal to the total pressure,
and thus is zero at edges where the water meets the free air.
In this case, set h to zero by using bounda_unknown combined with
bounda_time.
Postprocessing
For all printing, plotting etc. you normally get the hydraulic pressure head h
since it is the primary unknown solved in the storage equation.
The total pressure, static pressure and dynamic pressure
you can get by the post_calcul option.
Naming conventions
To connect better to conventional naming, we rematk that the capacity depends
on the porosity n and water compressibility b:
C = n b
and for the permeability:
where ki is the permeability in i-direction.
Consolidation analysis
Look in the 'Consolidation' section of the 'Interaction analyzes
and advanced analyzes' chapter in the end of this manual on how
to perform consolidation analyzes (combined groundwater flow
with soil stress analyzes).
2.5 Wave equation
The primary unknowns are the wave_scalar s and its first time
derivative wave_fscalar s (
as TOCHNOG only solves first order in time equations, the first time
derivative of s also becomes primary unknown in order to turn this second
order in time equation into a set of first order in time equations).
Further notation: x space coordinate, t time and c speed of sound.