A CONTINUUMMECHANICAL MODEL FOR FESIMULATIONS OF STEEL COSTRUCTIONS IN FIRE
ABSTRACT This paper presents a thermoviscoplastic constitutive model within the framework of continuum mechanics for the description of the behavior of structural steel subjected to fire loading. The model is represented through a set of ordinary differential equations and belongs to the class of socalled unified theories which do not separate plastic and creep strains. The material parameters are identified for the structural steel S 235 on the basis of tension and compression tests at constant temperature with varying strain rates and verified against transient creep test with varying temperature. For effective numerical simulations with the finite element program ABAQUS the model is prepared with a stress algorithm and consistent tangent operator. The simulations are compared with experimental data of fire resistance tests on columns. 1 ITRODUCTIO For the design of steel structures in fire the common material model of EC312 [1] is formulated as an onedimensional rateindependent relationship between stress and mechanical strain. It can be shown that the material behavior of carbon steel is well described for monotonically increasing temperatures and constant loading. But in case of fire, structures are also exposed to varying thermal and mechanical boundary conditions requiring additional considerations. For the simulation of these processes a model should be chosen that exhibits inelastic properties and is capable to describe creep and relaxation at any heating or cooling condition. Therefore, a thermoviscoplastic material model for the simulation of structural steel members under fire conditions has been developed.
 Citations (2)
 Cited In (0)

Article: A remark on the application of the NewtonRaphson method in nonlinear finite element analysis
[Show abstract] [Hide abstract]
ABSTRACT: Usually the notion NewtonRaphson method is used in the context of nonlinear finite element analysis based on quasistatic problems in solid mechanics. It is pointed out that this is only true in the case of nonlinear elasticity. In the case of constitutive equations of evolutionarytype, like in viscoelasticity, viscoplasticity or elastoplasticity, the MultilevelNewton algorithm is usually applied yielding the notions of global and local level (iteration), as well as the consistent tangent operator. In this paper, we investigate the effects of a consistent application of the classical NewtonRaphson method in connection with the finite element method, and compare it with the classical MultilevelNewton algorithm. Furthermore, an improved version of the MultilevelNewton method is applied.Computational Mechanics 06/2005; 36(2):100116. · 2.43 Impact Factor  SourceAvailable from: Kirk Valanis[Show abstract] [Hide abstract]
ABSTRACT: A new measure of intrinsic time is introduced which broadens the endochronic theory and lends it a wider predictive scope. Idealized plastic models are shown to be constitutive subsets of the general theory and the phenomenon of yield is proved to be a consequence of a particular definition of the intrinsic time measure in terms of the plastic strain tensor. Various versions of the classical plasticity theory are shown to be asymptotic cases of the endochronic theory. In particular the kinematic hardening model, the isotropic hardening model as well as their combinations, are derivable directly from the general theory. In addition, the translation vector of the yield surface in stress space is found to be a constitutive property given by a linear functional of the history of the plastic strain. The Prager and Ziegler rules are immediately obtainable as special cases. It is also shown that the essential features of plastic response under conditions of 'stress reversals', are contained in the constitutive equation whose form remains invariant of the deformation history. It is believed that this is the first time that one single constitutive equation has been shown to predict correctly the essential features of plastic response under conditions of loading, unloading and reloading. However, when the intrinsic time measure is precisely equal to the norm of the increment of the plastic strain tensor, the constitutive unity, spoken of above, no longer exists.Archives of Mechanics 04/1978; 32:68. · 0.57 Impact Factor
Page 1
7th International Conference on Structures in Fire
M. Fontana, A. Frangi, M. Knobloch (Eds.)
Zurich, Switzerland, June 68, 2012
A CO?TI?UUMMECHA?ICAL MODEL FOR FESIMULATIO?S OF
STEEL CO?STRUCTIO?S I? FIRE
Sebastian Hauswaldt*, Manfred Korzen* and Stefan Hartmann**
* BAM Federal Institute for Materials Research and Testing, Unter den Eichen 87, 12205 Berlin,
Germany. emails: sebastian.hauswaldt@bam.de, manfred.korzen@bam.de
** Institute of Applied Mechanics, Clausthal University of Technology, AdolphRoemerStr. 2a,
38678 ClausthalZellerfeld, Germany. email: stefan.hartmann@tuclausthal.de
Keywords: Constitutive Equation, Viscoplasticity, Endochronic Plasticity, UMAT, Finite Element
Method, Fire Resistance
Abstract. This paper presents a thermoviscoplastic constitutive model within the framework of
continuum mechanics for the description of the behavior of structural steel subjected to fire loading. The
model is represented through a set of ordinary differential equations and belongs to the class of socalled
unified theories which do not separate plastic and creep strains. The material parameters are identified
for the structural steel S 235 on the basis of tension and compression tests at constant temperature with
varying strain rates and verified against transient creep test with varying temperature. For effective
numerical simulations with the finite element program ABAQUS the model is prepared with a stress
algorithm and consistent tangent operator. The simulations are compared with experimental data of fire
resistance tests on columns.
1 I?TRODUCTIO?
For the design of steel structures in fire the common material model of EC312 [1] is formulated as
an onedimensional rateindependent relationship between stress and mechanical strain. It can be shown
that the material behavior of carbon steel is well described for monotonically increasing temperatures and
constant loading.
But in case of fire, structures are also exposed to varying thermal and mechanical boundary
conditions requiring additional considerations. For the simulation of these processes a model should be
chosen that exhibits inelastic properties and is capable to describe creep and relaxation at any heating or
cooling condition.
Therefore, a thermoviscoplastic material model for the simulation of structural steel members under
fire conditions has been developed.
2 A THERMOVISCOPLASTIC MATERIALMODEL
Within the framework of small deformations the deviatoric part TD of the stress tensor T is assumed
to be additively decomposed into two parts, i.e.
with an equilibrium stressTeq and an overstressTov with the deviatoric decomposition
1 3 ( )
=−
TTT Itr
with the trace
( ).
=
T
ii
trT
The spherical part of the stress tensor is assumed to be defined using the trace operator
classical thermoelastic relationship
=+
TTT
DD
eq
D
ov
(1)
D
(.) tr
via the
Page 2
Sebastian Hauswaldt, Manfred Korzen and Stefan Hartmann
( )T3()
m
trKtr
=
E
(2)
with the mechanical strain tensor
the absolute temperatureθ and reference temperature 0,
of the bulk modulus K and thermal expansion coefficient .
In view of assigning specific material properties to the stress components in eq. (1), see [3], Teq will
be assumed to be a rateindependent functional whereas Tov has to be a ratedependent functional of the
process history.
Based on the theory of endochronic plasticity [4] the equilibrium stress is assumed to be described by
[]
m0
α θθ
=−−
EE1
(3)
θ the strain tensor E and the material parameters
α
where ()ɺdenotes the material time derivative, with the strain deviator
the limit shear modulus Gt and the hardening parameter
differential equation
2[2](0)0
β
=−−=
T
ɺ
E
ɺ
TET
ɺ
z
D
eq
DD
eq
DD
eqst
GGwith
(4)
,ED the initial shear modulus Gs,
β The variable z is defined through the
.
with the additional material parameter κ constrained to0
rateindependent functional for the equilibrium stress. It is characterized by the fact that the strain is not
decomposed into elastic and plastic parts and that no yield surface and corresponding loading conditions
have to be defined as it is the case in classical elastoplastic constitutive models with a yield function.
The overstress is assumed to be defined by the differential equation
2
κ
Gs
=−
E
ɺ
T
ɺ
ɺ
z
DD
eq
(5)
1
κ≤< (see Fig. 1). Eqs. (4) and (5) define a
with the instantaneous shear modulus G0 and relaxation timeλ , which is given as a function of the
overstress
D
ov
D
ov
DD
eq
D
ov0
1
λ
2Gwith (0)0
= −+−=
T
ɺ
TE
ɺ
T
ɺ
T
(6)
D
ov2
D
ov01
3
a
)aa 1 tanh
−
a
λ
+
=+
T
(T
. (7)
Taking into account that TɺD
of viscoelasticity with respect to the equilibrium stress, i.e. TD
or for a relaxation process at constant strain as time is going to infinity.
All material parameters, i.e. K,
temperature . θ
eqis defined by (4), eq.(6) can be interpreted as a modified Maxwell model
ovvanishes asymptotically for slow motions
,
α Gs, Gt,
,
β
, κ G0, a0, a1, a2 and a3 may depend on the
3 U?IAXIAL STRESS CASE
For analyzing the proposed constitutive model in more detail, we apply it to the situation of simple
extension with a uniaxial stress state.
Taking into account the relationships
Page 3
we arrive at we arrive at we arrive at we arrive at we arrive at we arrive at we arrive at
w
response of the equilibrium stress
with the
Fig. Fig.Fig.
a)a)
elasticity E
for different input functions characterized by the fact for different input functions characterized by the fact
1 ε ɺ
ε ɺ
0,
=
0,
creep for creep for creep for creep for creep for σ
σ ɺ
σ =
0
=
and nd nd an
with the mechanical s
response of the equilibrium stress
with the with thewith the with the associated associatedassociatedassociatedassociatedassociated material parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters E material parameters Ematerial parameters Ematerial parameters Ematerial parameters Es, E
Figure 1. Figure 1. Figure 1. Figure 1. Figure 1. Figure 1. Stress
elasticity Eelasticity E
for different input functions characterized by the fact
1 discontinuously the magnitude odiscontinuously the magnitude o discontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude o discontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude o discontinuously the magnitude odiscontinuously the magnitude or type of input rate
ith the mechanical s
response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress
A springA spring
2a. 2a.
Please note Please note
elasticity E
for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact that they change after a period of constant strain rate
type of input rate
rate from rate from rate from rate from ε ɺ
ith the mechanical s ith the mechanical s ith the mechanical sith the mechanical sith the mechanical s ith the mechanical s ith the mechanical sith the mechanical sith the mechanical strain
A springA springA spring A springdashpot representation of the whole onedashpot representation of the whole one dashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole one dashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedimensional model
Please note
elasticity E elasticity E elasticity E0.
Please note Please note Please note Please note
Stress Stress
. The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of the total stress
Figure 2. a) Figure 2. a) Figure 2. a) Figure 2. a) Figure 2. a) Figure 2. a) Spring
σσ
ɺeq
σσσσσσσσσσσσσσσσσσσσσσσσσσσσσεεεεεεεεεεεεεεεεεεεεεεεε
ɺ
εεεεεεεσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσ
σσ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σσ
σ
σσ
σ
σσ
σ
σσσσ
σ
σσ
σ
σσ
σ
σσσσ
σ
σσ
σ
σσ
σ
σσ
σ
σσ
σ
σσ
σ
σσσσ
σ
σσ
σ
εε
ε
εεεε
ε
ε
0
ε
0
ε
ɺ
εε
ε
εε
ε
εε
ε
εε
ε
εεεε
ε
εε
ε
σσ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σ
ɺ
σσ
σ
σσ
σ
σσ
σ
σσσσ
σ
σσ
σ
σσ
σ
σσσσ
σ
σσ
σ
σσ
σ
σσ
σ
σσ
σ
σσ
σ
σσσσ
σ
σσ
σ
strain curves strain curves strain curves strain curves strain curves strain curves of the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading a of the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading at constant temperature
that for fast l that for fast lthat for fast lthat for fast lthat for fast lthat for fast lthat for fast lthat for fast loadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of
the total stress the total stress the total stress the total stress the total stress the total stress the total stress
that they change after a period of constant strain rate
type of input ratetype of input rate. Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to
ε ɺ to to
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmeqeqeqeqeqeqeq eq eqeqeqeqeqeqeqeqeqeqeqeqeqeqeq eqeqeqeqeqeqeqeq
ov
ov
ov
ovov
ov
ovov
ov
ovov
ov
ovovovov
ov
ovov
ov
ovov
ov
ovov
ov
ovov
ov
ovov
ov
ov
ov
ovov
ov
ovov
ov
ovov
ov
ovovovov
ov
ovov
ov
ov ov
ov
ovov
ov
ovov
ov
ovov
m
m
m
mm
m
mm
m
mm
m
mmmm
m
mm
m
mm
m
mm
m
mm
m
mm
eq
eq
eq
eqeq
eq
eqeq
eq
eqeq
eq
eqeqeqeq
eq
eqeq
eq
eqeq
eq
eqeq
eq
eqeq
eq
eqeq
ov
ov
ov
ovov
ov
ovov
ov
ovov
ov
ovovovov
ov
ovov
ov
ovov
ov
ovov
ov
ovov
ov
ovov
eqeqeqeqeqeqeq eqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq eqeqeqeqeqeq
==============================================================
= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −+++++++++++++++++++++++++++++++−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−===============================
train train
SSebastianebastianebastianebastian
ββββββββββββββββββεεεεε
ɺ
εεεεεεεεεεεεεσσσσσσσσσσσσσ
εεεεε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
E
ε
t m
E
ε
t mt m
λλλ
−−−−−−−−−−−−−−−+++++++++++++++−−−−−−−−−−−−−−−
1
εε
ε
εε
ε
εε
ε
εε
ε
εε
ε
εε
ε
α θα θ
α θ
α θα θ
σ
σeq
θθ
θ
θθ
θ
θθ
i.e. eq.i.e. eq.
, Et,
SpringSpringSpring
1
EEEEEEEEEEEEEEEEEEEEEEEE
1
GGGGGGGGGGGGGGGGGGGGGGGG
1
KKKKKKKKKKK9K9K9K9K9K9K9K9K9K9K9K9K9K
1
EEEEEEEEEEEEEEEEEEEEEEEE
1
GGGGGGGGGGGGGGGGGGGGGGGG
1
KK9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K
sssssssssssssssssssssssssssssssssssssssssstttttttttttttttttttttttttttttttttttttttttt
sgn(sgn( sgn(sgn( sgn()))))
EEEEEEEEEEEEEEEEEEEEEEEE
========−−−−−−−−−−−−−−−−
i.e. eq.i.e. eq.(9
m
an increaincreaincreaincrease of strainse of strainse of strainse of strainse of strainse of strainse of strainrate from
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
=====================+++++++++++++++++++++=====================+++++++++++++++++++++
sssssssssssss eq eqeq eqeqeqeqeqeqeqeqeqeq t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t m
ε
t mt m
33333333333333333333339999999993333333333333333333333
β
β
ββ
β
ββββ
β
βε
ε
εε
ε
ɺ
εεεε
ε
εσ
σ
σσ
σ
σ
κ
GG
mmmmmmmmmmmmmmm eqeqeqeqeqeqeqeqeqeqeqeqeqeqeq
EEEEEEEEEEEEEEEEEEEEEwithwith withwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwith
dashpot representation dashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representation
Hauswaldt, M Hauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, Manfred
−−−−−−−−−−
sgn( )
9
K9
K9
K9
K
K
933333
[
α θ
α θ
eq,
sgn( )
1
−
GGG
type of input ratetype of input ratetype of input ratetype of input ratetype of input ratetype of input rate
sgn( )
−
α θ
α θ
sgn( )
−−−−
]
]and the total strainand the total strainand the total strain
9), for monotonic loading is depicted in F
andand ββ
to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance. to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.
0
θ
oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of
the total stress σ
that they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain rate
. Particular focus has to be drawn to . Particular focus has to be drawn to
1 ε ɺ
σ
κ
, κκ and
σ
s
anfredanfred
σ
σ
σ
rate from
and andand
σ
σ
σ
σ
σ
and the total strain
), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F ), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in Fig.
Korzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and S Korzen and Stefan
and the total strainand the total strainand the total strainand the total strainand the total strainand the total strainε
b) b) Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.
dimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional model (Eqs.
b)
oadings the model responds always with the instantaneous modulus of
σ at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.
that they change after a period of constant strain ratethat they change after a period of constant strain rate
. Particular focus has to be drawn to
1 ε .
oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of
at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.
that they change after a period of constant strain rate
. Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to relaxation
ε in in in σ
tefan tefan HartmannHartmannHartmannHartmannHartmann
oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of
at constant temperature is depicted in Fig.
that they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain rate
withwithwith
σ 
oadings the model responds always with the instantaneous modulus of
at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig. 2
direction (direction (direction (direction (direction (direction (cf.
oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of
(Eqs. (Eqs. (Eqs. (9
(0)(0)(0)(0)0000
(0)(0)(0)(0)0000
(9)
t constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperature.
)(101010))
cf. [5]). [5]). [5]). [5]). The material The material
ig. 1 together
is presented in is presented in is presented in is presented in is presented in is presented in is presented in is presented in
The material
1 together 1 together 1 together 1 together 1 together
relaxationrelaxationrelaxationrelaxationrelaxation
The material The material The material The material
(10)(10)(10)
(8)(8)
(9)(9)
2b
Page 4
44
20
S
Composition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with a
identified
values at the
equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.
The instantaneous
aforementioned tests
assumption ofassumption ofassumption ofassumption ofassumption ofassumption ofassumption ofassumption of th
20
S 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens were
identified
values at thevalues at thevalues at thevalues at thevalues at thevalues at thevalues at the end of hold timesend of hold timesend of hold timesend of hold timesend of hold timesend of hold timesend of hold timesend of hold timesend of hold timesend of hold times
The instantaneous
aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests (see
EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?TIFICATIO? OF THE MA
The The
, 300, 4, 300, 4, 300, 400, 500 and 00, 500 and 00, 500 and 00, 500 and 00, 500 and 00, 500 and 00, 500 and 20, 300, 4
In Fig.In Fig.In Fig.In Fig.In Fig. 3
identifiedidentifiedidentifiedidentified equil
The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous
The material parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters as
equilequilequilibrium stressibrium stressibrium stressibrium stressibrium stressibrium stressibrium stress
Table 2. MeasuTable 2. MeasuTable 2. MeasuTable 2. MeasuTable 2. MeasuTable 2. MeasuTable 2. MeasuTable 2. Measured instantaneous red instantaneous red instantaneous red instantaneous red instantaneous red instantaneous red instantaneous red instantaneous red instantaneous Young’s
0.150.150.15
3 you can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressstrain diagram with the strain rates for
based on eq. (
tend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibrium
Figure 3. StressFigure 3. StressFigure 3. StressFigure 3. StressFigure 3. StressFigure 3. StressFigure 3. StressFigure 3. Stressstrain
CC
the temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of Poisson’s ratio
werewere
Si
0.20.2
Young’s modulusYoung’s modulus
(see (see Tab.Tab.Tab. 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus are
TemperatureTemperatureTemperatureTemperature
20
300
400
500
600600
600
performedperformedperformedperformedperformedperformed at BAM
spark emissispark emissispark emissispark emissispark emission spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noted.
Si
SSebastianebastianebastianebastian
600
Young’s modulusYoung’s modulusYoung’s modulusYoung’s modulusYoung’s modulusYoung’s modulusYoung’s modulusYoung’s modulus
300
400
500
600
°C with°C with°C with°C with°C with varying
at BAMat BAMat BAMat BAM on a servo
Mn
0.490.490.49
spark emissispark emissi
MnMn
based on eq. (based on eq. (based on eq. (
tend toward equilibrium
traintrain
Temperature
20
300
400
500
functions offunctions offunctions offunctions of
varyingvaryingvaryingvarying strain rate
TemperatureTemperature
Table 4. Chemical Composition STable 4. Chemical Composition S
P
0.0190.0190.019 0.0160.0160.0160.016 0.019
diagramiagramiagramiagram
Hauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, Manfred
Table 4. Chemical Composition S
P SS
strain diagram with the strain rates for
based on eq. (based on eq. (based on eq. (based on eq. (99) for for for the tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperature.
TIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATERIAL
functions offunctions offunctions of the
Table 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition S
strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for a
νννννννννννννθθθθθθθθθθθθθ
213000
204500
194500
177000
152500152500152500152500152500
of strainof strainof strainof strain
E0
= 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054
EE0
213000
204500
194500
177000177000177000177000
on a servoon a servoon a servoon a servoon a servo
0
Young’s Young’s Young’s Young’s modulus and modulus and modulus and modulus and modulus and modulus and modulus and calculated
213000
204500
194500194500194500
the
strain ratestrain ratestrain ratestrain ratestrain rates
as a function ofas a function ofas a function ofas a function ofas a function ofas a function ofas a function ofas a function ofas a function ofas a function of
213000
204500204500
the temperature temperature temperature temperature temperature temperature temperature were identified on the basis of
and hold times
hydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machine
ed.
oisson’s ratiooisson’s ratiooisson’s ratiooisson’s ratiooisson’s ratiooisson’s ratiooisson’s ratio according to
213000
anfredanfred
0.050.050.05
controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600 °C.
CrCr
Korzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and Stefan
hydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machine. In Table 1 the Chemical
s andand
171502
177680
173688
163012
144678144678144678144678
Mo
0.010.010.01
KK
171502
177680
173688
163012163012163012
MoMo
statesstatesstates
171502
177680
173688173688
were identified on the basis of
hold timeshold timeshold timeshold timeshold times of 9000of 9000of 9000of 9000of 9000 seconds
according to according to according to according to according to according to according to [7
171502
177680
were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of tensile
235235
0.070.070.07
are
calculatedcalculatedcalculatedcalculated bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.
NiNi
whichwhichwhichwhich are
temperature
calculated calculated calculated calculated calculated calculated from these values
7], i.e., i.e., i.e.
tefantefan HartmannHartmannHartmannHartmannHartmann
in %.in %.in %.
temperaturetemperaturetemperaturetemperaturetemperaturetemperature is
<0.01<0.01<0.01<0.01
TERIALTERIALTERIALTERIALTERIALTERIAL PARAMETERSPARAMETERSPARAMETERSPARAMETERSPARAMETERSPARAMETERSPARAMETERSPARAMETERSPARAMETERSPARAMETERS
AlAl
82332
78151
73939
67108
57548575485754857548
. It was assumed that
are are interpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of the interpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of the
GG0
82332
78151
73939
671086710867108
It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that
82332
78151
7393973939
0.050.050.050.05
a 600600600 °C tensile test°C tensile test°C tensile test°C tensile test°C tensile test°C tensile test°C tensile test°C tensile test and t
from these values from these values from these values from these values from these values from these values from these values from these values from these values from these values with the
82332
78151
CuCu
is
seconds
In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical
determined determined determined determined determined determined determined from the
secondssecondsseconds. T
0.0030.0030.003
NN
The tests he tests he tests he tests he tests of the
°C.°C.
tensiletensiletensile
the stressthe stressthe stressthe stressthe stress
from the from the
with the with the with the with the
tests tests
of the of the of the
and tand tand the
from the from the from the
(11)(11)(11)
tests at at
he
Page 5
corresponding
assumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independent
The overstress has to be identified on the basis of the difference between equilibrium stress and measured
total stress according
parameterparameterparameterparameterparameterparameterassas0.01
=
as0.01as0.01as0.01as isis cconstant andonstant andonstant andonstant andonstant andonstant andonstant and temperaturetemperaturetemperaturetemperaturetemperaturetemperaturetemperatureindependent.independent.independent.independent.independent.independent.independent.
corresponding corresponding corresponding
assumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independent.
The overstress has to be identified on the basis of the difference between equilibrium stress and measured
total stress accordingtotal stress accordingtotal stress accordingtotal stress accordingtotal stress accordingtotal stress according
0 a
The The The material parameters of the
corresponding corresponding corresponding corresponding parameter parameter parameter parameter parameter parameter functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig. 4. The parameter
The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured
total stress according
0=
material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the
Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature.
The overstress has to be identified on the basis of the difference between equilibrium stress and measured
total stress accordingtotal stress accordingtotal stress accordingtotal stress according toto eqeq.. (10).(10).(10). The materialThe materialThe materialThe materialThe materialThe materialThe materialparameters aparameters aparameters aparameters aparameters aparameters aparameters a1
The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured
Figure Figure Figure Figure 5.
SSebastianebastianebastianebastian
. Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the over
Hauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, Manfred
equilibriumequilibriumequilibriumequilibriumequilibriumequilibrium stress are stress are stress are stress are stress are identified identified identified identified identified identified using
anfredanfred Korzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and Stefan
overoverstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperature
1, a2
tefantefan HartmannHartmannHartmannHartmannHartmann
using using using
2 and aand aand aand a3
eq.eq. (9). The paramet
4. The parameter4. The parameter4. The parameter4. The parameter4. The parameter4. The parameter4. The parameter4. The parameter4. The parameter
are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5.
(9). The paramet(9). The paramet(9). The paramet(9). The paramet(9). The paramet(9). The paramet(9). The paramet(9). The parameters
atureature.
.
ers
κκ =
0.7 0.7 0.7 0.7 0.7 is
ers withwithwith
. The The The
the the
is
Page 6
Sebastian Hauswaldt, Manfred Korzen and Stefan Hartmann
5 IMPLEME?TATIO? OF THE CO?STITUTIVE MODEL I?TO FESOFTWARE
The stress state at time tn+1 develops from
with the time increment t △ and the stress rate
The spherical part of the stress tensor is
.
1
nn
t
+ =+
TTT
△
(12)
.
∆=
TT △t
1,1
()3()
nm n
tr Ktr
++
=
TE
.
(13)
According to eq. (4)
,1,
+
=−∆TTT
D
eq
D
eq n
D
eq ncan be determined using a Backward Euler method leading
to
?
L
,1
1
β
22 0 :
=
DD
eq n
T
D
eq
D
n
D
eqst
GzG
+
∆− + ∆− − ∆=
ETET
△
(14)
Here
1
+
∆=−
EEE
DD
n
D
nis prescribed. From eq. (5) follows the incremental formulation of the arclength
2
2
s
2
3
4
κ
G
κ
G
=∆∆−∆∆+∆⋅∆
⋅
⋅
EEETTT
△
DDDD
eq
D
eq
D
eq
s
z
. (15)
According to eq.(6), the overstress
,1
D
ov n+
T
can be calculated with
,10,1
D
ov n
T
D
n
DD
eq n
T
G
t
λ
+
λ
++
=+∆−
TE
△
.
(16)
For effective numerical simulations with the finite element method in addition to the stress
formulation we need the consistent tangent operator. The tangent operator is needed to develop the
structure stiffness matrix in a finite element analysis.
Since the stress computation is based on an iterative procedure, the implicit function theorem has to
be applied requiring the application of the chainrule applied to eq. (14), see [6] and the literature cited
therein.
?
L
E
?
L
∂
∂
∂
T
+=
∂
T
d
0
E
D
eq
D
eq
d
. (17)
?
L E can be found using Gâteauxderivatives. With the derivative of the arclength with respect to the
strain tensor
∂∂
1
1 2
2
△
z
()[ ]H2 [ ]H
3
κ
G
+
=∆−∆
EET
△
DD
eqn
s
D z
(18)
we obtain
1
,1
1
3
ˆ(
)[ ]
H
22[ ]
1 1
3
β
22[ ]H
β
κ
G
+
+
=+−⊗
−+ ∆−⊗∆−∆
EI I H
TTEET
△
△
nst
D
eq n
D
eq
D
n
DD
eqt
s
z
DLGG
G
z
I
.
(19)
Page 7
Furthermore Furthermore Furthermore Furthermore Furthermore Furthermore Furthermore Furthermore the derivat
is required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expression∂
After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. (17)
numerical solver with
rightrightright
The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of the
is is derived from eq.derived from eq.
The model
FORTRAN
tangent operatortangent operatortangent operatortangent operatortangent operatortangent operatortangent operator
FORTRAN
tangent operatortangent operator
66
strain
FFig. 6big. 6big. 6big. 6b.
a)a)
of of of 100100100 MPa
numerical solver with numerical solver with numerical solver with
righthand sides.hand sides.hand sides.hand sides.hand sides.hand sides.
FORTRAN FORTRAN FORTRAN FORTRAN FORTRAN
strainstrainstraintemperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests of S
Figure 6. Figure 6. Figure 6. Figure 6. Figure 6. a)
tthe tangent operator ofhe tangent operator of
numerical solver with numerical solver with numerical solver with numerical solver with numerical solver with numerical solver with numerical solver with numerical solver with
TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP
In Fig.In Fig.In Fig.In Fig.In Fig. 6
Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests
MPa MPa MPa have been have been have been have been have been have been carried out. carried out. carried out. carried out. carried out. carried out. carried out. The temperature rate was 7.2
DDDDDDDDDDDDDDDDDDDDDDDDGGGGGGGGGGGGGGGGGGGGGGGG
derived from eq.
The model The model The model The model The model The model is adapted to
subroutinesubroutinesubroutinesubroutinesubroutines
?LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEETTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEHHHHHHHHHHHHHHHHHHHHH[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ] .
derived from eq.derived from eq.derived from eq.derived from eq.derived from eq.derived from eq.derived from eq. (1
?()[ ])[ ])[ ])[ ])[ ])[ ] =)[ ] =
he tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator of
((((((
∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆+ ∆−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−∆−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+++++++++++++++++++++++++++++++++++++++
a) Simulation of Simulation of Simulation of Simulation of Simulation of Simulation of Simulation of transient
Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests on S
D
eqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq
D
eq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq neq n
D
eqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq
D
nnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
D
eqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq
D
D
DD
D
DDDD
D
DD
D
DD
D
DD
D
DD
D
DD
D
DDDD
D
DD
D
D
tttttttttttttttttttttttttttttt
subroutine
have been and are have been and are have been and are have been and are have been and are have been and are have been and are have been and are have been and are
6a simulationa simulationa simulationa simulationa simulationa simulationa simulations
the derivatthe derivatthe derivatthe derivatthe derivative
D
DD
D
DDDD
D
DD
D
DD
D
DD
D
DD
D
DD
D
DDDD
D
DD
D
D
D
DD
D
DDDD
D
DD
D
DD
D
DD
D
DD
D
DD
D
DDDD
D
DD
D
D
D
DD
D
DDDD
D
DD
D
DD
D
DD
D
DD
D
DD
D
DDDD
D
DD
D
D
D
DD
D
DDDD
D
DD
D
DD
D
DD
D
DD
D
DD
D
DDDD
D
DD
D
D
D
DD
D
DDDD
D
DD
D
DD
D
DD
D
DD
D
DD
D
DDDD
D
DD
D
D
is adapted to is adapted to is adapted to is adapted to is adapted to is adapted to is adapted to
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTEEEEEEE(E(E(E(E(E(E(EEEEEEEEEEEEEEEEEEHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHHHHHH[ ]H[ ]H
(16)
1 1
β
3
β
3
ββ
3
β
3
βββ
3
βββ
3
β
3
βββ
3
βββ
3
ββββββββββββββ
ββββββββββββββββββββββββββββββββ
(17) (17)
LU LU LU decompositiondecompositiondecompositiondecompositiondecompositiondecompositiondecompositiondecomposition
D
ovovovovovovovovovovovovovovovovovovovovovovovovovovovov
D
ov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov nov n
D
eqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqnnnnnnnnnnnnnnnnnnnnnnnnnnnn
).
SSebastianebastianebastianebastian
ve
1 11 11 1
3333333333322222222222222222222
==
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
which includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stress
used used used forforfor FEFESimulations.Simulations.Simulations.Simulations.Simulations.Simulations.Simulations.
ransient ransient ransient ransient c
ofof the arche arche arche arc
D z
△
D z
△
D z
△△
DDDDDDDDDDDDDDDDGGGGGGGGGGGGGG
of transient creep tests of transient creep tests of transient creep tests of transient creep tests of transient creep tests of transient creep tests of transient creep tests of transient creep tests of transient creep tests of transient creep tests of transient creep tests of transient creep tests at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates at constant heating rates are
∂∂∂∂
z Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz Gz GGGGGGGGGGGGGGGGGGGG
the equilithe equilithe equilithe equilithe equilibrium stressbrium stressbrium stressbrium stressbrium stressbrium stressbrium stressbrium stress C
the the overstress overstress overstress overstress overstress overstress
(
(((((((((((((((((((((((
)[ ]
)[ ]H)[ ]H)[ ]
H
)[ ]H)[ ]
H
)[ ]
H
)[ ]H)[ ]H)[ ]
H
)[ ]
H
)[ ]H)[ ]
H
)[ ]
H
)[ ]H)[ ]
H
)[ ]
H
)[ ]H)[ ]H)[ ]
H
)[ ]
H
)[ ]H)[ ]
H
)[ ]
[ ]
[ ]H[ ]H[ ]
H
[ ]H[ ]
H
[ ]
H
[ ]H[ ]H[ ]
H
[ ]
H
[ ]H[ ]
H
[ ]
H
[ ]H[ ]
H
[ ]
H
[ ]H[ ]H[ ]
H
[ ]
H
[ ]H[ ]
H
[ ]
?
L
LLL
∂∂∂
T
TTT
κκκκκκκκκκκκκκκκκκκκκκκκκκκκκκκκκκκκ
ssssssssssssssssssssssssssssssssssss
the the the finite elementfinite element
which includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stresswhich includes the implementation of the stress
∆∆
∆
∆
∆∆
∆∆
∆∆
∆
∆∆
∆
∆∆
∆∆
∆
==
=
=
==
==
==
=
==
=
==
==
=
−−
−
−
−−
−−
−−
−
−−
−
−−
−−
−
∆∆
∆
∆
∆∆
∆∆
∆∆
∆
∆∆
∆
∆∆
∆∆
∆
++
+
+
++
++
++
+
++
+
++
++
+
∆∆
∆
∆
∆∆
∆∆
∆∆
∆
∆∆
∆
∆∆
∆∆
∆
TT
T
T
TT
TT
TT
T
TT
T
TT
TT
TT
T
T
EE
E
E
EE
EE
EE
E
EE
E
EE
EE
EE
E
E
TT
T
T
TT
TT
TT
T
TT
T
TT
TT
TT
T
T
,,,,,,,,,,,,,111111111111111111111111110000000000000
++++++++++
creep reep reep tests
creep tests on Screep tests on Screep tests on Screep tests on Screep tests on Screep tests on Screep tests on S 460 at 170
Hauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, Manfred
D
eq
eq
eqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq
D
D
DDDDDDDDDDDDDDDDDD
eq
eq
eqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq
△△
)[ ])[ ])[ ])[ ])[ ])[ ] =)[ ] =
creep tests on S
lengthlengthlengthlength with respect to thewith respect to thewith respect to thewith respect to thewith respect to thewith respect to thewith respect to thewith respect to thewith respect to thewith respect to the equilibrium
D
DDDDDDDDDDDDDDDDD
D
DDDDDDDDDDDDDDDDD
D
eq
,,,,,,,,,,,11111111111
The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2The temperature rate was 7.2
D
eqfollowsfollowsfollowsfollowsfollows
finite elementfinite elementfinite elementfinite elementfinite elementfinite element
ests ests of S
∂
∂
TD
∂
∂∂∂∂∂∂∂∂
∂
∂T
∂
since eq.since eq.since eq.since eq.since eq. (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several (22) represents a system of equations with several
of Sof S 235 at 235 at 235 at 235 at 100
1
△△
3
?
LLLLLLLLLLLLL
EEEEEE
eqeq
1
z
zz
zG
GG
G
?
∂∂∂∂∂∂∂
D
dd
anfredanfred
22222222222222222222222221111111111111111111111111
T
dd
λλ +
κ
κ
κκ
κ
κκ
κ
κ
κ
κκ
κ
22
G
G
T
EEEEEE
D
eqeq
TTTTTTTTTEEEEEEEEE
λλ
△△
program ABAQUSrogram ABAQUSrogram ABAQUSrogram ABAQUSrogram ABAQUSrogram ABAQUSrogram ABAQUSrogram ABAQUSrogram ABAQUSrogram ABAQUSrogram ABAQUS [2]
460 at 170460 at 170460 at 170460 at 170460 at 170 MPa [8].
Korzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and Stefan
s
= −= −= −∂
LLLLL
D
CCCCCCCC
D
eq
eq
eqeqeqeq
D
eq
eq
eqeqeqeq
t
b)
100100 MPa MPa MPa with different heating rateswith different heating rates
MPa [8].MPa [8].MPa [8].MPa [8].
+
==
D
DDDD
D
DDDD
GG
dddddddddddd
?
II
b)
on Son S
κκ
κ
−−−−−−−⊗⊗⊗⊗⊗⊗⊗−−−−−−−
equilibriumequilibriumequilibriumequilibriumequilibriumequilibrium stress incrementstress incrementstress incrementstress incrementstress incrementstress incrementstress incrementstress incrementstress increment
1
33
with different heating rateswith different heating rateswith different heating rateswith different heating rateswith different heating rateswith different heating rateswith different heating rateswith different heating rateswith different heating rateswith different heating rateswith different heating rates b) Results of transient
235 235
K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of
2
2
s
1
tefantefan HartmannHartmannHartmannHartmannHartmann
can becan becan becan becan be determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of determined by the use of
235 transientransientransientransientransient creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress
[2] using the UMAT interface.
algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated
using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface.
are presentedpresented
of Sof Sof S 460
△△z
presentedpresentedpresented.
b) Results of transient b) Results of transient b) Results of transient b) Results of transient b) Results of transient b) Results of transient b) Results of transient b) Results of transient b) Results of transient b) Results of transient
460460 [8]
II
. The simulated The simulated
[8] depicted in depicted in depicted in depicted in depicted in depicted in
The simulated The simulated The simulated The simulated The simulated The simulated
(20)(20)(20)
(21)(21)(21)
(22)(22)(22)
(24)(24)(24)
a
Page 8
Sebastian Hauswaldt, Manfred Korzen and Stefan Hartmann
600 °C was reached after 5000 seconds. The following cooling phase with a temperature rate of 
7.2 K/min. ended at 120 °C when natural cooling was not fast enough.
Temperature, uniaxial stress and strain were recorded and compared to results of a simulation of the
test as depicted in Fig. 7. In the heating phase the calculated strain increases to higher values then in the
transient creep test. At higher temperatures when cooling down the simulated strains are in good
agreement with the experiment.
Figure 7. Transient creep test and simulation of S 235.
7 COLUM? TESTS
Furthermore, component simulations are compared with the results of two column fire tests. The
steel columns with a height of 3.56 m were fixed at the top while the bottom end was free to move
vertically. The mechanical load applied at the center of the bottom of the column was constant. By the use
of a servohydraulic testing machine an elastic foundation was applied additionally [10]. Axial loading and
stiffness of the foundation are presented in Tab. 3.
Table 3. Test Program Columns
Specimen Load
HEA 140 492 kN
HEB 180 1052 kN
Stiffness
47 kN/mm
69.5 kN/mm
Page 9
EN
111111 measuring
The temperature fi
using the measured surface temperatures at
thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according to
using the measured surface temperatures at
thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according
Figure 8.
simulation csimulation csimulation c simulation csimulation csimulation csimulation c) T
thermoviscoplastic material
application point at the bottom of simulation and experiment are compared in Fig.
time and temperature of maximum displacement and zero passage are in time and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are in time and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are in time and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are in good agreement.
ENEN 13631363
measuring
The temperature fi
using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at Dirichlet boundary conditionsusing the measured surface temperatures at
Figure 8. Figure 8. Figure 8. Figure 8. HEA
a)a)
a)a)
thermoviscoplastic material
application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig. application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig. 10. It can be seen that
The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to
1363
measuring
The temperature fiThe temperature fi The temperature fiThe temperature fiThe temperature fiThe temperature fi The temperature fiThe temperature field of the eld of the eld of the eld of the eld of the eld of the
With the use of the calculate temperature field the column tests were simulated with the presented
thermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic material thermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic material model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load
With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented
a)
The furnace was heated up according to
13631
measuringmeasuringmeasuring stationsstationsstations stations stations ((see Fig.see Fig.see Fig.see Fig.
HEAHEAHEA 140
Temperature field emperature field emperature field emperature field emperature field emperature field emperature field emperature field at deformed column
The furnace was heated up according to
1 standardstandardstandardstandardstandard [9]. [9]. [9]. The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30
8a8a).
columnscolumnscolumnscolumnscolumns has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element
Dirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditions a
The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to the so called uniform
The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30
140140: a) a) TemperatureTemperatureTemperatureTemperatureTemperatureTemperatureTemperature m
SSebastianebastianebastianebastian
Figure Figure Figure Figure
b)
Figure Figure Figure Figure 9
at deformed column at deformed column at deformed column at deformed column at deformed column at deformed column at deformed column at deformed column at deformed column at
b)
The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30
.
has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element method
measuring easuring easuring easuring easuring stations b)
1010. Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement a)
Hauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, Manfred
9. Steel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEA
The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30
tations b)tations b)tations b)
at 1600
the so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformtemperature
anfredanfred
tations b)
160016001600 sec.
b)
b)
c) c)
V. V. M
Korzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and Stefan
b)
Mises s
sec. sec. of the simulation of the simulation of the simulation of the simulation of the simulation of the simulation of the simulation of the simulation d)
b)
ises sises stress
a) HEAHEAHEA
to [1].[1].[1].
tresstresses ofes ofes of
140 b) HEB140 b) HEB140 b) HEB140 b) HEB140 b) HEB140 b) HEB 180
140140 b)
tefantefan HartmannHartmannHartmannHartmannHartmann
b) b) HEBHEBHEB 180
temperaturetemperaturetemperaturetemperaturetemperaturetemperaturetime
deformed columndeformed columndeformed column
d) Deformed columnDeformed columnDeformed columnDeformed columnDeformed columnDeformed columnDeformed columnDeformed column
d) d)
180180.
good agreement.good agreement.good agreement.good agreement.good agreement.good agreement.good agreement.good agreement.
and the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of
deformed columndeformed columndeformed columndeformed columndeformed column
180180.
timetime
10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that
curvecurve
thermocouples thermocouples thermocouples thermocouples thermocouples thermocouples thermocouples thermocouples at
methodmethodmethodmethod (
at 255at 255at 255 sec.
curvecurve according to according to according to according to according to according to according to
sec.
after test. after test.after test. after test. after test.
(ABAQUSABAQUS ABAQUSABAQUSABAQUS ABAQUS)
sec. of the of the of the of the
at
Page 10
Sebastian Hauswaldt, Manfred Korzen and Stefan Hartmann
8 SUMMARY A?D CO?CLUSIO?S
A socalled unified thermoviscoplastic constitutive model for the simulation of the behavior of
structural steel is presented. Due to the modular concept with an additive decomposition of the stress into
an equilibrium stress and an overstress specific material properties of rateindependent classical plasticity
and rate dependent viscoelasticity can be associated to experimental observations.
Based on this concept the identification of the temperaturedependent material parameters for
structural steel S235 are realized within a series of straincontrolled tensile and compression tests with
varying strainrates at constant temperature between 20 °C and 600 °C.
The modular concept of the constitutive model influences also the numerical solution strategy.
Whereas the numerical integration of the overstress might be realized by an explicit algorithm the
corresponding procedure for the equilibrium stress are treated via an implicit BackwardEuler algorithm.
This stabilizes essentially the computational procedure. The necessary consistent tangent operator splits
into two parts as well. Whereas the tangent operator for the overstress is written analytically via Gâteaux
derivative, the corresponding operator of the equilibrium stress is solved numerically. The numerical
implementation has been programmed as a user subroutine UMAT for the FEprogram ABAQUS.
Comparing simulation results with experimental data it is shown that stresscontrolled transient creep
tests are described with sufficient accuracy. The comparison of experimental data of fire resistance tests
on columns with numerical simulations show that the presented constitutive model is applicable without
problems and is capable to describe the column behavior quite reasonably. In contrast to rateindependent
models as in [1] the presented model is able to describe properly the material behavior as a function of the
heatingrate.
REFERE?CES
[1]
EN 1993, Eurocode 3: Design of steel structures  Part 12: General rules – Structural fire
design, CEN, Brussels, Belgium, December 2010.
ABAQUS 6.111: Abaqus Analysis User’s Manula Version 6.111, Dassault Systèmes Simulia
Corp., Providence, RI, USA, 2011.
Haupt, P.: “On the Mathematical Modelling of Material Behaviour in Continuum Mechanics,
Acta Mechanica”, 100, 129 154, 1993.
Valanis, K.C.: “Fundamental Consequences of a New Intrinsic Time Measure Plasticity as a
Limit of the Endochronic Theory”, Archives of Mechanics 32, 171191, 1980.
Korzen, M.: “Constitutive Equations for Structural Steel Subjected to Fire”, Applications of
Structural Fire Engineering – Proceedings of International Conference, F. Wald, K. Horová, J.
Jirku (eds.), Prague, 2011.
Hartmann, S.: "A remark on the application of the NewtonRaphson method in nonlinear finite
element analysis", Computational Mechanics, 36 (2), 100116, 2005.
Richter, F.: Physikalische Eigenschaften von Stählen und ihre Temperaturabhängigkeit,
Stahleisen Sonderberichte Heft 10, Verlag Stahleisen, Düsseldorf, Germany, 1983
Schneider, R.: „Untersuchungen zum zeitabhängigen mechanischen Materialverhalten von S460
im Brandfall“, Diss. , Darmstadt, Germany, 2011
EN 1363: Fire Resistance Tests  Part 1: General Requirements
[10] Korzen, M.; Magonette G. and Buchet P.: “Mechanical Loading of Columns in Fire Tests by
Means of the Substructuring Method,” Zeitschrift für Angewandte Mathematik und Mechanik,
79, 617–618, 1999.
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]