material modeling: - a game for scientists or an effective tool … · motivation a simple example...
TRANSCRIPT
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Material Modeling:A Game for Scientists
or an Effective Tool to Improve Structures
Holm Altenbach
Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät fürMaschinenbau, Otto-von-Guericke-Universität Magdeburg, Germany
15 May 2015 Altenbach Material Modeling 1
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Table of Contents
1 Motivation
2 A Simple Example
3 Historical Remarks
4 Creep-Damage Modeling
15 May 2015 Altenbach Material Modeling 2
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Outline
1 Motivation
2 A Simple Example
3 Historical Remarks
4 Creep-Damage Modeling
15 May 2015 Altenbach Material Modeling 3
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
General Statement of European Commission
Research & Innovation > Key Enabling Technologies >Research in Materials > Modelling Materials1
• The future of the European industry is associatedwith a strong materials modelling capacity.
• An efficient modelling approach is needed toshorten the development process ofmaterials-enabled products.
Among the areas
Nanotechnologies
Advanced Materials & Manufacturing
Biotechnology
Coal & Steel1http://ec.europa.eu/research/industrial_technologies/modelling-
materials_en.html15 May 2015 Altenbach Material Modeling 4
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Materials Science Approach
Analysis of microstructure
Material productionTesting
Description
Structural analysisManufacturing
Operation time
Failure
Material improvement
Strain
Str
ess
σ = f (ε)RVE
No RVE
Source: O. Prygorniev, PhD Thesis, University of Magdeburg, 2015 (subm.)15 May 2015 Altenbach Material Modeling 5
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Outline
1 Motivation
2 A Simple Example
3 Historical Remarks
4 Creep-Damage Modeling
15 May 2015 Altenbach Material Modeling 6
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Continuum Mechanics - Basics
V
A
cutting principle (method of sections)
axiom of reciprocal action (Newton’s Third Law)Continuum Mechanics governing equations
geometrical relations, e.g. strain-displacement equationsequilibrium equations, equations of motion
15 May 2015 Altenbach Material Modeling 7
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Continuum Mechanics - Non-polar
dA
dF
dF
only force actions
symmetric stress tensor
only translations
15 May 2015 Altenbach Material Modeling 8
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Basic Course Engineering Mechanics
Static equilibriumForcesMoments
Equations of motionBalance of momentumBalance of moment of momentum
Dependent or independent relations?2
2Truesdell, C. (1964). Die Entwicklung des Drallsatzes. ZAMM44(4/5):149–158
15 May 2015 Altenbach Material Modeling 9
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Continuum Mechanics - Polar
replacements
dA
dFdM
dMdF
force and moments actions
symmetric and nonsymmetric stress tensors
translations and rotations (independent!)
15 May 2015 Altenbach Material Modeling 10
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Material Modeling
in both situations we have a statically indeterminate system
establishment of constitutive equations and (maybe)evolution equations for closing the gap between thenumber of equations and the number of unknowns
constitutive equations interlinking in the simplest casestresses and strainsvarious approaches
top - down (from the more general to the simpler cases)bottom - up (stepwise generalization)rheological models
15 May 2015 Altenbach Material Modeling 11
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Outline
1 Motivation
2 A Simple Example
3 Historical Remarks
4 Creep-Damage Modeling
15 May 2015 Altenbach Material Modeling 12
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
First Steps - Archimedes of Syracuse
Born c. 287 BC, Syracuse,Sicily
Died c. 212 BC, Syracuse
Residence Syracuse, Sicily
Fields Mathematics, Physics,Engineering, Astronomy,Invention
Known for Archimedes’Principle, Archimedes’ screw,Hydrostatics, Levers,Infinitesimals
15 May 2015 Altenbach Material Modeling 13
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Lever
The earliest remaining writings regarding levers date fromthe 3rd century BC and were provided by Archimedes:Give me a place to stand, and I shall move the earth with alever.This is a remark of Archimedes who formally stated thecorrect mathematical principle of levers (quoted by Pappusof Alexandria).It is assumed that in ancient Egypt, constructors used thelever to move and uplift obelisks weighting more than 100tons.
15 May 2015 Altenbach Material Modeling 14
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Leonardo da Vinci
Born April 15, 1452, Vinci, Florence
Died May 2, 1519 Amboise, Touraine
Nationality Italian
Fields Many and diverse fields of arts andsciences
Known for Flying machine
15 May 2015 Altenbach Material Modeling 15
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Hooke’s Contributions
ceiiinossssttuv (1676)
Ut tensio, sic vis (1678)
As the extension, so the force.
F
∆l
Hooke’s law (original)
Robert Hooke (18 July 1635 – 3 March 1703)
15 May 2015 Altenbach Material Modeling 16
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Hooke’s Contributions
ceiiinossssttuv (1676)
Ut tensio, sic vis (1678)
As the extension, so the force.
σ
ε
Hooke’s law
Robert Hooke (18 July 1635 – 3 March 1703)
15 May 2015 Altenbach Material Modeling 16
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Analogy - Elastic Spring
For systems that obeyHooke’s law,
the extension produced isdirectly proportional
to the loadF = −kx
Modern form of the Hooke’s law
σσσ = Eεεε
σ stress, ε strain, E Young’s modulus
15 May 2015 Altenbach Material Modeling 17
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Bernoulli’s Contributions
1686postulate of the angularmomentum
Jakob Bernoulli (6 January 1655 - 16 August 1705)
15 May 2015 Altenbach Material Modeling 18
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Newton’s Contributions
Newton’s Axioms
ddt
(mv) = 0 (statics)
ddt
(mv) = F (kinetics)
actio = reactio
superposition
Isaac Newton (4 January 1643 - 31 March 1727)
15 May 2015 Altenbach Material Modeling 19
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Euler’s Contributions
Euler’s laws of motion
Balance of momentum
Balance of moment ofmomentum
Assumptions
Independence oftranslations and rotations
Force and moment actions
Leonhard Euler (15 April 1707 - 18 September 1783)
15 May 2015 Altenbach Material Modeling 20
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Cauchy’s Contributions
Definition of Stress
Stress σ is a measure of the average amount of force F exertedper unit area A . It is a measure of the intensity of the totalinternal forces acting within a body across imaginary internalsurfaces, as a reaction to external applied forces and bodyforces. It was introduced into the theory of elasticity by Cauchyaround 1822.
σ =FA
Augustin Louis Cauchy (21 August 1789 - 23 May 1857)15 May 2015 Altenbach Material Modeling 21
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Young’s ContributionRemarkThe concept was developed in 1727 byLeonhard Euler and the first experiments thatused the concept of Young’s modulus in itscurrent form were performed by the Italianscientist Giordano Riccati in 1782 - predatingYoung’s work by 25 years.
Definition
An elastic modulus, or modulus of elasticity, is themathematical description of a material’s tendency to bedeformed elastically (i.e., non-permanently) when a force isapplied to it. The elastic modulus of an object is defined as theslope of its stress-strain curve in the elastic deformation region.
Thomas Young (13 June 1773 - 10 May 1829)15 May 2015 Altenbach Material Modeling 22
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Poisson’s Contributions
Isotropic Linear Elastic Behavior (3D)
two or one material parameters
σσσ = α tr(εεε)I + βεεε + γεεεT
σσσ = σσσT ⇒ σσσ = K tr(εεε)I + 2µεεε
K bulk modulus, µ shear modulus
K =E
3(1 − 2ν), µ = G =
E2(1 + ν)
ν - parameter or fixed?
Siméon-Denis Poisson (21 June 1781 - 25 April 1840)15 May 2015 Altenbach Material Modeling 23
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Cosserat Brothers
EugéneCosserat
Cosserat Model (1896, 1909)
extended continuum model
balance of forces
balance of angular momentum
no constitutive equations
François Cosserat (26 November 1852 - 22 March 1914),Eugéne Cosserat (4 March 1866 - 31 May 1931)
15 May 2015 Altenbach Material Modeling 24
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Hilbert’s Problems
6th Problem (1900)
Axiomatic Formulation ofMechanics
unsolved up to now
David Hilbert (23 January 1862 - 14 Februars 1943)
15 May 2015 Altenbach Material Modeling 25
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Noll’s and Truesdell’s Contributions
Contributions to Mathematics and Mechanics
Rational Mechanics
Theory of Simple Materials
Rational Thermodynamics
History of Mechanics
Walter Noll (7 January 1925)Clifford Abmbrose Truesdell III (18 February 1919 - 14 January2000)
15 May 2015 Altenbach Material Modeling 26
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Anatolii Isakovich Lurie
Born 19 July 1901 Mogilev,Died 12 February 1980 Leningrad,
Three-dimensional Problems ofthe Theory of Elasticity. Moscow.GITTL, 1955, 492 pp.
Theory of Elasticity. Moscow,Nauka, 1970, 940 pp.(Springer 2005)
Nonlinear Theory of Elasticity.Moscow, Nauka, 1980, 512 pp.(North-Holland Series in AppliedMathematics and Mechanics,Volume 36, 1990)
15 May 2015 Altenbach Material Modeling 27
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Maksymilian Tytus Huber
Born 4 January 1872 Kroscienko,Died 9 December 1950 Kraków,Professor at Lwów Polytechnic,Warsaw Polytechnic, Gdansk Poly-technic and Akademia Górniczo-Hutnicza,Solid mechanics,Mechanical engineering,Maxwell (1865) - Huber (1904) -von Mises (1923) - Hencky (1924)criterion
15 May 2015 Altenbach Material Modeling 28
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Michał Zyczkowski
Born 12 April 1930 Kraków,Died 24 May 2006 Kraków,Professor at Cracow Polytechnic,Full member of Polish Academy of Sci-ences and Polish Academy of Learning,Foreign Corresponding Member of theAustrian Academy of Sciences,Solid mechanics,Plasticity, Optimization
15 May 2015 Altenbach Material Modeling 29
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Representation of Continuum Mechanics
From the previous statement one can conclude:
Differential equations can thus be employed in solvingproblems in continuum mechanics.
Some of these differential equations are specific to thematerials being investigated and are called constitutiveequations, while others capture fundamental physical laws,such as conservation of mass (continuity equation), theconservation of momentum (equations of motion andequilibrium), and energy (first law of thermodynamics).Continuum mechanics deals with physical quantities, which areindependent of any particular coordinate system in which theyare observed. These physical quantities are then representedby tensors, which are mathematical objects that areindependent of coordinate system. These tensors can beexpressed in coordinate systems for computationalconvenience.
15 May 2015 Altenbach Material Modeling 30
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Modern Representation
Material independent equations(as usual as integral relations)
Balance of massBalance of momentumBalance of moment of momentumBalance of energyBalance of entropy
Material dependent equationsConstitutive equationsEvolution equations
ConditionsBoundary Conditions
Kinematic conditionsStatic conditionsMixed conditions
Initial ConditionsDisplacementsVelocities
15 May 2015 Altenbach Material Modeling 31
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Material Dependent Equations
Specific (individual) response of the given material onarbitrary load.Modeling principles
Inductive approachfrom the simplest to more complex modelsDeductive approachfrom the general frame to special cases
IdentificationExperimental observationsMathematical analysisTheory of symmetry (Curie-Neumann’s principle)
15 May 2015 Altenbach Material Modeling 32
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Outline
1 Motivation
2 A Simple Example
3 Historical Remarks
4 Creep-Damage Modeling
15 May 2015 Altenbach Material Modeling 33
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Aim (I)
Development of phenomenological constitutive equationsthat describe inelastic behavior at elevated temperature,
To characterize hardening, recovery, and softeningprocesses a composite (fraction) model with creep-hardand creep-soft constituents is introduced,
In the case of heat resistent steels the volume fraction ofthe creep-hard constituent is assumed to decreasetowards a saturation value,
Such an approach describes well the primary creep as aresult of stress redistribution between constituents andtertiary creep as a result of softening (decrease of thevolume fraction)
15 May 2015 Altenbach Material Modeling 34
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Aim (II)
To describe the whole creep curve a damage parameter inthe sense of continuum damage mechanics is introduced,
The material parameters and the response functions in themodel are calibrated against published experimentalcurves for X20CrMoV12-1 steel,
To verify the model predictions with experimental creepcurve under stress change conditions and the stress-straincurve under constant strain rate are compared,
The consideration of both softening and damageprocesses is necessary to characterize the long termstrength in a wide stress range,
The model can be generalized to the case of multi-axialstress state,
15 May 2015 Altenbach Material Modeling 35
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
References
K. Naumenko, H. Altenbach & A. Kutschke: A combinedmodel for hardening, softening, and damage processes inadvanced heat resistant steels at elevated temperature. -Int. J. Damage Mech. 20(2011). - pp. 578-597
K. Naumenko, A. Kutschke, Y. Kostenko & T. Rudolf:Multi-axial thermo-mechanical analysis of power plantcomponents from 9-12% Cr steels at high temperature. -Engng Frac. Mech. 78(2011)8. - pp. 1657-1668
15 May 2015 Altenbach Material Modeling 36
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Microstructure of Advanced Chromium Steels
Group of 9% to 12% Cr-steels
Complex microstructure with grains, precipitates, subgrains anddislocations.Source: H. Chilukuru: On the microstructural basis of creep strength andcreep-fatigue interaction in 9-12% Cr steels for application in power plants,PhD-thesis, University Erlangen-Nuremberg, 2007
15 May 2015 Altenbach Material Modeling 37
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Basic Idea - Microstructure Observation
M23C6-carbideMX-precipitate
Prior Austenite-grain boundary20 µm
200 nm
Lath-Martensite-grain boundarSubgrain boundary
20 µm
Packet boundary
Block boundary
Gaffard et al. (2004) Masuyama (2001)Polcik et al. (1998) Kimura (2006)
15 May 2015 Altenbach Material Modeling 38
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
References
V. Gaffard, J. Besson, A.F. Gourgues-Lorenzon: Creep failuremodel of a 9Cr-1MoNbV (P91) steel integrating multipledeformation and damage mechanisms. - ECF 15 (Stockholm)2004. - 8 p.
Polcik, P., Straub, S., Henes, D. and Blum, W.: Simulation of theCreep Behaviour of 9-12% CrMo-V Steels on the Basis ofMicrostructural Data. In: Strang, A., Cawley, J. & Greenwood,G.W. (eds), Microstructural Stability of Creep Resistant Alloys forHigh Temperature Plant Applications, 1998, CambridgeUniversity Press. - pp. 405-429.
F. Masuyama: History of power plants and progress in heatresistant steels. - ISIJ (Iron and Steel Institute of Japan)International 41(2001)6. - pp. 612-625
Kimura, K.: 9Cr-1Mo-V-Nb Steel. In: Martienssen, W. (ed.),Creep Properties of Heat Resistant Steels and Superalloys,2006, Springer. - pp. 126-133
15 May 2015 Altenbach Material Modeling 39
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Characteristic Material Behavior
0 0.02 0.04 0.05 0.08 0.1 0.12 0.14
100
200
300
400
500
600
0 0.05 0.1 0.15 0.2 0.25
102
103
104
105
106
ε [-]
σ[M
Pa]
Experimental dataafter Röttger 1997
600◦C
530◦C
500◦C
creep strain
norm
aliz
edcr
eep
rate
|σ| = 150MPa (compression)σ = 150MPa (tension)|σ| = 175MPa (compression)σ = 175MPa (tension)|σ| = 185MPa (compression)σ = 185MPa (tension)|σ| = 196MPa (compression)σ = 196MPa (tension)
Experimental data after STRAUB (1995)
ε [-]
σ [MPa] 1 8151000 1200 1300
stress controlled, symmetric loading cycle,after Bunch and McEvily (1987)
plastic material behavior -hardening and softening
creep behavior - only first andthird creep stage observable
cyclic behavior - width of stressstrain loop increases, stressstrain loop is shifted ratcheting
15 May 2015 Altenbach Material Modeling 40
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Basic Idea - Binary Mixture Approach (I)
σs σ σh
F
Stress and Strain Rate Relations for Iso-strain Assumption
σ = (1 − ηh)σs + ηhσh
ηh - volume fraction of the inelastic hard phase
ε = εs = εh
15 May 2015 Altenbach Material Modeling 41
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Basic Idea - Binary Mixture Approach (II)
Young’smoduli
nonlinear elementinelastic soft
nonlinear elementinelastic hard
volume fraction
1D Visualization - Iso-strain Approach
kinematic backstress follows for different inelasticproperties, if volume fraction is kept constant it results in abackstress model (Frederick & Armstrong, 2007)
if the volume fraction of the creep hard constituentdecreases towards a saturation value the creep rateincrease after the minimum creep rate is reached
15 May 2015 Altenbach Material Modeling 42
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Basic Idea - Binary Mixture Approach (III)
Young’smoduli
nonlinear elementinelastic soft
nonlinear elementinelastic hard
volume fraction
Behavior of the Constituents
soft: εs =σ
E+ f (|σs|)
σs
|σs|
hard: εh =σ
E+
σh − σ
σh∗− σ
|ε|
E - macroscopic Young’s modulus,f - stress response function
15 May 2015 Altenbach Material Modeling 43
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Derivation of the Model (I)
The aim is to express the unknowns whether by
known variables,
terms of material model parameters to identify or
internal state variables
1D Equation - Fraction Model
Basic features
because of the iso-strain condition εcr = εs,
for constant stress, as in creep tests, εcr = εs = f (|σs|)σs
|σs|,
σ = ηsσs + ηhσh, σ - applied load,
the sum of the volume fractions is equal to 1, 1 = ηs + ηh,
therefore ηs can be replaced by 1 − ηh,
σ = (1 − ηh)σs + ηhσh
15 May 2015 Altenbach Material Modeling 44
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Derivation of the Model (II)
Introducing New Variables
Volume fraction reduces with time: Γ =ηh
ηs
ηs0
ηh0
, 1 ≥ Γ ≥ Γ∗,
ηs0
ηh0
= ch, ch - material model parameter to identify
With Γ the unknown ηh can be expressed as ηh =Γ
Γ + ch,
Therefore one can write σ = (1 −Γ
Γ + ch)σs +
Γ
Γ + chσh
By algebraic manipulations one can obtainσch = σsch + Γ (σh − σ)
By utilizing the overstress concept the variable β can beintroduced chβ = σh − σ
Finally one obtains σs = σ − βΓβΓ can be termed backstress or kinematic stress
15 May 2015 Altenbach Material Modeling 45
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Derivation of the Model (III)
Inner State Variables β and Γ
β describes the stress accumulation in the inelastic hardphase, as internal state variable an evolution equation isneeded
one can show by proper algebraic manipulations that fromthe iso-strain condition and by the chosen rate formulationfor the inelastic hard phase a formulation of the
Frederick-Armstrong type follows, β =Ech
εcr(
1 −β
β∗
)
Γ is controlling the volume change of the inelastic hardphase and is motivated by microstructural changes
the idea for the evolution equation is taken from literature:Γ = As(Γ∗ − Γ )εcr
Source: K. Naumenko et al. Engng Frac. Mech. 78(2011)8, 1657-1668
15 May 2015 Altenbach Material Modeling 46
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Derivation of the Model (IV)
Summary
εcr = f (σ − βΓ )
β =Ech
εcr(
1 −β
β∗
)
β evolution equation can be integrated and written in termsof εcr:
β = β∗
(
1 − e−E
chβ∗εcr)
Γ = As(Γ∗ − Γ )εcr
Γ evolution equation can be integrated and written in termsof εcr:
Γ = Γ∗ + (1 − Γ∗)(
e−AsΓ∗
εcr)
15 May 2015 Altenbach Material Modeling 47
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Derivation of the Model (V)
Required response functions
the stress response function f of the inelastic soft phaseneeds to be defined
a temperature response function that describes thetemperature dependence
β∗ and Γ∗ are stress level dependent and thereforeresponse functions need to be found
Identified Response Functions
viscosity function: f (x) = a0e−αT sinh(Bx)
β∗-hardening saturation β∗(x) = H∗x
Γ∗-softening saturation Γ∗(x) =aΓ
1 + bΓe−x
cΓ
15 May 2015 Altenbach Material Modeling 48
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Derivation of the Model (VI)
Continuum Damage Mechanics Approach
evolution equation:
ω = r (ω)h(σ)|εcr|
ε∗(σ)
with:
r (ω) = lωl−1
l ,
ε∗(σ) = εbr +aε
1 + b−
|σ|cε
ε
,
h(σ) =σ + |σ|
2σ
15 May 2015 Altenbach Material Modeling 49
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Derivation of the Model (VII)
Identified Response Functions
temperature response function: g(T ) = a0e−αT
stress response function: f (x) = g(T ) sinh(B
E (T )x)
β∗-hardening saturation β∗(x) = H∗x
Γ∗-softening saturation Γ∗(x) =aΓ
1 + bΓe−
xcΓ
15 May 2015 Altenbach Material Modeling 50
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Calibration Process (I)
Temperature and Stress Response Function
g(T ) = a0e−αT , f (σ) = g(T ) sinh
[
BE (T )
σ
]
stress, [MPa]
min
imal
cree
pst
rain
rate
,[
1 h
]
15 May 2015 Altenbach Material Modeling 51
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Calibration Process (II)
Stress Response Function and Backstress
g(T ) = a0e−αT , f (σ) = g(T ) sinh
[
BE (T )
σ
]
β = β∗
(
1 − e−E
chβ∗ε
cr)
creep strain, [-]
cree
pst
rain
rate
,[
1 h
]
15 May 2015 Altenbach Material Modeling 52
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Calibration Process (III)
Stress Response Function, Backstress and Softening
g(T ) = a0e−αT , f (σ) = g(T ) sinh
[
BE (T )
σ
]
β = β∗
(
1 − e−E
chβ∗ε
cr)
, Γ = Γ∗ + (1 − Γ∗)(
e−AsΓ∗
εcr)
creep strain, [-]
cree
pst
rain
rate
,[
1 h
]
15 May 2015 Altenbach Material Modeling 53
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Calibration Process (IV)
Backstress and Softening
g(T ) = a0e−αT , f (σ) = g(T ) sinh
[
BE (T )
σ
]
β = β∗
(
1 − e−E
chβ∗ε
cr)
, Γ = Γ∗ + (1 − Γ∗)(
e−AsΓ∗
εcr)
creep strain, [-]
cree
pst
rain
rate
,[
1 h
]
15 May 2015 Altenbach Material Modeling 54
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Calibration Process (V)
Backstress and Softening, but ch and As Averaged
g(T ) = a0e−αT , f (σ) = g(T ) sinh
[
BE (T )
σ
]
β = β∗
(
1 − e−E
chβ∗ε
cr)
, Γ = Γ∗ + (1 − Γ∗)(
e−AsΓ∗
εcr)
creep strain, [-]
cree
pst
rain
rate
,[
1 h
]
15 May 2015 Altenbach Material Modeling 55
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Calibration Process (VI)
β∗ and Γ∗
g(T ) = a0e−αT , f (σ) = g(T ) sinh
[
BE (T )
σ
]
β = β∗
(
1 − e−E
chβ∗εcr)
, Γ = Γ∗ + (1 − Γ∗)(
e−AsΓ∗
εcr)
β∗(σ) = H∗σ, Γ∗(σ) =aΓ
1 + bΓe−
σcΓ
σ, [MPa]σ, [MPa]
β∗, [
MP
a]
Γ∗, [
-]
15 May 2015 Altenbach Material Modeling 56
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Calibration Process (VII)
Extended Experimental Data
various stress levels (80 MPa to 460 MPa)
various temperature levels (400◦C up to 650C◦)
σ, [MPa]
cree
pst
rain
rate
,[
1 h
]
creep strain, [-]
cree
pst
rain
rate
,[
1 h
]
15 May 2015 Altenbach Material Modeling 57
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Calibration Process (VIII)
Introducing a Second Backstress
creep strain, [-]creep strain, [-]
cree
pst
rain
rate
,[
1 h
]
cree
pst
rain
rate
,[
1 h
]
550◦C, 280 MPa - 310 MPa 600◦C, 150 MPa - 230 MPa
600◦C, 235 MPa - 320 MPa550◦C - 650◦C, 230 MPa
15 May 2015 Altenbach Material Modeling 58
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Comments Concerning the Calibration Process
Conclusions
The Frederick-Armstrong-type backstress law followsnaturally from the proposed model
The evolution of microstructure is taken into account
The stress accumulation in only one inelastic hard phaseseems to not be sufficient
With the improved backstress formulation all creep curveswere described better
Future Tasks
Modify response functions, e.g. temperature
Simulate the stress strain behavior outside the calibrationrange
Analyze the anomalous ratchetting behavior of 12%-Crsteels
15 May 2015 Altenbach Material Modeling 59
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Extension to 3D
Set of Equations (Small Strains, Isotropic Behavior)
creep rate tensor εεεcr =32
f (σvM
1 − ω)
sssσvM
−1
2Gddt
(Γβββ),
sss = sss − Γβββ, σvM =
√
32
tr(sss)2
backstress tensor βββ = Ah(23εεεcr − εvM
βββ
β∗(σvM))
scalar softening Γ = As(Γ∗(σvM) − Γ )εcrvM
scalar damage ω = r (ω)h(σσσ)εcr
vM
ε∗(σvM), h(σσσ) =
12
(σI + |σI |)σI
K. Naumenko, H. Altenbach & A. Kutschke: A Combined Modelfor Hardening, Softening, and Damage Processes in AdvancedHeat Resistant Steels at Elevated Temperature, InternationalJournal of Damage Mechanics, 2011
15 May 2015 Altenbach Material Modeling 60
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Verification - Creep Curves
0 0.05 0.1 0.15 0.2 0.25
102
103
104
105
106
creep-strain
norm
aliz
edcr
eep
rate
|σ| = 150MPa (compression)σ = 150MPa (tension)|σ| = 175MPa (compression)σ = 175MPa (tension)|σ| = 185MPa (compression)σ = 185MPa (tension)|σ| = 196MPa (compression)σ = 196MPa (tension)
model-compression
model-tension
Experimental data after STRAUB (1995)
15 May 2015 Altenbach Material Modeling 61
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Verification - Creep Curves (Continued)
Explanatory Notes
0 0.05 0.1 0.15 0.2 0.25
102
103
104
105
106
creep-strain
norm
aliz
edcr
eep
rate
|σ| = 150MPa (compression)σ = 150MPa (tension)|σ| = 175MPa (compression)σ = 175MPa (tension)|σ| = 185MPa (compression)σ = 185MPa (tension)|σ| = 196MPa (compression)σ = 196MPa (tension)
model-compression
model-tension
Experimental data after STRAUB (1995)
The presented creep curvesdo not exhibit a secondarycreep stage with a constantcreep rate.
The difference of the creeprate slope between tensionand compression, above acertain time, is obvious.
The time when the creep rate under tension is changingrapidly is stress level dependent.
The difference between compression and tension isexplained by damage, which is controlled by micro-voids.Under compression micro-voids are not observable.
15 May 2015 Altenbach Material Modeling 62
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Verification - Cyclic Creep
0 0.05 0.1 0.15 0.2 0.25
102
103
104
105
106
150
170
196
creep strain
norm
aliz
edcr
eep
rate
|σ| = 150MPa (compression)|σ| = 196MPa (compression)
Exp. data (var. stress)Model (var. stress)
Experimental data after STRAUB (1995)
time, h
|σ|,
MP
a
15 May 2015 Altenbach Material Modeling 63
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Verification - Cyclic Creep (Continued)
0 0.05 0.1 0.15 0.2 0.25
102
103
104
105
106
150
170
196
|ε|cr
|εcr|/ε 0
|σ| = 150MPa (compression)|σ| = 196MPa (compression)
Exp. data (var. stress)Model (var. stress)
Experimental data after Straub [1995]
|ε|cr
|σ|,
MP
a The creep behavior undercyclic loading is wellreproduced by thedeveloped model.
The experiment shows that the level of the creep responsefor constant stresses 150 MPa and 196 MPa is not thesame level for the periodically constant stresses 150 MPaand 196 MPa, which is reproduced by the model.
The deviation after the changed stress level from 196 MPato 150 MPa will be investigated further.
15 May 2015 Altenbach Material Modeling 64
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Verification - Stress Strain Diagramms
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Strain
100
200
300
400
500
600S
tres
s
500◦C, model simulation530◦C, model simulation600◦C, model simulation500◦C, Experimental data (Straub et al., 1997)530◦C, Experimental data (Straub et al., 1997)600◦C, Experimental data (Straub et al., 1997)
15 May 2015 Altenbach Material Modeling 65
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Verification - Stress Strain Diagramms (Continued)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Strain
100
200
300
400
500
600
Str
ess
500◦C, model simulation530◦C, model simulation600◦C, model simulation500◦C, Experimental data (Straub et al., 1997)530◦C, Experimental data (Straub et al., 1997)600◦C, Experimental data (Straub et al., 1997)
The stress-strain behaviorunder a constantstrain-rate is wellreproduced by thedeveloped model.
As highlighted the transition from the linear to nonlinearbehavior is too strict predicted by the model.
A reason could be the slope of the increasing back-stress,which is maybe overestimated.
15 May 2015 Altenbach Material Modeling 66
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Summary (I)
Actual Output
In the range of calibration the models work fine and are able toreproduce the material behavior.
Physical processes on the microstructure level areconsidered, but the model parameters are determined withthe help of macroscopic creep tests.
Complex creep curves, hardening, softening, damage arewell predicted by the models.
The models show some predictive strength if loadingconditions differ from the range of calibration.
Numerical implementations are possible.
15 May 2015 Altenbach Material Modeling 67
Motivation A Simple Example Historical Remarks Creep-Damage Modeling
Summary (II)
Still Open Questions
Softening is partly overestimated by the model forstress-strain behavior.
For lower temperatures the models predicts too stronghardening.
The transition from linear elastic to nonlinear stress-strainbehavior is too strict.
Next Steps
Extend the database of calibration, replace/modify responsefunctions
15 May 2015 Altenbach Material Modeling 68