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Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys Daniel Leidermark

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Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys Daniel Leidermark
DEPARTMENT OF MANAGEMENT AND ENGINEERING
Mechanical Behaviour of Single-Crystal
Nickel-Based Superalloys
Master Thesis carried out at Division of Solid Mechanics
Linköping University
January 2008
Daniel Leidermark
LIU-IEI-TEK-A--08/00283--SE
Institute of Technology, Dept. of Management and Engineering,
SE-581 83 Linköping, Sweden
Framläggningsdatum
Presentation date
2008-01-28
Publiceringsdatum
Publication date
2008-02-04
Språk
Language
Avdelning, institution
Division, department
Division of Solid Mechanics
Dept. of Management and Engineering
SE-581 83 LINKÖPING
Rapporttyp
Report category
Svenska/Swedish
X Engelska/English
Licentiatavhandling
ISBN:
ISRN: LIU-IEI-TEK-A--08/00283--SE
X Examensarbete
C-uppsats
D-uppsats
Serietitel:
Title of series
Övrig rapport
Serienummer/ISSN:
Number of series
URL för elektronisk version
URL for electronic version
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-10722
Titel
Title
Mechanical behaviour of single-crystal nickel-based superalloys
Författare
Author
Daniel Leidermark
Sammanfattning
Abstract
In this paper the mechanical behaviour, both elastic and plastic, of single-crystal
nickel-based superalloys has been investigated. A theoretic base has been established
in crystal plasticity, with concern taken to the shearing rate on the slip systems.
A model of the mechanical behaviour has been implemented, by using FORTRAN, as a
user defined material model in three major FEM-programmes. To evaluate the model
a simulated pole figure has been compared to an experimental one. These pole figures
match each other very well. Yielding a realistic behaviour of the model.
Nyckelord:
Keyword
material model, single-crystal, superalloy, crystal plasticity, LS-DYNA, ABAQUS, ANSYS,
FORTRAN, pole figure
V
Abstract
In this paper the mechanical behaviour, both elastic and plastic, of singlecrystal nickel-based superalloys has been investigated. A theoretic base has
been established in crystal plasticity, with concern taken to the shearing
rate on the slip systems. A model of the mechanical behaviour has been
implemented, by using FORTRAN, as a user defined material model in three
major FEM-programmes. To evaluate the model a simulated pole figure has
been compared to an experimental one. These pole figures match each other
very well. Yielding a realistic behaviour of the model.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
VII
Preface
This work was done during the autumn of 2007 as a master thesis at Linköping
University. I would like to thank my two supervisors, Kjell Simonsson and
Sören Sjöström, for all their help and hints during the work of this master
thesis. A big appreciating for the support and interesting discussions with all
the PhD colleagues at the division. Also the financial support from the KME
programme is appreciated. A big thanks to Jonas Larsson at Medeso AB for
testing the material model in ANSYS. I would like to thank the people at
SIEMENS in Finspång, for letting me be "one of the team" during the three
weeks I spent there. Big thanks to Johan Moverare and Ru Peng who solved
the mystery of the pole figure, that had haunted me for weeks. A special
thanks to my family who have supported and pushed me all the way from
the time that I was a child to now. And finally my girlfriend Maria, who I
like to thank for always being there for me.
Daniel Leidermark
Linköping, January 2008
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
IX
Nomenclature
Variable
Ak
a
b
a1 , a2
Ce
cm
C1 , C2
c
D
Ee
E
F
Fe
Fp
G
Gs
Gαr
hαβ
hβ
h0
I
K
KI , KII , KIII
L
m
M 1, M 2
Nf
nα
q αβ
q
q
R
s
sα
S
T
Vk
W
Description
Associated thermodynamic forces
Hardening parameter
Material parameter
Crystal orientations
Elastic tangent stiffness tensor
Constants material array
Material parameter
Material parameter
Rate of deformation tensor
Elastic Green-Lagrange strain tensor
Modulus of elasticity
Total deformation gradient
Elastic deformation gradient
Plastic deformation gradient
Shear modulus
Reference slip resistance
Slip resistance of each slip system
Strain hardening rate
Single slip hardening rate
Reference hardening rate
Unit tensor
Bulk modulus
Stress intensity factor in Mode I, II, III
Velocity gradient
Slip rate sensitivity
Structural tensors
Fatigue life
Normal vector of each slip system
Latent-hardening matrix
Latent-hardening ratio
Heat flux
Load ratio
Specific entropy
Slip direction of each slip system
2:nd Piola-Kirchhoff stress tensor
Temperature
Internal state variables
Spin tensor
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
X
Variable
α
γ̇0
∆γ α
∆γ αmax
∆ǫ
∆KI
∆t
εy
η
J
λ
µ
ω
∇T
P int
ρ
σ
σu
σy
τ
τα
Φ
φ
Ψ
Ω
Ω̄
Ω̄iso
Ω0
Description
Slip system
Reference shearing rate
Shearing rate of each slip system
Maximum shearing rate
Strain amplitude
Range of the stress intensity factor in Mode I
Timestep
Strain in y-direction
Elastic parameter
Jacobian determinant
Lamé constant
Lamé constant
Mandel stress tensor
Temperature gradient
Internal power
Density
Cauchy stress tensor
Ultimate stress
Tensile load in y-direction
Kirchhoff stress tensor
Resolved shear stress
Thermodynamic dissipation
Plastic lattice rotation
Helmholtz free energy
Current configuration
Intermediate configuration
Isoclinic intermediate configuration
Reference configuration
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
XI
Contents
1 Introduction
1.1 Gas turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Superalloys . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
3
2 Fatigue
2.1 Low-cycle fatigue . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Thermomechanical fatigue . . . . . . . . . . . . . . . . . . . .
5
5
5
3 Crack propagation
7
4 Crystal structure
9
5 Theory
5.1 Tangent stiffness tensor
5.2 Kinematics . . . . . .
5.3 Crystal plasticity . . .
5.3.1 Virgin material
5.4 Thermodynamics . . .
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6 Implementation
6.1 Elastic material model . . . . . .
6.1.1 Validation . . . . . . . . .
6.2 Crystal plasticity material model
6.2.1 Validation . . . . . . . . .
6.3 Flowchart . . . . . . . . . . . . .
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7 Discussion
33
7.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
XII
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
The interior of a stationary power generating gas turbine . .
The In-Phase thermomechanical fatigue cycle . . . . . . . .
The Out-of Phase thermomechanical fatigue cycle . . . . . .
Crack loaded in different Modes . . . . . . . . . . . . . . . .
FCC crystal structure . . . . . . . . . . . . . . . . . . . . .
The (111) plane in the unit cell . . . . . . . . . . . . . . . .
Material description, with a plastic lattice rotation. . . . . .
Material description, without a plastic lattice rotation. . . .
Rotation of the crystal orientation . . . . . . . . . . . . . . .
The cube loaded uniaxially by a tensile load . . . . . . . . .
Stereographic projection and (001) standard pole figure . . .
Crystal orientations in correspondence to the global coordinate system used by Kalidindi and (011) pole figure . . . . .
(001) pole figure of the deformed cube . . . . . . . . . . . .
Flowchart of analysis done by LS-DYNA . . . . . . . . . . .
Flowchart of analysis done by ABAQUS . . . . . . . . . . .
Flowchart of analysis done by ANSYS . . . . . . . . . . . .
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List of Tables
1
2
3
4
Slip systems . . . . . . . . . . . . .
Material parameters for pure nickel
Material parameters for copper . .
Initial slip hardening parameters for
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copper
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— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
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1
1
Introduction
In gas turbines the operating temperature is very high. The temperature is
so high that regular steels will melt. Therefore superalloys are often used
to manage the high temperature. The superalloys treated in this report are
single-crystal superalloys, which have even better properties against temperature then their coarse-grained polycrystal cousins. The thermal efficiency
increases with the operating temperature of a gas turbine and therefore the
temperature is increasing with every new turbine that is developed. When
the temperature is getting higher and higher the components of the turbine
will be more and more exposed to fatigue, which will limit the lifetime of the
turbine components. At a certain point the turbine components will reach
the crack initiation point due to fatigue and the crack will then propagate
through the single-crystal with little resistance. The designer wants to produce better and more efficient gas turbines which can manage higher and
higher temperatures. This requires that under the development of new gas
turbines there are tools and directives which take all of these aspects into
consideration.
How do the components of the turbine handle certain temperatures and load
cycles? How long is the life of the components? When will a crack be initialised and propagate? The first thing is to look into the material and see
how it behaves. This is done by developing a constitutive model of the superalloy that will handle all of these aspects.
SIEMENS Industrial Turbomachinery AB in Finspång, Sweden, develops
and manufactures gas turbines for a wide range of applications. SIEMENS is
participating in a research programme that aims at solving material related
problems associated with the production of electricity based on renewable fuels and to contribute in the development of new materials for energy systems
of the future. This programme, called Konsortiet för Materialteknik för termiska energiprocesser (KME), was founded in 1997 and consists at present
of 8 industrial companies and 12 energy companies participating through
Elforsk AB in the programme. Elforsk AB, owned jointly by Svensk Energi
(Swedenergy) and Svenska Kraftnät (The Swedish National Grid), started
operations in 1993 with the overall aim to coordinate the industry’s joint
research and development.
The here presented master thesis has been carried out as a first step in
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
2
1 INTRODUCTION
one of the KME-projects, namely KME-410 Thermomechanical fatigue of
notched components made of single-crystal nickel-base superalloys. The basic
goal of this project is to develop constitutive and life prediction models for
superalloys in gas turbines. At this early stage of the project the lifetime
estimation has not yet been addressed and the focus has instead been placed
on the the basic constitutive model, which so far has been made in two
versions: one elastic model and one crystal plasticity model.
1.1
Gas turbines
The function of a gas turbine is to supply electric power, to propel heavy
machinery or transport vessels such as ships and aircrafts. A gas turbine
basically consists of a compressor, a combustor and a turbine, see Figure 1.
The incoming air is compressed in the compressor to increase the pressure of
the air. The compressed air then enters the combustion chamber, where it is
mixed with the fuel and ignited. These hot gases will then flow through the
turbine and by doing so make the turbine rotate. The temperature of the
turbine components can range from 50◦ C to 1500◦C [1]. The turbine drives
the compressor by the jointly connected shaft. In jet engines the hot gases are
then passed through a nozzle, giving an increase in thrust as it is returned to
normal atmospheric pressure. For stationary power generating gas turbines
there is an extra power turbine which generates electricity, instead of the
nozzle.
Compressor
Power turbine
Combustor
Turbine
Figure 1: The interior of a stationary power generating gas turbine, with
permission from SIEMENS
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
1.2 Superalloys
1.2
Superalloys
A superalloy is an alloy that exhibits excellent mechanical strength and creep
resistance at high temperatures. A superalloy also has very good corrosion
and oxidation resistance. The word alloy means to combine or bind together.
In this context it means to combine materials with beneficial properties and
in doing so receive a material that has a combination of these properties.
Nickel-based superalloys are alloys based on nickel. Nickel is used as the base
material on account of its face-centered cubic (FCC) crystal lattice structure,
which is both ductile and tough, its moderate cost and low rates of thermally
activated processes. Nickel is also stable in the FCC form when heated from
room temperature to its melting point, i.e., there are no phase transformations. Compared with other typical aerospace materials, like titanium and
aluminium, nickel is a rather dense material, which is due to its small interatomic distances.
There are often more than 10 different alloying materials in a superalloy,
each with their specific enhancing property. The alloying materials reside in
different phases, which for a typical nickel-based superalloy are [2]
1. The gamma phase, γ. This phase exhibits the FCC crystal lattice
structure and it forms a matrix phase, in which the other phases reside.
Common materials of this phase are nickel, iron, cobalt, chromium,
molybdenum, ruthenium and rhenium.
2. The gamma prime phase, γ ′ . This ordered phase is promoted by additions of aluminium, titanium, tantalum, niobium and presents a barrier
to dislocations. The role of this phase is to confer strength to the superalloy.
3. Carbides and borides. These segregate to the grain boundaries of the
γ phase, as a grain boundary strengthening element. Carbon, boron
and zirconium often reside in this phase.
There are also other phases in certain superalloys. However these should be
avoided, because they do not promote the properties of the superalloys.
The historical development of superalloys started prior to the 1940s, these
superalloys were iron-based and cold wrought. In the 1940s the investment
casting was introduced of cobalt-based superalloys, by which the operating
temperature was raised significantly. These were mainly used in aircraft jet
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
3
4
1 INTRODUCTION
engines and land turbines. During the 1950s the vacuum melting technic
was developed allowing a fine control of the chemical composition of the superalloys, which in turn led to a revolution in processing techniques such
as directional solidification of alloys and of single-crystal superalloys. In
the 1970s powder metallurgy was introduced to develop certain superalloys,
leading to improved property uniformity due to the elimination of microsegregation and the development of fine grains. In the later part of the 20th
century the superalloys have become commonly used for many applications.
Single-crystal superalloys are alloys that only consist of one grain. They
have no grain-boundaries, hence grain-boundary strengtheners like carbon
and boron are unnecessary. Grain-boundaries are easy diffusion paths and
therefore reduce the creep resistance of the superalloys. Due to the nonexistence of grain-boundaries single-crystal superalloys possess the best creep
properties of all superalloys.
Nickel-based superalloys are used in aircraft and industrial gas turbines as
blades, disks, vanes and combustors. Superalloys are also used in rocket
engines, space vehicles, submarines, nuclear reactors, military electric motors,
chemical processing vessels, and heat exchanger tubing.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
5
2
Fatigue
The gas turbine will be exposed to load- and temperature cycling. This leads
to fatigue of the components.
2.1
Low-cycle fatigue
Low-cycle fatigue (LCF) will take place when the temperature is below the
creep regime or when a component is affected by isothermal cycling. Where
the stress is high enough for plastic deformation to occur in a component,
then the stress is not as useful as the strain when it comes to describing
this. Hence, low-cycle fatigue is usually characterised by the Coffin-Manson
relation, expressed here in both elasticity and plasticity
∆ǫ =
C2 b
Nf + C1 Nfc
E
(1)
where ∆ǫ is the strain amplitude, E is the modulus of elasticity, Nf is the
fatigue life and C1 , C2 , b, c are material parameters. Equation (1) is also
known as universal slope .
2.2
Thermomechanical fatigue
When it comes to fatigue of high temperatures and temperature cycling it
is called thermomechanical fatigue (TMF). This is taken into concern when
the material is in the creep regime. There are two essential types of TMF
cycles: In-Phase cycle and Out-of Phase cycle.
• In-Phase cycle
This is when the strain and the temperature are cycled in phase, see
Figure 2. A typical example is a cold spot in a hot environment, which
at high temperature will be loaded in tension and at low temperature
loaded in compression.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
6
2 FATIGUE
σ
Tmax
ε
Tmin
Figure 2: The In-Phase thermomechanical fatigue cycle
• Out-of Phase cycle
This is when the strain and the temperature are cycled in counterphase,
see Figure 3. A typical example is a hot spot in a cold environment,
which at low temperature will be loaded in tension and at high temperature loaded in compression.
σ
Tmin
ε
Tmax
Figure 3: The Out-of Phase thermomechanical fatigue cycle
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
7
3
Crack propagation
After a number of load cycles, a fatigue crack may initiate and then propagate through the superalloy. In a coarse-grain polycrystal material the crack
will be slowed down due to the grain boundaries. In single-crystal superalloys there are no grain boundaries, implying that the crack propagation will
encounter very little resistance. There are three specific crack modes, which
are seen in Figure 4.
a)
b)
c)
Figure 4: Crack loaded in a) Mode I, b) Mode II and c) Mode III. From [3],
with permission from Dahlberg T.
For each of the three crack modes there is one corresponding stress intensity
factor
√
KI = σyy πaf
(2)
√
KII = τxy πag
(3)
√
(4)
KIII = τyz πah
where a is the crack length and f , g, h are functions of geometry and type
of loading. In cyclic loading the crack growth can be described by Paris’ law
da
= C (∆KI )n
dN
(5)
where C and n are material properties and ∆KI is the range of the stress intensity factor due to crack growth in Mode I. For a more detailed description
of the crack growth, the effect of the load ratio has to be included. Paris’
law is then modified by the R-value, which is the load ratio between the
minimum stress σmin and the maximum stress σmax , expressed as
R=
σmin
σmax
(6)
For more information the reader is referred to e.g. Suresh [4].
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
9
4
Crystal structure
The material used in gas turbines is nickel-based superalloys. The material
properties of nickel in the high temperature domain are very good. Nickel
does not change its properties with the temperature as much as other materials, which makes it a good base material in superalloys. There is a wide
range of other alloying materials present in superalloys as well. With nickel
as the base material the superalloys have the same crystal lattice structure
as nickel, namely face-centered cubic (FCC), see Figure 5.
Figure 5: FCC crystal structure
The FCC structure is a very close-packed structure with a coordination number of 12 [5], which is the maximum. The coordination number is the number
of atoms surrounding each particular atom in the structure. Inelastically, the
material deforms primarily along the planes which are most tightly packed,
these are called close-packed planes. The FCC structure has 4 close-packed
planes which, in Miller indices, are of the family {111}.

(111)



(1̄11)
{111}
(11̄1)



(1̄1̄1)
The unit cell of the crystal structure, with plane (111), is seen in Figure 6.
The axes of the unit cell, labelled a1 , a2 and a3 , define the crystal orientation
with respect to the global coordinate system. It is most likely that the crystal
orientation do not coincide with the global coordinate system.
In each of these planes there are three slip directions, disregarding the negative directions. These directions are the most close-packed directions in each
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
10
4 CRYSTAL STRUCTURE
a3
a2
a1
Figure 6: The (111) plane in the unit cell
planes. The slip directions are of the family < 110 >.

 [110] [1̄10]
[101] [1̄01]
< 110 >

[011] [01̄1]
These planes and their respective slip directions constitute 12 slip systems,
which are shown in Table 1. Note that the respective slip directions and
normal directions are orthogonal.
α
1
2
3
4
5
6
s¯α
1
√ [11̄0]
2
1
√ [1̄01]
2
1
√ [011̄]
2
1
√ [101]
2
1
√ [1̄1̄0]
2
1
√ [011̄]
2
Table 1: Slip systems
n¯α
α
s¯α
1
1
√ [111]
7 √ [1̄01]
3
2
1
1
√ [111]
8 √ [01̄1̄]
3
2
1
1
√ [111]
9 √ [110]
3
2
1
1
√ [1̄11]
10 √ [1̄10]
3
2
1
1
√ [1̄11]
11 √ [101]
3
2
1
1
√ [1̄11]
12 √ [01̄1̄]
3
2
n¯α
1
√ [11̄1]
3
1
√ [11̄1]
3
1
√ [11̄1]
3
1
√ [1̄1̄1]
3
1
√ [1̄1̄1]
3
1
√ [1̄1̄1]
3
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
11
5
Theory
Basic knowledge in continuum mechanics [6] and material mechanics [7] is
essential for the understanding of the following section. All the theory is made
in a large deformation context with general tensor notation and Cartesian
coordinates. Note: ẋ = dx
dt
5.1
Tangent stiffness tensor
The tangent stiffness tensor has been adopted from Schröder et al. [8] and
reworked to suit the behaviour of a superalloy. Nickel-based superalloys are
elastically anisotropic when in single-crystal form. Hence the stiffness is
dependent on the crystal orientation relative to the loading direction. The
tangent stiffness tensor yields
C e =λI ⊗ I + µ(I⊗I + I⊗I) + 2η(M 1 ⊗ M 1 + M 2 ⊗ M 2
+ M 1 ⊗ M 2 − I ⊗ M 1 − I ⊗ M 2)
(7)
where λ, µ are the Lamé constants, η is an additional third elastic parameter
and M 1 , M 2 are structural tensors that depend on the crystal orientations
a1 , a2 accordingly to
M 1 = a1 ⊗ a1
(8)
M 2 = a2 ⊗ a2
(9)
The operator ⊗, called dyadic product, assembles two vectors to a 2:nd order
tensor, two 2:nd order tensors to a 4:th order tensor, etc. In Cartesian
coordinates the dyadic product takes the form
(a ⊗ a)ij = ai aj
(10)
(M ⊗ M )ijkl = Mij Mkl
(11)
(M ⊗M )ijkl = Mik Mjl
(12)
(M ⊗M )ijkl = Mil Mjk
(13)
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
12
5 THEORY
5.2
Kinematics
When the body is deformed the material description is changed. As seen in
Figure 7, the body undergoes a deformation from the reference configuration
(Ω0 ) to the current configuration (Ω). Instead of taking the direct way,
with the use of the total deformation gradient F , the other way through the
isoclinic intermediate configuration (Ω̄iso ) and the intermediate configuration
(Ω̄) can be taken [9]. The first step is performed by shearing of the lattice,
due to the plastic deformation gradient F p . The lattice then undergoes a
plastic lattice rotation φ. Finally, the lattice is both elastically stretched
and rotated by the elastic deformation gradient F e .
n¯α
nα
F
sα
s¯α
Ω0
Ω
Fe
Fp
n¯α
n¯α0
φ
s¯α0
Ω̄
s¯α
Ω̄iso
Figure 7: Material description, with a plastic lattice rotation.
The total deformation gradient is thus divided into an elastic part and a
plastic part, through the following multiplicative decomposition.
F = F e φF p
(14)
In the subsequent discussion the plastic lattice rotation φ is set equal to
the unit tensor, so the material exhibits no plastic lattice rotation. There— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
13
5.2 Kinematics
fore the material description in Figure 8 is used instead, hence the isoclinic
intermediate configuration becomes the intermediate configuration.
n¯α
F
nα
sα
s¯α
Ω0
Ω
Fp
Fe
n¯α
Ω̄
s¯α
Figure 8: Material description, without a plastic lattice rotation.
The multiplicative decomposition is then expressed as [10]
F = F eF p
(15)
The velocity gradient can then also be expressed in an elastic part and a
plastic part.
−1
−1
−1
L = Ḟ F −1 = F˙ e F e + F e F˙ p F p F e
(16)
From Equation (16) the following can be defined
−1
Le = F˙ e F e
(17)
−1
−1
Lp = F e F˙ p F p F e
(18)
L̄p = F˙ p F
(19)
p−1
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
14
5 THEORY
where Le , Lp are the elastic and plastic velocity gradient, respectively, defined in the current configuration (Ω) while L̄p is the plastic velocity gradient
defined in the intermediate configuration (Ω̄).
The velocity gradient can be divided into one symmetric part and one skewsymmetric part.
L=
1
1
L + LT +
L − LT = D + W
2
2
(20)
where D is the rate of deformation tensor (symmetric) and W is the spin
tensor (skew-symmetric). These two can each be divided into an elastic part
and a plastic part, accordingly to
D = De + Dp
(21)
W = We + Wp
(22)
where
1 p
1 e
T
T
L + Le , D p =
L + Lp
2
2
1 e
1 p
T
T
We =
L − Le , W p =
L − Lp
2
2
De =
(23)
(24)
The elastic Green-Lagrange strain tensor Ē e measured relative to the intermediate configuration is defined as
1 eT e
Ē e =
F F −I
(25)
2
The relationship between the elastic rate of deformation tensor D e defined in
the current configuration and the elastic Green-Lagrange strain rate tensor
Ē˙ e defined in the intermediate configuration is given by a push-forward or a
pull-back operation [11]
De = F e
−T
T
−1
Ē˙ e F e , Ē˙ e = F e D e F e
(26)
The 2:nd Piola-Kirchhoff stress tensor S̄ defined in the intermediate configuration can be expressed by a pull-back of the Kirchhoff stress tensor τ from
the current configuration by the following
−1
S̄ = F e τ F e
−T
T
⇒ τ = F e S̄F e = J σ
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
(27)
15
5.2 Kinematics
where J = det F e . As can be seen, in order to receive the Cauchy stress
tensor the Kirchhoff stress tensor is scaled by the Jacobian determinant [11].
The internal power P int , when a body is deformed, is defined as
Z
int
P =
σ :DdV
(28)
Ω
The internal power can be divided into an elastic part and a plastic part
Z
Z
e
int
P =
σ :D dV + σ :Dp dV
(29)
Ω
Ω
The elastic part is transformed by a regular pull-back to the intermediate
configuration, accordingly to
Z
Z
e
σ :D dV = S̄: Ē˙ e dV̄
(30)
Ω
Ω̄
and the plastic part is transformed, due to symmetry of σ and D, as
Z
Z
Z
Z
T
p
p
e
p e
σ :D dV = σ :L dV = S̄: F L F dV̄ =
: L̄p dV̄ (31)
ω
Ω
Ω
Ω̄
Ω̄
ω
where
is the Mandel stress tensor. The Mandel stress tensor is a nonsymmetric tensor and it is defined in the intermediate configuration. From
Equation (31) it can be seen that the Mandel stress tensor in relation to the
2:nd Piola-Kirchhoff stress tensor is given by
T
= F e F e S̄
(32)
ω
This can be further developed with the insertion of Equation (27), so that it
relates to the Kirchhoff stress tensor by
T
= Fe τFe
−T
(33)
ω
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
16
5 THEORY
5.3
Crystal plasticity
In a tension test of a single-crystal the axial load that initiates plastic flow
depends on the crystal orientation. A shear stress acting in the slip direction,
on the slip plane, must be produced by the axial load. It is this shear stress,
called the resolved shear stress, that initiates the plastic deformation. It is
expressed by Schmid’s law, with the Kirchhoff stress tensor, as
τ α = nα τ sα
(34)
The slip occurs on the slip systems that exhibit the greatest resolved shear
stress. If only one slip system is active, the other slip systems have a smaller
resolved shear stress than the initial critical stress and due to this slip does
not occur on these systems. This is called single slip. Secondary slip systems
can also be activated, this is then called multi slip, but these are not considered in this report.
During deformation of a sample, either in tension or compression, the crystal
orientation will rotate. As seen in Figure 9 the normal direction n¯α will
rotate away from the axial axis in tension and toward it in compression.
The slip systems defined in Chapter 4 are defined in the intermediate configuration. The slip directions may be transformed into the current configuration
by
sα = F e s¯α
(35)
Since s¯α and n¯α are unit vectors and orthogonal to each other it follows that
n¯α · s¯α = nα · sα = 0
(36)
Hence, the transformation for the normal vector can be defined as
nα = n¯α F e
−1
(37)
where sα and nα are orthogonal to each other but no longer unit vectors.
In this work it has as, a first step, been decided to use a crystal plasticity
model that has already been developed and which is used in the scientific
society. This is done to receive results at an early stage in the project. The
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
17
5.3 Crystal plasticity
n¯α
nα
nα
sα
s¯α
sα
a)
b)
Figure 9: Rotation of the crystal orientation in a) tension b) compression
model will be developed further in a more thoroughly study in the future.
Hence, the following model is adopted from the work of Kalidindi [12] and
Hopperstad [13].
As mentioned above, plastic deformation occurs due to slip on the active slip
systems [14], in the current configuration this is expressed as
X
Lp =
γ˙α sα ⊗ nα
(38)
α
where γ˙α is the shearing rate on the slip system α. With the use of Equation
(35) and (37) the plastic deformation can be expressed in the intermediate
configuration as
X
L̄p =
γ˙α s¯α ⊗ n¯α
(39)
α
The plastic part of the internal power in the current configuration can be
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
18
5 THEORY
expressed with Equation (38), taking into account that dV = J dV̄ , as
Z
Z X
Z X
p
α
α
α
˙
σ :L dV =
γ τ :(s ⊗ n ) dV̄ =
γ˙α τ α dV̄
(40)
Ω
Ω
Ω
α
α
or in the intermediate configuration, with Equation (39), as
Z
Z X
Z X
p
: L̄ dV̄ =
γ˙α :(s¯α ⊗ n¯α ) dV̄ =
γ˙α τ α dV̄
ω
ω
Ω̄
Ω̄
Ω̄
α
(41)
α
where τ α is the resolved shear stress. As Equation (40) and (41) yields the
same result, the resolved shear stress can be expressed from both of them, as
ω
τ α = τ :(sα ⊗ nα ) =
:(s¯α ⊗ n¯α )
(42)
It can be shown from Equation (42) that the following is true
τ α = sα τ nα = s¯α n¯α
(43)
ω
which represents Schmid’s law in both the current- and the intermediate configuration.
The shearing rate on the slip system α is in this work assumed to obey the
following viscoplastic relation [12]
γ˙α = γ˙0
|τ α |
Gαr
m1
sgn (τ α )
(44)
where γ˙0 is the reference shearing rate, m is the slip rate sensitivity and Gαr
is the slip resistance on the slip system α.
The hardening rate on slip system α is calculated by
X
G˙αr =
hαβ γ˙β (45)
β
where hαβ is the strain hardening rate on slip system α due to shearing on
slip system β, if α = β then hαα is the self-hardening rate of slip system α
and if α 6= β then hαβ is the latent-hardening rate of slip system α caused
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
19
5.3 Crystal plasticity
by slip on system β.
The strain hardening rate is defined as
hαβ = q αβ hβ
(46)
where q αβ is the matrix that describes the latent-hardening of the singlecrystal and hβ is the single slip hardening rate. In the latent-hardening
matrix q αβ the systems {1, 2, 3}, {4, 5, 6}, {7, 8, 9} and {10, 11, 12} are coplanar. The ratio between the latent-hardening rate and the self-hardening rate
are unity, for coplanar slip systems. The non-coplanar systems depend on
the latent-hardening ratio parameter q (typically 1 ≤ q ≤ 1.4), which represents a stronger latent-hardening effect in the intersecting slip systems [12].
If q = 1, then only self-hardening is obtained. The latent-hardening matrix
consists of


A qA qA qA
 qA A qA qA 

(47)
q αβ = 
 qA qA A qA 
qA qA qA A
where


1 1 1
A= 1 1 1 
1 1 1
The single slip hardening rate hβ is composed of
a
Gβr
β
h = h0 1 −
Gs
(48)
(49)
where h0 is the reference hardening rate, Gs is the reference slip resistance
and a is a slip system hardening parameter.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
20
5 THEORY
5.3.1
Virgin material
For an initially virgin material with only one slip system activated, e.g. α =
1, there is only one resolved shear stress, hence
τα = τ1
(50)
There exists an initial slip resistance on each slip plane, prior to deformation.
The slip resistance of the active slip system is
Gαr = G1r
(51)
The other slip systems are not updated due to slip on these systems, but
they grow because of the latent-hardening from the active slip system α.
The shearing rate on the active slip system can then be expressed as
γ˙1 = γ˙0
|τ 1 |
G1r
m1
sgn τ 1
(52)
Because it is only one activated slip system the strain hardening rate, with
β = α, is expressed as
a
Gαr
αα
αα α
αα
h = q h = q h0 1 −
(53)
Gs
The latent-hardening matrix q αα then yields unity in all positions, representing that on the active slip system only self-hardening is obtained. The
hardening rate Ġ1r for this slip system then yields
1 m1
1 a
G
|τ
|
r
1 1
sgn τ (54)
Ġr = h0 1 −
γ˙0
Gs G1r
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
21
5.4 Thermodynamics
5.4
Thermodynamics
The Helmholtz free energy is assumed given by
Ψ = Ψ(Ē e , Vk )
(55)
where Ē e is the elastic Green-Lagrange strain tensor in the intermediate
configuration and Vk are the internal state variables, which account for the
loading history of the material. Differentiating this yields
dΨ
∂Ψ ˙ e ∂Ψ ˙
=
Ē +
Vk
dt
∂Vk
∂ Ē e
(56)
The 2:nd principle of thermodynamics can be expressed with the use of
Helmholtz free energy, leading to the Clausius-Duhem inequality [7]
∇T
≥0
(57)
σ :D − ρ Ψ̇ + sṪ − q
T
where ρ is the density of the current configuration, s is the specific entropy,
T is the temperature, q is the heat flux and ∇T is the temperature gradient.
For isothermal conditions motivating the adopted form of the Helmholtz
free energy and with a decoupled thermal and mechanical dissipation, with
ρ0 = J ρ, the following mechanical dissipation inequality is received
τ :D − ρ0
dΨ
≥0
dt
(58)
Equation (58) can be further developed with the insertion of Equation (56)
and by separating D into an elastic part and a plastic part which can be
taken from Equations (30) and (31), respectively. This gives
ω
S̄: Ē˙ e +
: L̄p − ρ0
∂Ψ ˙ e
∂Ψ ˙
Vk ≥ 0
e Ē − ρ0
∂Vk
∂ Ē
(59)
Considering only elastic deformations and since Clausius-Duhem inequality
holds for any particular Ē˙ e it follows that
S̄ = ρ0
∂Ψ
∂ Ē e
(60)
and, as a result the following relation is received
: L̄p − Ak V˙k ≥ 0
(61)
ω
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
22
5 THEORY
where the thermodynamic forces associated with the internal variables are
Ak = ρ0
∂Ψ
∂Vk
(62)
From this the dissipation is received
ω
Φ=
: L̄p − Ak V˙k ≥ 0
(63)
By, as a specific case letting, the Helmholtz free energy take the form (no
internal variables)
Ψ=
1 e e e
C : Ē : Ē
2ρ0
(64)
Equation (60) then yields
S̄ = ρ0
∂Ψ
= C e : Ē e
∂ Ē e
(65)
The adopted constitutive formulation must be associated with a positive
dissipation. Under the given conditions requirement is
ω
Φ=
: L̄p ≥ 0
(66)
With the insertion of Equation (41) and (44) this gives
1
ω
Φ=
p
: L̄ =
X
α
γ˙α τ α
= γ˙0
X |τ α | m
α
Gαr
|τ α | ≥ 0
(67)
which implies that the 2:nd principle of thermodynamics is fulfilled.
However, since the hardening of the material is surely associated with a
microstructural storage of energy, it follows that the use of the above model
in a thermomechanical analysis will overestimate the heat production, due
to the inelastic flow. Thus, in the future, a more thermomechanical realistic
model will be developed.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
23
6
Implementation
The material model has been implemented in the three FEM-programmes
LS-DYNA [15], ABAQUS [16] and ANSYS [17], which are world wide spread
and extensively used in industry.
At first only an elastic material model was developed. This was later developed further into a crystal plasticity material model. The elastic material
model has been implemented in all three FEM-programmes, while the crystal
plasticity material model so far only has been implemented in LS-DYNA. For
the two models, the following 11 parameters must be given as input data
cm(1) = K
cm(2) = G
cm(3) = λ
cm(4) = µ
cm(5) = η
cm(6) = a1 (1)
cm(7) = a1 (2)
cm(8) = a1 (3)
cm(9) = a2 (1)
cm(10)= a2 (2)
cm(11)= a2 (3)
Bulk modulus
Shear modulus
Lamé constant
Lamé constant
Elastic parameter
Crystal orientation
Crystal orientation
Crystal orientation
Crystal orientation
Crystal orientation
Crystal orientation
The bulk modulus and the shear modulus are calculated from λ and µ accordingly to an isotropic elastic behaviour (Hooke’s law). They are only needed
in LS-DYNA for calculating an estimate of the critical time step (needed
for explicit analysis), and they are thus not used in the material models.
When performing a simulation, the only thing to do is to give these 11 input parameters. Based on the given input the material models will calculate
the internal forces and for an implicit analysis the tangent stiffness tensor
too. The material parameters concerning the slip hardening are incorporated
in the implemented code. These will be input parameters in the input file
in the near future, hence all parameters will be specified in the input data file.
The density of the material is also set in the input data file but is not used
in the material models. The total deformation gradient is calculated by the
specific FEM-programme in use. The material models were implemented in
FORTRAN 77.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
24
6 IMPLEMENTATION
6.1
Elastic material model
The elastic material model calculates the Cauchy stress tensor for the next
time step (n + 1). The deformation gradient consists only of the elastic part,
thus the total deformation gradient equals the elastic deformation gradient.
The analysis is done implicitly.
Pidgin code
1
E n+1 =
F Tn+1 F n+1 − I .
2
S n+1 = C e E n+1 .
1
σ n+1 =
F n+1 S n+1 F Tn+1 .
det F n+1
6.1.1
Validation
To evaluate that the elastic material model calculates the right result a validation of the obtained modulus of elasticity was made. Different moduli of
elasticity will be received for different crystal orientations. A cube with sides
of length 1 m was uniaxially loaded in the y-direction by a tensile load of
σy = 1 · 106 P a, see Figure 10.
z
σy
y
x
Figure 10: The cube loaded uniaxially by a tensile load
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
25
6.1 Elastic material model
Material properties of pure nickel were used, since its moduli of elasticity in
the crystal orientations [100], [110] and [111] are known, see e.g. [2]. These
are
E[100] = 125 GP a
E[110] = 220 GP a
E[111] = 294 GP a
Other material parameters that were used in the analysis are specified in
Table 2.
Table 2: Material parameters for pure nickel
η
λ
-147 GP a 13.5 GP a
µ
ρ
118.5 GP a 8.902 · 103 kg/m3
From the FEM-programme the following uniaxial strains were obtained for
the three crystal orientations
εy
εy
εy
[100]
[110]
[111]
= 7.98702 · 10−6
= 4.52995 · 10−6
= 3.33786 · 10−6
The moduli of elasticity were then calculated by
σy = Eεy
(68)
yielding the following results
E[100] = 125.203 GP a
E[110] = 220.753 GP a
E[111] = 299.593 GP a
which match those from above very well.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
26
6 IMPLEMENTATION
6.2
Crystal plasticity material model
The crystal plasticity material model calculates the Cauchy stress tensor for
the next time step, but also the slip resistance and the plastic deformation
gradient are calculated. The crystal plasticity material model has only been
implemented for explicit analysis in LS-DYNA.
Pidgin code
loop, k : 1 → 12
F en+1,k = F n+1,k F pn+1,k−1
1
T
e
Ē n+1,k = (F en+1,k F en+1,k − I)
2 e
S̄ n+1,k = C e Ē n+1,k
−1
T
ω
n+1,k
= F en+1,k F en+1,k S̄ n+1,k
loop, α : 1 → 12
¯α
n+1,k n
ω
α
τn+1,k
= s¯α
αmax
αmax
∆γn+1
= γ̇n+1
∆tn+1 = ∆tn+1 γ̇0
end loop
α ! m1
τ max n+1,k
αmax
sgn
τ
n+1,k
α
Gr,nmax
F pn+1,k − F pn+1,k−1
−1
αmax αmax
=
F pn+1,k−1 = γ̇n+1
s̄
⊗ n̄αmax
∆tn+1
αmax αmax
⇒ F pn+1,k = I + ∆γn+1
s̄
⊗ n̄αmax F pn+1,k−1
end loop
pαmax
L̄n+1
F en+1 = F n+1 F pn+1
−1
T
= F en+1 F en+1 S̄ n+1
1
−T
σ n+1 =
F en+1
e
det F n+1
e ¯α
α
s =F s
ω
n+1
eT
n+1 F n+1
ω
nα = n¯α F e
−1
Gαr,n+1 = Gαr,n +
X
β
q αβ h0
Gβr,n
1−
Gs
!a
β γ̇n+1 ∆tn+1
where n is the current time step. The first loop is over all the slip planes,
where k symbolises which plane that is observed. In this loop the plastic
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
6.2 Crystal plasticity material model
deformation gradient is updated and the slip system that experience the
greatest resolved shear stress is determined. When the all planes have been
dealt with the plastic deformation gradient reaches its final value. This is
used to update the elastic deformation gradient and the Cauchy stress is calculated. The slip directions and the normal directions are also transformed
into the current configuration by the elastic deformation gradient and finally
the slip resistance is updated.
Initially the plastic deformation gradient F pn+1,k−1 equals F pn,k−1 and the
initial value for F pn,k−1, with n = 0 and k = 1, is the unit tensor I. This
corresponds to that the intermediate configuration coincide with the reference
configuration initially.
6.2.1
Validation
The validation of the crystal plasticity material model is of a more complicated matter compared with the elastic material model. The slip systems
are transformed to the current configuration, which reflects that the body
is deformed. The normal directions of the deformed slip systems can then
be evaluated by a stereographic projection [18], and in doing so a so called
pole figure is received. The (001) pole figure means that the normal direction (001) of the crystal is orientated in the centre of the pole figure. The
stereographic projection is created by letting all the normal directions of the
planes be extended to an imaginary reference sphere. A projection plane is
placed tangent to this sphere. At the other side of the sphere and orthogonal
to the projection plane a point is marked. This point is called point of projection, from this point lines are drawn to intersect the normal directions at
the sphere radius. These lines then cut the projection plane at a number of
places, that mark the relative position of the planes of the crystal structure,
see Figure 11.
The simulated pole figure is then compared with an experimental pole figure
done by X-ray diffraction, of a sample of the same material that is deformed
in the same way as in the simulation. These two should coincide to reflect
that the simulation gives the same result as the experiment.
A single-crystal copper cylinder was investigated by Kalidindi [12], which
was compressed in room temperature by an axial strain rate of −0.001 s−1 .
The material properties and slip hardening parameters of copper, that were
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
27
28
6 IMPLEMENTATION
Projection plane
Reference sphere
P′
P
Point of
projection
a)
b)
Figure 11: a) Stereographic projection and b) (001) standard pole figure.
From [18]
used in the experiment, can be seen in Table 3 and 4, respectively. The slip
resistance Gαr is initially set equal to Gr0 , which is the initial value of the slip
resistance on each slip plane.
Table 3: Material parameters for copper
η
λ
µ
ρ
-104 GP a 20 GP a 75 GP a 8.93 · 103 kg/m3
The deformation of the copper cylinder was studied both computationally
and experimentally. Pole figures were drawn for both applications and compared in Kalidindi’s work, the received normal directions of the planes are
(101), (1̄01), (1̄10), (110) and in the centre (011), see Figure 12.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
29
6.2 Crystal plasticity material model
Table 4: Initial slip hardening parameters for copper
a
2.5
Gr 0
16 MP a
Gs
190 MP a
h0
250 MP a
m
q
γ˙0
0.012 1.4 0.001 s−1
y
(1̄10)
z
a3
a2
(110)
(011)
x
45o
a1 ,x
a)
y
b)
(1̄01)
(101)
Figure 12: a) Crystal orientations in correspondence to the global coordinate
system used by Kalidindi b) (011) pole figure. From [12]
With the pole figures from Figure 12 as reference, the simulated pole figure
were to be duplicated. The same material properties for the copper were
used, but the geometry was a bit different. It was a cube with sides of
length 1 m, instead of a cylinder, and, further, it consisted of only one
element. This would not change the slip plane deformation, due to the singlecrystal structure and a state of homogeneous deformation. A stereographic
projection was made in MATLAB using the normal directions of the slip
planes from the deformed cube, with the global z-axis in the centre. The
crystal orientations, in the simulation, are expressed in the global coordinate
system, hence the use of the (001) pole figure to evaluate the directions. This
simulated pole figure was then compared to the (001) standard pole figure in
Figure 11b) to get the corresponding crystal orientations of the slip planes.
These are (101), (1̄01), (1̄1̄0) and (11̄0), see Figure 13. To compare this
pole figure with the one in Figure 12 one has to rotate it 45o around the
a1 -axis, due to that Kalidindi expresses the global coordinates in the crystal
orientations, and use the opposite normal directions of (1̄1̄0) and (11̄0). As
one can see they match each other very well, since (1̄1̄0) and (11̄0) are the
same planes as (110) and (1̄10) but with opposite normal directions. The
exception is the plane in the centre (011), this plane does not experience any
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
30
6 IMPLEMENTATION
slip because the load is parallel to the normal direction. Hence, by use of
Schmid’s law, the resolved shear stress is equal to zero on this plane and due
to this the shearing rate is also zero. So in the search for the slip system that
exhibit the greatest resolved shear stress, then this one is not included in the
material model, hence no point corresponding to (011) is present in Figure
13.
a2
(1̄01)
(101)
a1
(1̄1̄0)
(11̄0)
Figure 13: (001) pole figure of the deformed cube
6.3
Flowchart
The specific FEM-programmes have different interfaces with the two implemented material models. There are some "boxes" that are used for all
FEM-programmes, and these are
• Neutral Material Model
Contains either the elastic material model or the crystal plasticity material model. Calculates the corresponding stress state and for the
plasticity model also the plastic deformation gradient and the slip resistance.
• Const
This calculates the tangent stiffness tensor (not yet for the crystal plasticity material model).
• Subroutines
Contains all the subroutines used in the main programmes, e.g. calculations of the inverse and dyadic product, Voight-transformations etc.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
31
6.3 Flowchart
Furthermore, for the specific implementations it is to be noted that the
flowcharts are structured as
• LS-DYNA
If the calculations are to be done implicitly then the material model is
called by two routines. These are umat50 and utan50. If the analysis
is to be done explicitly then only one command is used, umat50. The
utan50 is used to calculate the tangent stiffness tensor. When the
calculations are done implicitly then the timestepping algorithm is done
by Newton’s method, hence the need of the tangent stiffness tensor.
In explicit analysis the timestepping is done by the central difference
method. The flowchart for LS-DYNA can be found in Figure 14, where
the dashed rectangle is only used for implicit analysis.
umat50
Neutral Material Model
LS-DYNA
utan50
Const
Subroutines
Figure 14: Flowchart of analysis done by LS-DYNA
• ABAQUS
Only the elastic material model has been implemented for ABAQUS,
thus the analysis is done implicitly. ABAQUS call the material model
by use of umat. The corresponding flowchart of the algorithms can be
found in Figure 15.
Neutral Material Model
ABAQUS
umat
Const
Subroutines
Figure 15: Flowchart of analysis done by ABAQUS
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
32
6 IMPLEMENTATION
• ANSYS
Only the elastic material model has been implemented for ANSYS,
thus the analysis is done implicitly. ANSYS call the material model by
use of usermat. The corresponding flowchart of the algorithms can be
found in Figure 16.
Neutral Material Model
ANSYS
usermat
Const
Subroutines
Figure 16: Flowchart of analysis done by ANSYS
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
33
7
Discussion
The work presented in this report has been to model and implement a crystal
plasticity material model for single-crystal superalloys. In the literature there
are many kinds of crystal plasticity models, some handle many aspects and
others less. The model used in this work focuses mainly on the hardening of
the single-crystal due to deformation on the slip systems.
The first step in the modeling was to develop an elastic material model,
which was evaluated against three given moduli of elasticity in certain crystal directions. The second step was to develop a crystal plasticity material
model, which depended on the crystal structure and how it deformed. The
crystal plasticity material model was evaluated against an experimental pole
figure. As can be seen from the validations, both material models (elastic
and plastic) exhibits the correct behaviour of single-crystal superalloys.
These models can also be used for coarse-grain polycrystal materials. The
models are then applied to each grain, each with their own crystal orientation.
To get satisfying orientations these need to be randomised, so in the preprocessor there is to be some kind of randomiser for each grain.
7.1
Future work
This work is a project in the programme KME, as been said above, hence
this is the first step in studying the fatigue behaviour of single-crystal superalloys. There are many more aspects, than those given in this report, to be
considered. In the future the following steps will be carried out:
• The crystal plasticity material model will be implemented for ABAQUS
and ANSYS. It will also be implemented as an implicit model, implying
that also a consistent tangent stiffness tensor needs to be set up.
• The high temperature in gas turbines is an essential source to fatigue
and thus the temperature dependency need to be implemented as a
next step.
• Handle that secondary slip systems might become activated, overshooting effect.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
34
7 DISCUSSION
• Handle the back-stress on the slip planes, due to the Bauschinger effect,
and incorporate this in the model.
• Handle LCF in the model and later TMF.
• Handle fatigue crack propagation.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
35
References
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of Uncoated and Coated Nickel-base Superalloys, Linköping University,
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36
[15] LS-DYNA, 2008-01-25, http://www.ls-dyna.com/
[16] ABAQUS, 2008-01-25, http://www.simulia.com/products/abaqus_fea.html
[17] ANSYS, 2008-01-25, http://www.ansys.com/
[18] Hertzberg R.W., 1996, Deformation and Fracture Mechanics of Engineering Materials, John Wiley & Sons Inc.
— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —
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