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Rajagopal.pdf
1
REPORT TITLE: AN INVESTIGATION INTO THE MECHANICS
OF SINGLE CRYSTAL TURBINE BLADES WITH A VIEW
TOWARDS ENHANCING GAS TURBINE EFFICIENCY
FINAL SCIENTIFIC REPORT
REPORTING PERIOD START DATE:10/01/01
REPORTING PERIOD END DATE: 3/31/05
PRINCIPAL AUTHORS: K. R. Rajagopal and I. J. Rao
DATE REPORT WAS ISSUED: May 5, 2006
DOE AWARD NUMBER: DE-FC-01NT41344
NAME AND ADDRESS OF SUBMITTING ORGANIZATION: Texas
A&M University, College Station, TX-77843
SUB-CONTRACTOR: New Jersey Institute of Technology, Newark, NJ
2
Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States
Government. Neither the United States Government nor any agency thereof, nor any of their
employees, makes any warranty, express or implied, or assumes any legal liability or
responsibility for the accuracy, completeness, or usefulness of any information, apparatus,
product, or process disclosed, or represents that its use would not infringe privately owned rights.
Reference herein to any specific commercial product, process, or service by trade name,
trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement,
recommendation, or favoring by the United States Government or any agency thereof. The views
and opinions of authors expressed herein do not necessarily state or reflect those of the United
States Government or any agency thereof.
3
Abstract
The demand for increased efficiency of gas turbines used in power generation and aircraft
applications has fueled research into advanced materials for gas turbine blades that can withstand
higher temperatures in that they have excellent resistance to creep. The term “Superalloys”
describes a group of alloys developed for applications that require high performance at elevated
temperatures. Superalloys have a load bearing capacity up to 0.9 times their melting temperature.
The objective of the investigation was to develop a thermodynamic model that can be used to
describe the response of single crystal superalloys that takes into account the microstructure of
the alloy within the context of a continuum model. Having developed the model, its efficacy was
to be tested by corroborating the predictions of the model with available experimental data. Such
a model was developed and it is implemented in the finite element software
ABAQUS/STANDARD through a user subroutine (UMAT) so that the model can be used in
realistic geometries that correspond to turbine blades.
4
Table of Contents
Executive Summary
Report Details
Introduction
Results and Discussions
Conclusions
Graphical Material list
References
page 5
page 7
page 7
page 20
page 24
page 25
page 29
5
Executive Summary
The demand for increased efficiency of gas turbines used in power generation and aircraft
applications has fueled research into advanced materials for gas turbine blades. Higher
efficiencies are possible if turbine blades can be designed to withstand inlet temperature of the
order of 2200 oC or more. At such high temperatures, it is critical to use materials that have
excellent resistance to creep. The term “Superalloys” describes a group of alloys developed for
applications that require high performance at elevated temperatures. Superalloys have a load
bearing capacity up to 0.9 times their melting temperature. They retain their strength even after
long exposure time at high temperatures and they have good low temperature ductility as well.
The objective of the investigation was to develop a rational thermodynamic model that
can be used to describe the response of single crystal superalloys that takes into account the
microstructure of the alloy within the context of a continuum model. Having developed the
model, its efficacy was to be tested by corroborating the predictions of the model with available
experimental data. Such a model was developed and it is implemented in the finite element
software ABAQUS/STANDARD through a user subroutine (UMAT) so that the model can be
used in realistic geometries that correspond to turbine blades.
We develop a constitutive theory within a thermodynamic setting to describe the creep of
single crystal super alloys that gainfully exploits the fact that the configuration that the body
would attain on the removal of the external stimuli, referred to as the ‘‘natural configuration’’,
evolves, with the response of the body being elastic from these evolving natural configurations.
The evolution of the natural configurations is determined by the tendency of the body to undergo
a process that maximizes the rate of dissipation. Here, the elastic response is assumed to be
linearly elastic with cubic symmetry associated with the body which remains the same as the
configuration evolves. A form for the inelastic stored energy (the energy that is _trapped_ within
dislocation networks) is utilized based on simple ideas related to the motion of the dislocations.
The rate of dissipation is assumed to be proportional to the density of mobile dislocations and
another term that takes into account the damage accumulation due to creep. The model developed
herein is used to simulate uni-axial creep of oriented single crystal nickel-base super alloys. The
predictions of the theory agree well with the available experimental data for CMSX-4.
The numerical scheme developed is implemented in UMAT to study the creep behavior of
single crystal superalloys loaded along the <001> orientation. The UMAT is validated by using it
to obtain the results already obtained in previous sections using the semi-inverse approach. The
results obtained through the UMAT is compared with the results obtained using the semi-inverse
approach and the experimental results. A comparison of strain versus time curves for loading
along <001> orientation at temperatures θ = 750, 982 and 1000 oC obtained using the UMAT
and the semi-inverse method and experimental results. We also make a comparison for the
inelastic stored energy at temperatures θ = 750, 982 and 1000 oC. A similar comparison is made
for third component of the backstress tensor as knowledge of these quantities is critical in the
design of single crystal super alloy blades.
We start with a discussion of single crystal superalloys, their microstructure that needs
to be reflected by the model. This is followed by a discussion of the various models that are
currently in vogue to describe the response of single crystal superalloys and their inadequacy in
describing the creep characteristics of the material. Next we discuss the general thermodynamic
framework within which the model is to be placed and follow this with the development of the
6
model. The specific model that is picked is introduced next, and the response of the model along
certain crystallographic directions, for the range of temperatures of technical relevance, are
compared against available experimental data. We find that the model performs very well and
does not have the shortcomings of the models that are in use currently.
7
REPORT DETAILS
1. INTRODUCTION
1.1 Microstructure of superalloys
Single crystal nickel-base superalloys have been developed for gas turbine blade
applications. These alloys have superior thermal, fatigue and creep properties compared to
conventional cast alloys because grain boundaries have been eliminated. Typical first generation
alloys include CMSX-2, second generation alloys include CMSX-4, MC-2, TMS-63, and third
generation alloys include CMSX-10 with high Re content. A typical modern superalloy (e.g.
CMSX-4) for turbine blades is a single crystal, which contains particles, based on the ordered
γ ′ L12 structure, lying in a matrix based on a disordered face-centered cubic Ni3Al. The γ ′ phase
forms remarkably regular cubes packed in a rather regular cubic array and it occupies 65–70% of
the volume. The two-phase structure of a superalloy contributes essentially to its excellent creep
strength at high temperatures, the phase boundaries providing obstacles to dislocation motion.
The volume fraction of the γ ′ phase is an important factor in optimizing superalloy composition
to get the best creep strength. Usually a maximum in the creep strength is reached between 70%
and 80% volume fraction of γ ′ phase with further increase leading to a significant drop in
strength (see [9]). It has been observed that in the primary stage of creep in modern superalloys,
and during most of the secondary creep, plastic deformation is confined to the γ channels. The
γ′
particles act as impenetrable obstacles. The γ ′ phase has another very remarkable property.
Whereas most metals and alloys, including the γ matrix, have flow stresses that decrease steadily
with increasing temperature, alloys related to Ni3Al, and many other alloys with the L12 structure,
show flow stresses that can increase by a factor of 5 as the temperature increases from room
temperature to about 650 oC [37]. The high strength of γ ′ is especially valuable at high
temperatures. The lattice parameters of the γ matrix and the γ ′ precipitate are very similar, but
not identical. The creep deformed microstructure and many mechanical properties depend on the
lattice misfit. The presence of various alloying elements strongly affects the value of misfit. The
misfit could be positive or negative depending on the particular composition of the superalloy.
Moreover, the misfit changes with the kind of heat treatment the alloy is subject to and it also
varies with temperature [46]. The sign of the misfit plays an important role in the evolution of
microstructure as the material creeps (see [46]).
Another important microstructural property of superalloys is the ability of cubic γ ′
phase to transform into flat plates (‘‘rafts’’) under the influence of stress and temperature. This
directional coarsening is especially important in nickel base superalloys because the
morphological changes in the two phase microstructure alter the creep resistance of the material
in the stress and temperature range where these alloys are used in applications such as turbine
blades. It has been shown by Nabarro [38] that in the elastic regime, the thermodynamic driving
force for rafting is proportional to the applied stress, to the lattice misfit and to the difference of
8
elastic constant of the γ and γ ′ phases. It has also been found that the direction of rafting
depends upon the direction of loading and the sign of lattice misfit. Two types of rafting behavior
in <001> oriented nickel-base single crystals have been identified.
1. Type N-rafts develop transverse to the direction of the externally applied stress.
2. Type P-rafts develop parallel to the direction of the externally applied stress.
Type N behavior is usually associated with negative misfit alloys stressed in tension, or positive
misfit alloys stressed in compression. Conversely, type P behavior is associated with positive
misfit alloys stressed in tension, and negative misfit alloys stressed in compression. The
differences in the microstructural evolution associated with a change from positive to negative
misfit are an indication that the rafting is primarily dominated by internal stresses developed due
to the misfit. In fact, the γ – γ ′ interface plays an important role in the creep property of
superalloys [8]. The evolution of rafts with creep depends on the applied stress and operating
temperature. At temperatures beyond 950 oC, experiments (Reed et al. [64]) suggest that rafting
is complete during very early stages of creep deformation. Thus, from the modeling point of
view, we can take into account, only the effect which a fully rafted microstructure confers.
However, at lower temperatures, it is likely that the rafts evolve at a rate comparable with the rate
of the evolution of the strain, in which case a suitable criterion for the evolution of the
microstructure is needed.
1.2 Creep Behavior of Superalloys
The creep behavior of single crystal superalloys is highly anisotropic. The inherent
crystallography of single crystals leads to orientation dependent creep behavior. From a design
point of view, it is imperative to use an orientation, which utilizes maximum strength of the
superalloy. In fact it is known that the creep strength of a modern single crystal superalloy along
the <001> orientation , which is also the preferred grain growth direction is favorable compared
to the <011> or <111> orientations.
There have been numerous experimental investigations into the creep behavior and the
related microstructural aspects of the <001> oriented single crystal nickel based superalloys.
Apart from this, experiments have also focused on characterizing the behavior of single crystal
turbine blades with centrifugal loading away from the exact <001> orientation. Such situations
are common in actual practice where mis-alignment of up to 15o [33] could occur due to variety
of reasons. A number of studies have been devoted to studying the creep performance of <001>
oriented superalloy single crystals. At lower temperatures, particularly in the vicinity of 750 oC, a
considerable amount of primary creep can occur (see [28] and [11]). At temperatures between
850 oC and 1000 oC, loading along <001> yields a creep strain rate which increases
monotonically with creep strain (i.e., tertiary creep is dominant), there being no evidence of a
steady state regime (See [5], [39] and [66]). At temperatures beyond 1000 oC, Reed et al. [64]
reported that rafting of γ ′ phase occurs very rapidly and is complete in the very initial stages of
creep deformation. After this stage, strain rate decreases with increasing strain for a considerable
amount of time. Reed et al. [64] concluded that this strain hardening effect arises as a
consequence of rafting of γ ′ phase. The strain rate in this temperature range keeps decreasing
with increasing strain until a critical strain is reached. After the critical strain is reached, the creep
9
strain rate increase sharply with strain with failure occurring eventually. Moreover, this critical
strain was found to be essentially constant in the temperature range of 1050-1200 oC. Reed et al.
[64] observed that the rapid increase in the creep strain in the later stages of creep is associated
with highly localized deformations in the vicinity of the fracture surface. Furthermore, this creep
deformation is associated with creep cavitation occurring at, or in the vicinity of casting porosity
and topologically closed packed (TCP) phases.
As pointed out earlier, the inherent crystallography of single crystals leads to orientation
dependent creep behavior. The degree of anisotropy is strongly influenced by the temperature
(around 750-850 oC) and it is also known that at higher temperatures, the orientation dependence
of creep behavior is less strong. Several studies have been devoted to study the effect of
orientation on creep behavior of single crystal superalloys and identify the slip systems
responsible for the observed deformation behavior. Experiments carried out by Kear and Piearcey
[23] on first generation single crystal nickel based superalloy MAR-M200 revealed that creep
resistance close to the <001> and <111> orientations is substantially better than that close to the
<011> orientation in the temperature range 760-871 oC. They also found that orientation has
much less influence on creep life at 982 oC. It was also observed that at 760 oC, <001>
orientation has the best creep life, however at temperatures 872 and 982 oC, <111> orientation
displayed the best creep life. Significant amount of primary creep was observed close to the
<001> orientation at 760 oC, however primary creep was absent for the <011> and <111>
orientations. At temperatures beyond 760 oC, tertiary creep was dominant in all the orientations
studied. Another experimental study on creep of MAR-M200 at 760 oC was performed by
Leverant and Kear [27], wherein they studied the creep behavior of specimens oriented within 18o
of <001> orientation. They observed primary, secondary and tertiary creep regimes for all the
orientations and noted that the primary and steady state creep rates increase in the following
order: <001>, <001>/<011> boundary, orientations between <001>/<011> and <001>/<111>
boundaries, <001>/<111> boundary. A similar study performed by MacKay and Maier [29] on
another first generation single crystal nickel based superalloy MAR-M247 at temperature 774 oC
showed that crystals having orientations within 25o of the <001> orientation exhibited
significantly longer creep lives when their orientations were closer to the <001>/<011>
boundary of the stereographic triangle than to the <001>/<111> boundary. These observations
were in accordance with the results for the creep of MAR-M200 ([37, 41]), the only difference
being that MAR-M247 showed best creep life close to the <111> orientation whereas MARM200 showed best creep life close to the <001> orientation. Caron et al. [6] studied the effect of
orientation on creep lives of first generation single crystal nickel based superalloy CMSX-2 at
760 oC and 750 MPa. Their experiments showed that the best creep life occurs close to the <001>
orientation, however unlike the results for MAR-M200 and MAR-M247 [37, 41, 43], CMSX-2
exhibited very poor creep life close to the <111> orientation. Moreover, orientations away from
the <001> orientation (say by 20o) did not cause significant reduction in creep life as was the
case for MAR-M200 and MAR-M247 [37, 41, 43].
The effect of orientation on creep behavior of second generation single crystal nickel
based superalloy CMSX-4 was studied by Matan et al. [32]. They studied the creep behavior for
small misorientations away from the <001> orientation. Their investigation showed that at 750
o
C, significant amount of primary creep takes place, the extent of which depends strongly upon
small misorientations away from the <001>/<011> boundary of the stereographic triangle. At
950 oC, tertiary creep is dominant with very little primary creep. They also observed that
orientation dependence is less strong at 950 oC. Recent creep tests carried out by Gunturi et al.
[19] on CMSX-4 at 750 oC in crystallographic orientations distant from the <001> orientation
showed that orientations distant from the <001>/<111> boundary had relatively lower creep
lives while orientations closer to the <001>/<111> boundary had longer creep life.
10
1.3 Previous works on single crystals
Several models have been proposed to describe the response of single crystals. The
notion of Bravais lattice has been associated with the structure of single crystals at the atomic
level to model its behavior. The single crystals are not free of imperfections in that they have
dislocations and inclusions which are responsible for the permanent inelastic deformation of
single crystals. In polycrystals, the presence of grain boundaries complicate the material behavior
signifcantly.
There are several studies on the kinematical aspects of crystals, under the assumption of
uniformly distributed dislocations (see Bilby [3], Eshelby [13], Kondo [24], KrÄoner [25] and
Nabarro [36]). A dynamical theory based for single crystals based on the notion of a Cosserat
continuum via the introduction of directors was established by Naghdi and Srinivasa [41]. The
theory based on the notion of directors has several inherent difficulties such as introduction of
new balance laws containing terms which are not physically motivated. Moreover there are
associated difficulties with regard to specifying boundary conditions for quantities such as
directors. Naghdi and Srinivasa [40] have also introduced a measure of the influence of
dislocations on the plastic deformation of single crystals through the curl of the plastic
deformation gradient.
Early experiments on single crystals were carried out by Ewing and Resenhain [14],
Taylor and Elam [71, 72, 73], Piercy et al. [44], and Kocks [45]. Piercy et al. [44] and Kocks [45]
studied multiple slips in single crystals. Various mechanisms have been proposed to explain the
response of single crystals and a discussion on these can be found in papers by Asaro [1], Havner
[20], Taylor [70] and Van Buereu [98]. The most important issue to recognize, concerning the
modeling of a single crystal is that it is not a simple material in the sense of Noll [43] (see also
Truesdell and Noll [74]), that is the stress in the material can not be purely determined by the
history of the deformation gradient. Several methods have been proposed to capture this nonsimple behavior of the body (for example, the theory based on directors) but they are fraught with
difficulties. One of the goals of the current work is to understand why the theory of simple
materials fail for single crystals and to develop a rigorous approach to model such non-simple
materials.
1.4 Current models for creep and shortcomings
There have been several attempts to model the creep behavior of single crystal
superalloys. Phenomenological models have been developed both to describe the creep
deformation of <001> oriented single crystals and to describe the orientation dependence of
creep behavior. Dyson and Mclean [11] observed that the tertiary creep rate in most engineering
materials including conventional nickel based superalloys increases monotonically with
accumulated plastic strain. They concluded that strain softening is caused by active damage
mechanisms (cavitation and development of cracks on the surface) and accumulation of
dislocations. They excluded the possibility of γ ′ phase coarsening causing the strain softening as
was thought earlier. They also presented an empirical model wherein the strain rate is determined
by inelastic strain rather than time. Following their observations, several empirical models for
11
creep in superalloys have been proposed that take into account the kinetics of the dislocation
motion [10]. The models developed in [11] and [10] are isotropic models and hence cannot
capture orientation dependent creep behavior. An extension of the isotropic model to capture the
anisotropic creep behavior was developed by Ghosh and Mclean [16] and Ghosh et al. [15]. Since
the level of primary creep observed in most single crystal superalloys is small, they restricted
their extension of anisotropy to the analysis of tertiary creep only and not to the primary creep.
Their model accounted for tertiary creep based on the accumulation of mobile dislocations with
plastic strain. Reed et al. [64] extended the model developed by Ghosh and co-workers to include
the effect of rafting at high temperature.
Several other models for creep of superalloys have been proposed based on the
framework of continuum damage mechanics and single crystal plasticity. Bertram and
Olschewski [2] proposed an anisotropic constitutive model for describing creep behavior of single
crystal superalloys. They constructed a three dimensional model by a projection technique which
is essentially a generalization of the four-parameter Burgers model. Their model was restricted to
the undamaged material behavior of the primary and secondary creep phase. Qi and Bertram [48]
extended the model using the theory of continuum damage mechanics to incorporate the damage
induced in the material through the introduction of a fourth order tensor that assesses damage.
Recently Maclachlan and Knowles [30] have proposed a model based on single crystal
plasticity wherein they incorporate the damage induced due to creep through a fourth order
damage operator. Most of the models for creep of single crystal superalloys fail to take into
account the symmetry of single crystals and the fact that the symmetry does not change as the
single crystal undergoes inelastic deformation. Apart from this, most of these models are
empirical in nature and lacked a three dimensional framework. These models also lack
thermodynamical underpinnings. Although models based on single crystal plasticity are three
dimensional and they incorporate the symmetry of single crystals, such models require extensive
details of slip systems which are operating. Also, the model requires information about self and
latent hardening of slip systems that are active which lead to a overwhelming number of material
parameters (The model developed by Maclachlan and Knowles [30] has 42 material parameters).
The effect of high temperature rafting on creep was incorporated in models developed by Reed et
al. [64] and Maclachlan and Knowles [30] who used a dislocation hardening mechanism first
proposed by Gilman [17]. Apart from including the effect of rafting on creep in a macroscopic
way, there have been several works devoted to describe the kinetics of the rafting behavior itself
but most of them are limited to the elastic regime (see for example [63, 34, 35, 24, 25, 15, 54]).
Such approaches are fraught with difficulties as the elastic regime is very difficult to detect in
modern superalloys and rafting is always associated with significant amount of inelastic strain.
Such a drawback was emphasized by the work of Carry and Strudel [10, 11], Ignat and coworkers [33, 7] and others where it was shown that interaction of dislocations created during
creep deformation with γ − γ ′ interface play an important role in morphological changes of
precipitates. The models developed by Socrate and Parks [68]and Veron et al. [75] attempted to
account for the inelastic strain, but their work was empirical in nature, lacked a 3D framework
and was within the purview of small strain theory.
1.5 Goals of the current work
In this work, the aim is to develop a constitutive theory within the context of continuum
mechanics, to predict the creep deformation of single crystal superalloys. The goal of such
12
continuum theories is to describe the macroscopic behavior of a material without explicitly going
into the complex details at the microscopic level, while at the same time taking cognizance of the
microstructure , albeit in a homogenized sense.
The constitutive model is within a thermodynamic setting and it exploits the fact that the
configuration that the body would attain on the removal of external stimuli, referred to as “natural
configuration”, evolves, with the response of the body being non-dissipative (in more general
situations non entropy producing) from these evolving “natural configurations”. The evolution of
these natural configurations is determined by the tendency of the body to undergo a process that
maximizes the rate of dissipation.
As mentioned before, it is important to recognize that single crystals can not be modeled
by theory of simple materials. The lattice structure and the material symmetry remains the same
when a single crystal is subject to inelastic deformation. This experimental fact was recognized
by the seminal work of Taylor and Elam [71] in as early as 1923. The current work aims to take
cognizance of this essential empirical fact and develop a constitutive theory that complies with
this observation. Another goal of the current work is to evaluate the theories based on single
crystal plasticity which explicitly take into account elaborate details of the motion of dislocations
on various slip systems and model the self and latent hardening of these systems during the
inelastic deformation process. Such an elaborate detail is not needed to model the inelastic
behavior of single crystals and a phenomenological continuum mechanics based model which
take cognizance of the microstructure in a homogenized sense will suffice. As pointed out earlier,
one of the shortcomings of incorporating elaborate details regarding motion of dislocations on
slip systems and the hardening of slip systems is that one ends up with an overwhelming number
of material parameters.
The constitutive model that has been developed is implemented in finite element software
ABAQUS/STANDARD through a user subroutine (UMAT). The User Material has been
developed and tested to ensure that the built in implicit creep integration routine based on a first
order backward difference operator works well. The results obtained through the use of UMAT
and those obtained using semi-inverse method work well. At this point the UMAT is ready to be
used for solving real world problems.
2. DEVELOPEMT OF CONTINUUM MODEL FOR CREEP OF SUPERALLOYS
The model that has been developed is within the framework of multiple natural configurations
developed by Rajagopal and coworkers. For materials undergoing large deformations, Eckart
[12] seems to have been the first to recognize that many materials can possess multiple stress free
states (natural configurations), that he called variable relaxed states, and studied them in some
detail. However, he did not worry about symmetry considerations of the variable relaxed states
or the role that the evolution of the symmetry plays in the constitutive relation for the material.
Nor was Eckart interested in placing the evolution of material structure within a thermodynamic
framework. A detailed discussion of the central role played by natural configurations in a variety
of dissipative processes with associated symmetry changes and the change of the response
characteristics of the body has been considered by Rajagopal [49, 50]. In fact the notion of
“Natural configurations” is central to the development of constitutive theories in continuum
mechanics. The crucial role it plays in describing the response of a broad range of material
behaviors has not been recognized and delineated in detail until the recent work of Rajagopal [49,
13
50]. Invoking the notion of “natural configurations” has led to the development of constitutive
theories which has filled the lacunae that existed in incorporating the microstructural details from
a continuum perspective. It has led to development of constitutive theories with rigorous
thermodynamic underpinnings without resorting to any ad hoc means such as invoking “internal
variables” into the theory. The phenomenal success of such a framework in describing the
response of a disparate class of materials can be seen in a series of papers by Rajagopal and
coworkers: the response of multi-network polymers [77, 61], twinning [52, 53], traditional plastic
response [54, 55], solid to solid phase transitions [52, 53], shape memory alloys [56], viscoelastic
response [57], anisotropic response of liquids [58], crystallization of polymers [63], superplastic
response [51], response of asphalt mixtures [35], growth and adaptation of biological materials
[62] and response of superalloys [47]. The classical theories of elasticity and linearly viscous
fluid arise naturally as sub-cases. The framework of multiple natural configurations exploits the
fact that the configuration that the body would attain on the removal of external stimuli, referred
to as “natural configuration”, evolves, with the response of the body being non-dissipative (in
more general situations non entropy producing) from these evolving natural configurations. The
evolution of these natural configurations is determined by the tendency of the body to undergo a
process that maximizes the rate of dissipation. Different natural configurations are accessed
during different processes. It is suffice to say that the notion of natural configuration is a
primitive in our framework and it can be thought of as one of the state variables in the
constitutive theory.
2.1 Development of constitutive model
Let us now start with the reduced energy dissipation equation (see Green and Naghdi [18]):
where where T is the Cauchy stress,
ρ
is the density, ψ is the Helmholtz potential, η is the
specific entropy, θ is the absolute temperature, q is the heat flux vector, n is the rate of entropy
production and f is the rate of dissipation.
In what follows, the effect of temperature is ignored and an isothermal model is developed. The
motivation to develop an isothermal model stems from the fact that all the creep experiments are
done at a constant temperature. The material parameters that will appear in the constitutive model
will be different for different temperatures.
Splitting the entropy production part as that due to thermal effects and a part due to mechanical
dissipation lead us to,
This equation is the starting point for the development of an isothermal constitutive model.
The form for the stored energy of the crystalline materials is assumed to be
14
It is assumed that the Helmholtz potential can be decomposed in the following way:
where
is related to the elastic stored energy, and
is related to the inelastic stored energy. A standard rearrangement will yield
It is now assumed that the Cauchy stress is of the form
On using the form for the Cauchy stress, the reduced dissipation equation can be decomposed to
where
For a material that is elastically isotropic, T will be an isotropic function of
τ =0.
The rate of dissipation due to creep is assumed to be of the form
Vκ p ( t )
so that
15
i.e., the rate of dissipation depends both upon the rate at which the material stretches as well as
the rate at which the orientation changes. For a specimen loaded in any arbitrary orientation, the
crystal lattice rotates. However, for uniaxial creep loading along orientations <001>, <111> and
<011>, there is no associated rotation of the crystal lattice.
Reducing the decomposed reduced dissipation equation further yields
2.2 Specific constitutive relations
2.2.1 The Helmholtz potential
The specific form for the elastic stored energy consistent with a crystal having cubic symmetry is
where a, b and c are orthogonal unit vectors along the principal cubic axes and c11, c12 and c44 are
three independent parameters characterizing the elastic response.
2.2.2 Inelastic part of the stored energy
The form for the inelastic stored energy captures the part of mechanical work that is trapped in
the dislocation networks. Models due to Lee [26], Brown et al. [4] and Mason et al. [31] account
for this kind of energy storage mechanism in the body by multiplying the “plastic work” by an ad
hoc factor whose value is approximately 0.8. The current work utilizes a rigorous form for such
an energy storage mechanism without resorting to any ad hoc means. The energy that is stored in
dislocation networks is of tremendous importance during the inelastic deformation of single
crystal superalloys as such superalloys are multi phase materials with their microstructure
engineered in such a way to ensure solid solution hardening and precipitation hardening.
We will assume the following form for the inelastic part of the free energy
Mollica et al., [34] and subsequently used by Prasad et al., [47]:
ψ
developed by
16
where the scalar variable s = s(t) is referred to as “inelastic strain pathlength” and is defined as
ψ1 , ψ 2
and η are material constants and
Ep is a measure of accumulated inelastic strain with reference to reference configuration and it
can be described as a measure of the total amount of slip that has taken place on slip systems
which are active [42].
Also, a(s) is the density of the dislocation network (defined as the total length of dislocation lines
per unit volume [65]) and assumes the following form:
The motivation to choose such a form stems from the experimental observation that the
dislocation density increases with monotonic inelastic deformation and reaches a saturation value
after a while.
The time rate of change of inelastic part of stored energy is given by
where
(.)′
denotes the total derivative with respect to s.
The tensor
is the backstress tensor. On taking the time derivative of the expression for backstress tensor, it
can be seen that the backstress, would satisfy the following evolution equation:
17
which is a generalized version of the non-linear kinematic hardening rule [7]. Although no
experimental data concerning the inelastic stored energy is available, to our knowledge, for single
crystal superalloys, there are experimental data for polycrystalline metals [3, 76]. Apart from the
experiments, several attempts have been made to model the inelastic stored energy (see for
example, papers by Chaboche [13, 14] and Kamlah and Haupt [22]). The inelastic stored energy
might not be significant at high temperatures at which creep occurs as only a small fraction of
energy is stored (dissipation mechanisms being dominant).
2.2.3. Rate of dissipation
Following form for the rate of dissipation is assumed:
where K is a fourth order tensor reflecting cubic symmetry that is a function of the temperature,
the inelastic history of the material and the driving force:
where I4 is the fourth order identity tensor, and the fourth order tensor N has the
form
ai, bi and ci are the components of the orthogonal unit vectors a, b and c. The evolution equation
for the natural configuration is determined by the tendency of the body to undergo a process that
maximizes the rate of dissipation. The idea of maximization of rate of dissipation is not a
fundamental principle of thermodynamics. However it is also not ad hoc either. It is a
generalization of a notion due to Gibbs that an isolated system tends to a state of maximal
entropy. A further assumption is made that the way, the body gets to the state of maximal entropy
is by producing entropy at the maximal possible rate (see Rajagopal and Srinivasa [59, 60]).
Maximizing the rate of dissipation subject to the constraint tr(Dp) = 0 (inelastic deformation being
isochoric) gives the following equations for Dp and Wp:
and
18
The rate of dissipation functions are required to be non-negative. Our constitutive assumption for
the rate of dissipation ensures that it is non-negative. The rate of dissipation is dependent on the
fourth order tensor K which is anisotropic. It is assumed that two mechanisms contribute to the
rate of dissipation in the following way:
The first mechanism is related to the dissipation caused by mobile dislocations. There is a rapid
multiplication of dislocations in the γ matrix at the beginning of creep. It is observed that the γ ′
phase is “hard" for significant amount of creep strains and hence no dislocation activity is
associated with it. As the deformation increases, these dislocations start moving in the γ matrix.
Further deformation results in these dislocations being stuck and bowed. This causes “hardening”
of the material. In order to describe the loss of mobility of the dislocations with accumulating
“creep strain”, it is assumed that the mean velocity of the dislocations remains constant at
constant stress while the density of dislocations, which are mobile decay exponentially. That is,
only a fraction of the total remains mobile. Let f be this fraction. In accordance with the work of
Gilman [17], following form for the mobile fraction is assumed:
where
α2
is the attrition coefficient.
The density of mobile dislocations, am (s) is then given by
The first mechanism of dissipation is assumed to be proportional to the inelastic strain pathlength,
s(t) in the following way:
Apart from this, the dissipation mechanism, in general, also depends upon the driving force, as
most dislocation interactions become less inhibitory with increasing driving force.
The second mechanism associated with dissipation is related to the damage accumulation by
creep cavitation. With the creep strain accumulating, the material starts getting “damaged’ by
means of highly localized deformation in the vicinity of crack surfaces. As reported in numerous
experiments, this creep elongation at the later stage of deformation is associated with creep
cavitation occurring at, or in the vicinity of casting porosity and topologically closed-packed
phases. Hence this stage is marked by a rapid increase in the strain (“softening”). Moreover,
experiments clearly indicate that the effect of highly localized damage due to creep cavitation
becomes dominant only after a certain critical strain is reached. The second mechanism of
dissipation is assumed to be proportional to the inelastic strain pathlength, s(t) in the following
way:
19
Moreover, it is also assumed that this second mechanism of dissipation remains active throughout
the creep process but becomes dominant only in the later stages of creep. This second
mechanism, in general, is also dependent on the driving force.
The coefficients associated with the tensor K have the following form:
It should be noted that k11, k12 and k44 have the same form. The material parameters ψ 1 , ψ 2 and
η and which are associated with the inelastic stored energy do not change with the orientation of
the crystal. The form for the rate of dissipation function is motivated based on creep deformation
when the specimen is loaded along the <001> and the <111> directions. The coefficients k11 - k12
and k44 reflect the dissipation which takes place when the specimen is loaded in the <111> and
the <001> direction, respectively. Such a description is a simplistic one and we will see that it
captures the creep deformation of single crystal superalloys reasonably well. The dependence of
rate of dissipation on dislocation motion is much more intricate when the specimen is loaded
along arbitrary directions.
2.2.4. Instantaneous rate of energy storage, R
One of the important quantities in any inelastic process is the ratio of rate of energy stored to the
rate of work done by the externally applied tractions. The rate at which work is done during the
inelastic process by the applied tractions per unit mass is given by
The rate of dissipation per unit mass is given by
The instantaneous rate of energy storage, R is then given by
20
where the equation of balance of energy is utilized in the form of reduced energy dissipation
equation. Although no experiments have been conducted, to our knowledge, on single crystal
superalloys to measure such a quantity, there are some experimental results for polycrystalline
metals (see Williams [76]) which can provide some guidance in our choice for the same for single
crystal superalloys.
3. RESULTS AND DISCUSSIONS
CREEP OF SUPERALLOYS LOADED ALONG THE <001>, <111> AND <011> ORIENTATIONS
Creep deformation of single crystal superalloys under loading in any arbitrary direction is fully
three dimensional. However, uniaxial creep loading along the orientations <001>, <111> and
<011> gives rise to a simple deformation field and there is no associated rotation of the crystal
lattice.
We will assume the following form for the deformation in rectangular coordinates:
where (X, Y, Z) is a material point in the configuration ·R and (x; y; z), the corresponding material
point in the configuration ·t.
The deformation gradient associated with the above motion is given by
The symmetric and skew part of Lp are given by
which leads us to the governing equations for the motion:
where ()i denotes the i-th diagonal element of the respective second order tensors. The above
equations form the set of governing non-linear ordinary differential equations.
21
3.1 Results for loading along the <001> orientation
The creep strain was obtained by integrating the above equations over time. These equations are
solved using a solver (ODE15S) for initial value problems in MATLAB. The results were
obtained for a range of temperatures, which are pertinent to the problem under consideration. The
material parameters, in general, are functions of temperature. Simulation was carried out at three
di®erent temperatures at various stress values and the results are compared with the experimental
results for CMSX-4 ([69] and [21]).
The variation of creep strain, instantaneous rate of energy storage and inelastic stored energy with
time are presented for different temperatures at various stresses. The variation of creep strain with
time for θ = 750, 982 and 1000 oC are shown in Figs. 5, 6 and 7. It can be seen that the
predictions of the model agree well with the experimental data for the different temperatures
considered here. Figs. 8, 9 and 10 depict the variation of the inelastic stored energy with respect
to the inelastic strain pathlength. Also, Figs. 11, 12 and 13 represent the variation of third
component of backstress tensor with inelastic strain pathlength. Variation of the instantaneous
rate of energy storage with inelastic strain pathlength is also studied. It can be seen from Figs. 14,
15 and 16 that the material stores energy in the initial stages of deformation but as the
deformation proceeds most of the energy is dissipated.
Moreover, it can be observed that the fraction of energy stored decreases at higher temperatures.
That is, at higher temperatures most of the work done by the applied tractions is dissipated. This
is to be expected as the dislocation motion which is primarily the source of dissipation becomes
less and less inhibitory as the temperature increases. The inelastic stored energy increases with
inelastic deformation but its value seems to be attaining a saturation value. The driving force
doesn't seem to have significant effect on the variation of inelastic stored energy as the curves for
the different stresses lie almost on top of each other. The driving force however, has significant
effect on the rate of dissipation as dissipation mechanisms themselves are strongly dependent on
the driving force.
3.2 Results for loading along the <001>, <111> and <011> orientations
The set of ordinary differential equations are solved using an initial value problem solver in
MATLAB (ODE15s). The material parameters associated with coefficient k44 are fixed by
matching the results with experiments for loading along the <001> orientation. Similarly, the
parameters in the expression for k11-k12 are fixed by matching the results with experiments for
loading along the <111> orientation. Hence, the orientations <001> and <111> are complete for
determining the material constants for the current set of orientations studied. Once the complete
set of materials parameters are obtained from the <001> and <111> data set, these are used to
simulate the uniaxial creep along the <011> orientation. The results for single crystal nickel
based superalloy CMSX-4 for loading along the <001>, <111> and <011> orientations for
various values of stresses at 800 oC and 950 oC are obtained and compared with the available
experimental results ([67, 45]). It is worth emphasizing that the material parameters are fixed
with respect to one experiment and they are then used to predict the results for a different
experiment.
22
The variation of the creep strain with time is shown in Figs. 17-22. It can be seen that the
predictions of the model agree well with the experimental data for the temperatures considered
here. The experimental data for loading along <011> orientation acts as a test case for measuring
the efficacy of our model. Although no such data is available for 800 oC, we find that the
prediction of the model are satisfactory for loading along the <011> orientation at 950 oC (see
Fig. 22). Figs. 23 and 24 depict the variation of the inelastic stored energy with inelastic strain
pathlength. The inelastic energy increases with inelastic deformation and eventually attains a
saturation value. Neither the driving force nor the orientation seems to have any effect on the
variation of inelastic stored energy as the curves for different stresses and loading orientations lie
almost on top of each other. Figs. 25 and 26 show the variation of the third component of
backstress tensor with inelastic strain pathlength. For certain orientations, the driving force seems
to have little or no effect on the backstress, though for comparable stress values, the value of the
backstress is somewhat lower for the <011> orientation than for the <001> and the <111>
orientations. Variation of the instantaneous rate of energy storage with inelastic strain pathlength
is depicted in Figs. 27 and 28. It can be seen that the material stores energy in the initial stages of
deformation but as the deformation proceeds most of the energy is dissipated. For certain
orientations, the fraction of energy stored decreases as the stress increases. This is expected as the
dislocation motion which is primarily the source of dissipation becomes less and less inhibitory as
the stress increases.
23
IMPLEMENTATION IN FINITE ELEMENT SOFTWARE ABAQUS
The constitutive model that has been developed is implemented in finite element software
ABAQUS/STANDARD through a user subroutine (UMAT).
For the large deformation analysis based on continuum formulation, ABAQUS sends, for each
integration point, the values of total deformation gradient, F at the current time t, the deformation
gradient from the current natural configuration, Fe at current time t, current value of the Cauchy
stress, T and the state variables such as s and G at the current time step. It also sends an estimate
of the total deformation gradient at the next time step (t + ∆ t). The UMAT supplies the values of
the state variables such as s, Fe and G and Cauchy stress at time step t+ ∆ t and return it back to
the ABAQUS. This process continues until the a converged solution is obtained at time step t +
∆ t. In estimating the value of total deformation gradient at time t + ∆ t, ABAQUS uses the
value of the jacobian, J which is specified in the UMAT. An exact definition of the consistent
jacobian is necessary to ensure quadratic convergence however the jacobian is often
approximated in favor of a simpler algorithm and computational speed. This may result in the loss
of quadratic convergence. In the current work, we have also used an approximate jacobian which
is the same as the elasticity matrix for a face centered cubic crystal.
Let us now develop the numerical scheme based on implicit first order backward difference
formula (backward Euler method). For the sake of completeness let us now list the equations
describing the constitutive model. For the case when the rate of dissipation is assumed to be
isotropic, the equations describing the constitutive model reduces to
Using the definition for inelastic strain pathlength, we can get an explicit expression:
which leads to
Discreatizing the above equations using first order backward difference, we arrive at the
following algebraic non-linear equations.
24
This set of equations is solved using Newton-Raphson method.
CONCLUSIONS:
The numerical scheme developed in the previous section is implemented in UMAT to study the
creep behavior of single crystal superalloys loaded along the <001> orientation. The UMAT is
validated by using it to obtain the results already obtained in previous sections using the semiinverse approach. The results obtained through the UMAT is compared with the results obtained
using the semi-inverse approach and the experimental results. Figs. 29, 30 and 31 show a
comparison of strain versus time curves for loading along <001> orientation at temperatures θ =
750, 982 and 1000 oC obtained using the UMAT and the semi-inverse method and experimental
results. Figs. 32, 33 and 34 show a comparison for the inelastic stored energy at temperatures θ
= 750, 982 and 1000 oC. A similar comparison is shown for third component of the backstress
tensor in Figs. 35, 36 and 37. Figs. 38, 39 and 40 show the same comparison for instantaneous
rate of energy storage.
25
GRAPHICAL MATERIALS LIST
Figure 1: Natural configurations associated with the body
Figure 2: Noll’s rule for simple materials
Figure 3: Shearing of lattice of a single crystal
Figure 4: Creep of a specimen under constant load
Figure 5: Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 750 oC:
Comparison of the predictions of the model with experimental results of Svoboda and Lucas
[69] and Henderson and Lindblom [21].
Figure 6: Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 982 oC:
Comparison of the predictions of the model with experimental results of Svoboda and Lucas
[69] and Henderson and Lindblom [21].
Figure 7: Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 1000 oC:
Comparison of the predictions of the model with experimental results of Svoboda and Lucas
[69] and Henderson and Lindblom [21].
Figure 8: Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 750 oC: Predictions of the model.
Figure 9: Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 982 oC: Predictions of the model.
Figure 10: Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 1000 oC: Predictions of the model.
Figure 11: Third component of backstress vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, θ = 750 oC: Predictions of the model.
Figure 12: Third component of backstress vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, θ = 750 oC: Predictions of the model.
Figure 13: Third component of backstress vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, θ = 1000 oC: Predictions of the model.
Figure 14: Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4
for loading along the <001> orientation, θ = 750 oC: Predictions of the model.
Figure 15: Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4
for loading along the <001> orientation, µ = 982 oC: Predictions of the model.
Figure 16: Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4
for loading along the <001> orientation, θ = 1000 oC: Predictions of the model.
26
Figure 17: Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 800 oC:
Comparison of the predictions of the model with experimental results of Schubert et al.,
[67].
Figure 18: Strain vs. time for CMSX-4 for loading along the <111> orientation, θ = 800 oC:
Comparison of the predictions of the model with experimental results of Schubert et al.,
[67].
Figure 19: Strain vs. time for CMSX-4 for loading along the <011> orientation,
Predictions of the model.
θ
= 800 oC:
Figure 20: Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 950 oC:
Comparison of the predictions of the model with experimental results of MacLachlan et al.,
[30].
Figure 21: Strain vs. time for CMSX-4 for loading along the <111> orientation, θ = 950 oC:
Comparison of the predictions of the model with experimental results of MacLachlan et al.,
[30].
Figure 22: Strain vs. time for CMSX-4 for loading along the <011> orientation, θ = 950 oC:
Comparison of the predictions of the model with experimental results of MacLachlan et al.,
[30].
Figure 23: Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 ,
Predictions of the model.
θ
= 800 oC:
Figure 24: Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 ,
Predictions of the model.
θ
= 950 oC:
Figure 25: Third component of backstress vs. inelastic strain pathlength for CMSX-4 ,
800 oC: Predictions of the model.
θ
=
Figure 26: Third component of backstress vs. inelastic strain pathlength for CMSX-4 ,
950 oC: Predictions of the model.
θ
=
Figure 27: Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 ,
θ = 800 oC: Predictions of the model.
Figure 28: Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 ,
θ = 950 oC: Predictions of the model.
Figure 29: Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 750 oC:
Comparison of the results obtained from User Material in ABAQUS with results obtained
in MATLAB and experimental results of Svoboda and Lucas [69] and Henderson and
Lindblom [21].
27
Figure 30: Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 982 oC:
Comparison of the results obtained from User Material in ABAQUS with results obtained
in MATLAB and experimental results of Svoboda and Lucas [69] and Henderson and
Lindblom [21]
Figure 31: Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 1000
C: Comparison of the results obtained from User Material in ABAQUS with results
obtained in MATLAB and experimental results of Svoboda and Lucas [69] and Henderson
and Lindblom [21].
o
Figure 32: Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 750 oC: Comparison of the results obtained from User
Material in ABAQUS with results obtained in MATLAB.
Figure 33: Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 982 oC: Comparison of the results obtained from User
Material in ABAQUS with results obtained in MATLAB.
Figure 34: Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 1000 oC: Comparison of the results obtained from User
Material in ABAQUS with results obtained in MATLAB.
Figure 35: Third component of backstress vs. inelastic strain pathlength for CMSX-4, θ =
750 oC: Comparison of the results obtained from User Material in ABAQUS with results
obtained in MATLAB.
Figure 36: Third component of backstress vs. inelastic strain pathlength for CMSX-4, θ =
982 oC: Comparison of the results obtained from User Material in ABAQUS with results
obtained in MATLAB.
Figure 37: Third component of backstress vs. inelastic strain pathlength for CMSX-4, θ =
1000 oC: Comparison of the results obtained from User Material in ABAQUS with results
obtained in MATLAB.
Figure 38: Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4
for loading along the <001> orientation, θ = 750 oC: Comparison of the results obtained
from User Material in ABAQUS with results obtained in MATLAB.
Figure 39: Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4
for loading along the <001> orientation, θ = 982 oC: Comparison of the results obtained
from User Material in ABAQUS with results obtained in MATLAB.
Figure 40: Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4
for loading along the <001> orientation, θ = 1000 oC: Comparison of the results obtained
from User Material in ABAQUS with results obtained in MATLAB.
28
29
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35
36
37
38
39
0.25
Model 735 MPa
Model 800 MPa
Experiment 735 MPa
Experiment 800 MPa
0.2
Strain
0.15
0.1
0.05
0
0
1
2
3
t (sec)
4
5
6
6
x 10
Fig. 5. Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 750 oC:
Comparison of the predictions of the model with experimental results of Svoboda and Lucas [69]
and Henderson and Lindblom [21].
40
0.35
Model 167.5 MPa
Model 206.9 MPa
Model 248.2 MPa
Experiment 167.5 MPa
Experiment 206.9 MPa
Experiment 248.2 MPa
0.3
0.25
Strain
0.2
0.15
0.1
0.05
0
0
1
2
3
t (sec)
4
5
6
6
x 10
Fig. 6. Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 982 oC:
Comparison of the predictions of the model with experimental results of Svoboda and Lucas [69]
and Henderson and Lindblom [21].
41
0.16
Model 150 MPa
Model 200 MPa
Experiment 150 MPa
Experiment 200 MPa
0.14
0.12
Strain
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
4
t (sec)
5
6
7
6
x 10
Fig. 7. Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 1000 oC:
Comparison of the predictions of the model with experimental results of Svoboda and Lucas [69]
and Henderson and Lindblom [21].
42
1800
1600
1400
1200
ψp
1000
800
600
400
200
Model 735 MPa
Model 800 MPa
0
0
0.05
0.1
0.15
0.2
0.25
s
Fig. 8. Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading along the
<001> orientation, θ = 750 oC: Predictions of the model.
43
350
300
250
ψ
p
200
150
100
50
Model 167.5 MPa
Model 206.9 MPa
Model 248.2 MPa
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s
Fig. 9. Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading along the
<001> orientation, θ = 982 oC: Predictions of the model.
44
180
160
140
120
ψ
p
100
80
60
40
20
0
Model 150 MPa
Model 200 MPa
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
s
Fig. 10. Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading along the
<001> orientation, θ = 1000 oC: Predictions of the model.
45
7
3
x 10
Model 735 MPa
Model 800 MPa
2.5
α33
2
1.5
1
0.5
0
0
1
2
3
s
4
5
6
6
x 10
Fig. 11. Third component of backstress vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 750 oC: Predictions of the model.
46
6
2
x 10
1.8
1.6
1.4
α33
1.2
1
0.8
0.6
0.4
Model 167.5 MPa
Model 206.9 MPa
Model 248.2 MPa
0.2
0
0
0.05
0.1
0.15
0.2
0.25
s
Fig. 12. Third component of backstress vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 982 oC: Predictions of the model.
47
6
8
x 10
7
6
α
33
5
4
3
2
Model 150 MPa
Model 200 MPa
1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
s
Fig. 13. Third component of backstress vs. inelastic strain pathlength for CMSX-4 for loading
along the <001> orientation, θ = 1000 oC: Predictions of the model.
48
0.5
Model 735 MPa
Model 800 MPa
0.45
p
W (t)/W (t)
0.4
s
0.35
0.3
0.25
0.2
0
0.05
0.1
0.15
0.2
0.25
s
Fig. 14. Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, θ = 750 oC: Predictions of the model.
49
0.3
Model 167.5 MPa
Model 206.9 MPa
Model 248.2 MPa
0.25
s
p
W (t)/W (t)
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s
Fig. 15. Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, µ = 982 oC: Predictions of the model.
50
0.3
Model 150 MPa
Model 200 MPa
0.28
0.26
0.22
s
p
W (t)/W (t)
0.24
0.2
0.18
0.16
0.14
0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
s
Fig. 16. Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, θ = 1000 oC: Predictions of the model.
51
0.05
Theory 462 MPa
Theory 500 MPa
Theory 650 MPa
Theory 750 MPa
Experiment 462 MPa
Experiment 500 MPa
Experiment 650 MPa
Experiment 750 MPa
0.045
0.04
0.035
Strain
0.03
0.025
0.02
0.015
0.01
0.005
0
0
500
1000
1500
t (hours)
2000
2500
3000
Fig. 17. Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 800 oC:
Comparison of the predictions of the model with experimental results of Schubert et al., [67].
52
0.06
Theory 500 MPa
Theory 575 MPa
Theory 675 MPa
Experiment 500 MPa
Experiment 575 MPa
Experiment 675 MPa
0.05
Strain
0.04
0.03
0.02
0.01
0
0
200
400
600
800
1000
t (hours)
1200
1400
1600
1800
2000
Fig. 18. Strain vs. time for CMSX-4 for loading along the <111> orientation, θ = 800 oC:
Comparison of the predictions of the model with experimental results of Schubert et al., [67].
53
0.08
Theory 450 MPa
Theory 500 MPa
Theory 550 MPa
Theory 650 MPa
Theory 700 MPa
0.07
0.06
Strain
0.05
0.04
0.03
0.02
0.01
0
0
200
400
600
t (hours)
800
1000
Fig. 19. Strain vs. time for CMSX-4 for loading along the <011> orientation,
Predictions of the model.
1200
θ
= 800 oC:
54
0.09
Theory 180 MPa
Theory 250 MPa
Theory 320 MPa
Theory 350 MPa
Theory 450 MPa
Experiment 180 MPa
Experiment 250 MPa
Experiment 320 MPa
Experiment 350 MPa
Experiment 450 MPa
0.08
0.07
0.06
Strain
0.05
0.04
0.03
0.02
0.01
0
0
500
1000
1500
2000
2500
t (hours)
Fig. 20. Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 950 oC:
Comparison of the predictions of the model with experimental results of MacLachlan et al., [30].
55
0.2
Theory 250 MPa
Theory 300 MPa
Theory 320 MPa
Theory 350 MPa
Theory 400 MPa
Theory 450 MPa
Experiment 250 MPa
Experiment 300 MPa
Experiment 320 MPa
Experiment 350 MPa
Experiment 400 MPa
Experiment 450 MPa
0.18
0.16
0.14
Strain
0.12
0.1
0.08
0.06
0.04
0.02
0
0
500
1000
1500
2000
2500
t (hours)
Fig. 21. Strain vs. time for CMSX-4 for loading along the <111> orientation, θ = 950 oC:
Comparison of the predictions of the model with experimental results of MacLachlan et al., [30].
56
0.2
Theory 250 MPa
Theory 350 MPa
Theory 400 MPa
Theory 450 MPa
Experiment 250 MPa
Experiment 350 MPa
Experiment 400 MPa
Experiment 450 MPa
0.18
0.16
0.14
Strain
0.12
0.1
0.08
0.06
0.04
0.02
0
0
200
400
600
t (hours)
800
1000
1200
Fig. 22. Strain vs. time for CMSX-4 for loading along the <011> orientation, θ = 950 oC:
Comparison of the predictions of the model with experimental results of MacLachlan et al., [30].
57
300
250
180 MPa <001>
250 MPa <001>
320 MPa <001>
350 MPa <001>
450 MPa <001>
250 MPa <111>
300 MPa <111>
320 MPa <111>
350 MPa <111>
400 MPa <111>
450 MPa <111>
250 MPa <011>
350 MPa <011>
400 MPa <011>
450 MPa <011>
Inelastic stored energy
200
150
100
50
0
0
0.05
0.1
0.15
0.2
0.25
s
Fig. 23. Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 ,
Predictions of the model.
θ
= 800 oC:
58
120
100
Inelastic stored energy
80
462 MPa <001>
500 MPa <001>
650 MPa <001>
750 MPa <001>
500 MPa <111>
575 MPa <111>
675 MPa <111>
450 MPa <011>
500 MPa <011>
550 MPa <011>
650 MPa <011>
700 MPa <011>
60
40
20
0
0
0.01
0.02
0.03
0.04
0.05
s
0.06
0.07
0.08
Fig. 24. Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 ,
Predictions of the model.
0.09
θ
0.1
= 950 oC:
59
7
2.5
x 10
2
1.5
α33
462 MPa <001>
500 MPa <001>
650 MPa <001>
750 MPa <001>
500 MPa <111>
575 MPa <111>
675 MPa <111>
450 MPa <011>
500 MPa <011>
550 MPa <011>
650 MPa <011>
700 MPa <011>
1
0.5
0
0
0.01
0.02
0.03
0.04
0.05
s
0.06
0.07
0.08
Fig. 25. Third component of backstress vs. inelastic strain pathlength for CMSX-4 ,
Predictions of the model.
0.09
θ
0.1
= 800 oC:
60
6
9
x 10
8
7
180 MPa <001>
250 MPa <001>
320 MPa <001>
350 MPa <001>
450 MPa <001>
250 MPa <111>
300 MPa <111>
320 MPa <111>
350 MPa <111>
400 MPa <111>
450 MPa <111>
250 MPa <011>
350 MPa <011>
400 MPa <011>
450 MPa <011>
6
α
33
5
4
3
2
1
0
0
0.05
0.1
0.15
0.2
0.25
s
Fig. 26. Third component of backstress vs. inelastic strain pathlength for CMSX-4 ,
Predictions of the model.
θ
= 950 oC:
61
0.04
462 MPa <001>
500 MPa <001>
650 MPa <001>
750 MPa <001>
500 MPa <111>
575 MPa <111>
675 MPa <111>
450 MPa <011>
500 MPa <011>
550 MPa <011>
650 MPa <011>
700 MPa <011>
0.035
0.03
R
0.025
0.02
0.015
0.01
0.005
0
0.01
0.02
0.03
0.04
0.05
s
0.06
0.07
0.08
0.09
Fig. 27. Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 ,
800 oC: Predictions of the model.
0.1
θ
=
62
0.11
180 MPa <001>
250 MPa <001>
320 MPa <001>
350 MPa <001>
450 MPa <001>
250 MPa <111>
300 MPa <111>
320 MPa <111>
350 MPa <111>
400 MPa <111>
450 MPa <111>
250 MPa <011>
350 MPa <011>
400 MPa <011>
450 MPa <011>
0.1
0.09
0.08
R
0.07
0.06
0.05
0.04
0.03
0.02
0
0.05
0.1
0.15
0.2
0.25
s
Fig. 28. Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 ,
950 oC: Predictions of the model.
θ
=
63
Abaqus 735 MPa
Abaqus 800 MPa
Experiments 735 MPa
Experiments 800 MPa
Matlab 735 MPa
Matlab 800 MPa
Fig. 29. Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 750 oC:
Comparison of the results obtained from User Material in ABAQUS with results obtained in
MATLAB and experimental results of Svoboda and Lucas [69] and Henderson and Lindblom
[21].
64
Abaqus 167.5 MPa
Abaqus 206.9 MPa
Abaqus 248.2 MPa
Experiment 167.5 MPa
Experiment 206.9 MPa
Experiment 248.2 MPa
Matlab 167.5 MPa
Matlab 206.9 MPa
Matlab 248.2 MPa
Fig. 30. Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 982 oC:
Comparison of the results obtained from User Material in ABAQUS with results obtained in
MATLAB and experimental results of Svoboda and Lucas [69] and Henderson and Lindblom
[21]
65
Abaqus 150 MPa
Abaqus 200 MPa
Experiment 150 MPa
Experiment 200 MPa
Matlab 150 MPa
Matlab 200 MPa
Fig. 31. Strain vs. time for CMSX-4 for loading along the <001> orientation, θ = 1000 oC:
Comparison of the results obtained from User Material in ABAQUS with results obtained in
MATLAB and experimental results of Svoboda and Lucas [69] and Henderson and Lindblom
[21].
66
Abaqus 735 MPa
Abaqus 800 MPa
Matlab 735 MPa
Matlab 800 MPa
Fig. 32. Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading along the
<001> orientation, θ = 750 oC: Comparison of the results obtained from User Material in
ABAQUS with results obtained in MATLAB.
67
Abaqus 167.5 MPa
Abaqus 206.9 MPa
Abaqus 248.2 MPa
Matlab 167.5 MPa
Matlab 206.9 MPa
Matlab 248.2 MPa
Fig. 33. Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading along the
<001> orientation, θ = 982 oC: Comparison of the results obtained from User Material in
ABAQUS with results obtained in MATLAB.
68
Abaqus 150 MPa
Abaqus 200 MPa
Matlab 150 MPa
Matlab 200 MPa
Fig. 34. Inelastic stored energy vs. inelastic strain pathlength for CMSX-4 for loading along the
<001> orientation, θ = 1000 oC: Comparison of the results obtained from User Material in
ABAQUS with results obtained in MATLAB.
69
Fig. 35. Third component of backstress vs. inelastic strain pathlength for CMSX-4, θ = 750 oC:
Comparison of the results obtained from User Material in ABAQUS with results obtained in
MATLAB.
70
Abaqus 167.5 MPa
Abaqus 206.9 MPa
Abaqus 248.2 MPa
Matlab 167.5 MPa
Matlab 206.9 MPa
Matlab 248.2 MPa
Fig. 36. Third component of backstress vs. inelastic strain pathlength for CMSX-4, θ = 982 oC:
Comparison of the results obtained from User Material in ABAQUS with results obtained in
MATLAB.
71
Abaqus 150 MPa
Abaqus 200 MPa
Matlab 150 MPa
Matlab 200 MPa
Fig. 37. Third component of backstress vs. inelastic strain pathlength for CMSX-4, θ = 1000 oC:
Comparison of the results obtained from User Material in ABAQUS with results obtained in
MATLAB.
72
Abaqus 735 MPa
Abaqus 800 MPa
Matlab 735 MPa
Matlab 800 MPa
Fig. 38. Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, θ = 750 oC: Comparison of the results obtained from User
Material in ABAQUS with results obtained in MATLAB.
73
Abaqus 167.5 MPa
Abaqus 206.9 MPa
Abaqus 248.2 MPa
Matlab 167.5 MPa
Matlab 206.9 MPa
Matlab 248.2 MPa
Fig. 39. Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, θ = 982 oC: Comparison of the results obtained from User
Material in ABAQUS with results obtained in MATLAB.
74
Abaqus 150 MPa
Abaqus 200 MPa
Matlab 150 MPa
Matlab 200 MPa
Fig. 40. Instantaneous rate of energy storage vs. inelastic strain pathlength for CMSX-4 for
loading along the <001> orientation, θ = 1000 oC: Comparison of the results obtained from User
Material in ABAQUS with results obtained in MATLAB.
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