Single crystal materials are being used in gas turbine airfoils and are candidates for other hot section components because of their increased temperature capabilities and resistance to thermal fatigue. Development of a constitutive model which assesses the inelastic behavior of these materials has been studied in 2 NASA programs: Life Prediction and Constitutive Models for Engine Hot Section Anisotropic Materials and Biaxial Constitutive Equation Development for Single Crystals. The model has been fit to a large body of constitutive data for single crystal PWA 1480 material. The model uses a unified approach for computing total inelastic strains (creep plus plasticity) on crystallographic slip systems reproducing observed directional and strain rate effects as a natural consequence of the summed slip system quantities. The model includes several of the effects that have been reported to influence deformation in single crystal materials, such as shear stress, latent hardening, and cross slip. The model is operational in a commercial Finite Element code and is being installed in a Boundary Element Method code

Jordan, E. H.

Meyer, T. G.

Walker, K. P.

English

NASA Technical Reports Server

N88-22388
UNIFIED CONSTITUTIVE MODEL FOR SINGLE CRYSTAL
DEFORMATION BEHAVIOR WITH APPLICATIONS
K.P. Walker
T.G. Meyer
E.H. Jordan
Engineering Science Software, Inc. (Smithfield, RI)
Pratt & Whitney (East Hartford, CT)
University of Connecticut (Storrs, CT)
Single crystal materials are being used extensively in gas turbine airfoils
and are candidates for other hot section components because of their
increased temperature capabilities and resistance to thermal fatigue. Under
many operating conditions_ the thermal and mechanical loads are
sufficiently severe to cause inelastic material behavior. It is widely
recognized that design of such components for long fatigue life requires an
accurate assessment of that inelastic behavior. Howeverp no convenient
inelastic material models are currently available for structural analysis
of these anisotropic components.
Development of such a constitutive model for single crystal material has
been undertaken recently in two NASA sponsored programs: Life Prediction
and Constitutive Models for Engine Hot Section Anisotropic Materials
(NAS3-23939) and Biaxial Constitutive Equation Development for Single
Crystals (NAG3-512). A slip system based constitutive model for single
crystal materials is now in its final stages of development. The model has
been fit to a large body of constitutive data for single crystal PWA 1480
material and will also be tested against data for a second single crystal
material. The model uses a unified approach for computing total inelastic
strains (creep plus plasticity) on crystallographic slip systems
reproducing observed directional and strain rate effects as a natural
consequence of the summed slip system quantities. The model includes
several of the effects that have been reported to influence deformation in
single crystal materials. These include the contributions from slip system
stresses other than the Schmid shear stress_ latent hardening due to
simulataneous straining on all slip systems, and cross-sllp from the
octahedral to the cube sllp systems. The model is operational in a
commercial Finite Element code and is being installed in a Boundary Element
Method code.
Contract: NAS3-23939
Grant: NAG3-512
NASA LeRC Technical Monitors: Drs. G.R. Halford and R.L. Thompson
2-57
https://ntrs.nasa.gov/search.jsp?R=19880013004 2020-03-20T06:33:38+00:00Z
ADVANCES IN TURBINE AIRFOIL MATERIALS 
HAVE RESULTED IN HIGHER STRENGTH AND 
INCREASED D U RABl LlTY 
One of the more important recent developments in gas turbine blade 
materials has been the introduction of directionally solidified and single 
crystal castings. Among the advantages of these new materials is an 
increased creep and oxidation resistance which results from the elimination 
of grain boundaries. In addition, the elimination of grain boundary 
strengthening elements in single crystal material results in higher melting 
temperature and permits greater flexibility in achieving optimum heat 
treatments. the growth direction also improves 
the fatigue life by reducing the thermally induced stresses. 
The low elastic modulus in 
CONVENTIONALLY DIRECTION ALLY 
CAST SOLIDIFIED 
iTiiAivsv tnstLv 
ISOTROPIC) 
,.A C I I I  a -. -. 
\iau I nurib] 
SINGLE 
CRYSTAL 
(ANISOTRUPIC) 
2-58 
NICKEL BASE SINGLE CRYSTAL SUPERALLOYS HAVE
A FACE CENTERED CUBIC CRYSTALLINE STRUCTURE
The single crystal material being used in the current effort is PWA 1480.
It is a two phase Nickel based superalloy having a Face Centered Cubic
atomic arrangement in both the 7 matrix (mostly Nickel) and the 7'
strengthening phase. The cuboidal ?' is arranged in a regular array in the
matrix material. It is well known that this material deforms by slip on
specific crystallographic planes. This fact has been used in developlng a
constitutive model for PWA 1480.
_"'_ precipitate
I f_>l_>lTl<_ L'_//7' particles
2-59
SINGLE CRYSTAL COMPONENT PROPERTIES
ARE HIGHLY DIRECTIONAL (ANISOTROPIC)
A key factor in the design and behavior of columnar grained and single
crystal material is the variation of material properties with respect to
the natural material axes. For example, in PWA 1480 at 1200F, Young's
Modulus obtained from a tensile bar oriented along one of the material's
cubic axes (<001> Miller index) is approximantely 15 Msi while the value in
the cube diagonal direction (<111> Miller index) is approximately 39 Msi.
_ , _l _<111> E = 39 x 106 psif,,¢ /<001 > E
= 15 x 10 epsi <21:3> E = 28 x 106 psi
<011> E = 28 x 106 psii
I \ /
Single crystal Young's modulus varies by almost a
factor of 3 with direction
2-60
CLASSICAL MODELS GIVE POOR PREDICTIONS
OF YIELD STRESS VARIATION WITH ORIENTATION
While these elastic property variations are correctly predicted with most "
current structural analysis tools, the prediction of creep and plastic
behavior, which is often required for durability assessment, is not easily
accomplished. For example, the classical yield models of Lee & Zaverl
(1978) and Hill (1948) would predict the yield stress to decrease
continually from a maximum along the <001> axis to a minimum along the
<iii> direction. But experiment has shown that the yield strength has a
minimum for an orientiation in the middle of the stereographic projection.
Furthermore, it is now recognized that the classical approach of separately
calculating creep and plastic strains is difficult to apply in the
structural analysis of components that operate in transient thermal and
mechanical environments.
The constitutive model developed in the current effort attempts to
incorporate metallurgical observations regarding the deformation of single
crystals in a unified viscoplastic formulation. Inelastic strains
(accounting for "plasticity" and "creep" simultaneously) are computed on
crystallographic slip systems. The model thus achieves the required
directional properties as a natural consequence o£ summing the slip system
stresses and strains.
100
Specimen axis defined
by angles _ and
35.26 °
130_
Yield stress
17°_________L_ JO o
O° e'--_- 450
001
fcc cubeunit
010
, A 35.26 °
Expedment I
Yield stress /124.7 \ A
psi _( (_ / _
\ \ Ioo
0 ° # _ 45 °
2-61
CRYSTALLOGRAPHIC SLIP FORMULATION
The model has been formulated to use stresses and strains referred to any
geometric or "global" coordinate system which may be convenient for the
structural analysls of a component. As with any analysis involving an
anisotropic material, the relative orientation o£ the global and material
axes are also required. The model then transforms the global stresses and
strains onto a crystal coordinate system. These crystal system quantities
are subsequently resolved onto each of twelve octahedral and six cube stip
systems. The Schmld shear stress on each system is used to calculate the
inelastic shear stress on that system. Provision has been made in the model
to include the effect of all six components of slip system stress.
r__ Crystal system
z • Find stress in crystal
Global system ,_Ol]/"7_ system, x,y,z
z _ /Oxx OxvOxz\
ix ,o,o, r, =/ox.o..o.zJ
_-- \°xz°,.°zz//' .ooj
/,,_lr r, Schmid shear stress
X3(e3) X3
• Find shear stresse 1)
on twelve octahedral
slip system
_r=e_ •_._'
rth slip system
2-62
CRYSTALLOGRAPHIC SLIP FORMULATION
The general form of the equations governing the inelastic strain on each
slip system is shown below. The form is the familiar viscoplastic equation
employing two state variables (Walker 1981). The rate of change of the slip
system inelastic shear strain is a function of the applied slip system
shear stress, an internal back stress, and a drag stress. The back stress
and the drag stress each evolve with inelastic strains. An inelastic shear
strain rate is calculated in each slip system's coordinate axes and
transformed to the common crystal coordinate system where they are summed
to obtain a combined inelastic strain rate. Once this inelastic strain rate
is known, the rate of change o£ stress is easily obtained using the known
total strain rate and the stiffness matrix. Finally, the stress rate is
tranformed onto the global coordinate system.
_r
X 3
X=_ _Equilibrium
p - 1 f stress
• "Yr : K, Kr _---Drag stress
rth slip system E_> Evolution equations for _r and Kr
• r/2 0
Crystal
system (_kP)r-" (Qki T), (_iP)r (Qjl)r
• Gij Dijk, {Ekl _# " P'°Ct _ ""p'cube_"= -- '£kl;r -- 'Ek' 'r
r=l r=l
• Global system dij= Ri: dkl Rij
2-63
CONSTITUTIVE MODEL EMPLOYS VISCOPLASTIC
EQUATIONS ON OBSERVED MATERIAL SLIP PLANES
The viscoplastic model constants for the octahedral and cube slip systems
are not the same. The figure below shows the terms in the evolution
equation that were found to be active for the PWAI480 data base. The full
model however includes thermal recovery terms in the back stress rate
equation and an evolutionary equation for the drag stress which includes
latent hardening among like slip systems and cross hardening between the
cube and octahedral systems. Details of the mathematical formulation can
be found in the reports by Walker and Jordan (1985) and Swanson et. al.
(1987).
001
6 cube systems 12 octahedral systems
(r c _ COc)I rc _ coc id- 1
%= Zc"
('ro - oJo) I "ro - _o I P-1
_o = KoP
Cbc= Pl "YC-- P2 I _c I °_c _o = P6 _o - P71%1 _o
010
2-64
SINGLE CRYSTAL CONSTITUTIVE MODEL
BASED ON CRYSTALLOGRAPHIC SLIP THEORY
CAPTURES THE OBSERVED ORIENTATION AND
RATE DEPENDENT DEFORMATION BEHAVIOR
A large body of uniaxial cyclic stress - strain data has been obtained from
temperatures of 800F to 2100F using specimens oriented in the <001>,
<111>, <011> and <123> directions (Swanson et. al. 1978). Additional
torsional stress - strain data has also been obtained (Jordan and Walker,
1985). Strain rates were varied over four orders o£ magnitude from 1% per
sec. to .0001% per sec. The £igure below is typical o£ the correlation with
experiment. The slip system based model captures the orientation and rate
dependence quite veil.
<001 >
111>
1600°F steady state cyclic hysteresis loops
160000
96000
32000
Stress
in psi
-32000
-96000
-160000
- 1.40 -0.84 -0.28 0.28 0.84
Strain in percent
- _ <111> <001>
D
-- 1 pSl= 6.895 kPA _C
V
Strain retes
(% per seco_J)
A) 0.001
Bl 0.0025
c! o.oi
0) 0.1
E) O.S
, I I I I I
1.40
Stress
in psi
r--'-'----I160000-
ITheorvl <111> <001>
I "l o
c
• s . B
96000
32000
-32000
-96000
-160000 I I I I I
-1.40 -0.84 -0.28 0.28 0.84 1.40
Strain in percent
2-65
SLIP BASED CRYSTALLOGRAPHIC CONSTITUTIVE MODEL
IS NOW BEING TESTED UNDER THERMOMECHANICAL
LOADING CONDITIONS SIMILAR TO THOSE
ENCOUNTERED IN SERVICE
The constitutive model has been formulated for use in non-lsothermal
anaylyses and is currently being evaluated against a body of
thermomechanical loading tests as illustrated in the figure below. In this
test cycle, the temperature and imposed strain are varied simultaneously
while the stress is monitored. Good correlation has been achieved in the
high temperature portion of the cycle. However too much inelasticity is
predicted in the low temperature portion. Final adjustments to the model
are now being undertaken to address the low temperature response.
Strain and temperature
vary sinusoidally with time
Strain, Stress,
in/in ksi
E 02
E-02 2.0
0.3-
0.2 _t case cycle
0.1
0
o
Expe_me.tel "_-._%,,
- 0.2 TMF cycle
-0.3 ] I I I I I I I I I I I
0.8 1.0 1.2 1.4 1.6 1.8 2.0
E 03
Temperature, °F
1.5
1.0
0.5
0
-0.5
-0.3
Stress vs. mechanical strain
in thermomechanical fatigue test
-- Experiment
---- Simulation in
finite element code
-/ f_.-y
_S j .
_r:>'T -'_'S I I I 1 I
-0.2 -0.1 0 0.1 0.2 0.3
Strain, in/in £--02
2-66
SINGLE CRYSTAL MODEL
CAN EASILY BE INSTALLED AS A SUBROUTINE
IN ANY NONLINEAR FINITE ELEMENT PROGRAM
The single crystal model is now operational in the MARC finite element code
and is being installed in the "BEST3D" boundary element code under NASA
LeRC contract "Inelastic Analysis Methods for Hot Section Components",
NAS3-23697. It is also operational in a nonlinear structural analysis code
on the IBM PC-XT and is being installed on a 80386-WEITEK desktop/lap
personal computer. As part of the current contract effort, the code will
be demonstrated in a finite element analysis of a gas turbine hot section
component. The constitutive model has been written to be comparable with
any finite element code that uses the initial load vector approach for
material nonlearity.
• Start increment N; obtain Au from previous increment;
compute A_ = BAu
Performed
in FE
program
• Enter constitutive subroutine and compute D and _" Pass D and A_-
to FE program
_1o = D(AE - _T) - _(Au)
Anisotropic iJ _ Inelastic stress increment from
elasticity matr single crystal model depends
on _u
(_BT DBdV)_u= zIP+_BT ( exiT+ _IAu))dV IFE program solves
v v FE equilibrium equation
K:
stiffness
matrix
_ for displacement
Applied Initial load vector increment £u
load
vector
in J
Iterate until solution Au has converged
)Performed
• Update solution and proceed to next increment, N + 1 / in FE
program
2-67
THE SINGLE CRYSTAL CONSTITUTIVE RELATIONS CAN
BE EXTENDED TO OTHER CLASSES OF MATERIAL BY
MEANS OF SELF-CONSISTENT MODELS
This slip system based model for single crystal material offers the
opportunity to develop constitutive models for other classes of materials
by means of self-consistent methods (Walker, 1984). Directlonally
solidified materials consist of aligned columnar single crystal grains
which are oriented at random in the basal plane perpendicular to the
solidification direction. This random orientation of grains produces a
material with transversely isotropic properties. A self consistent method
for modeling this material is achived by surrounding a particular single
crystal columnar grain with a transversely isotopic material. Using
methods proposed by Eshelby (1957), the properties of the surrounding
transversely isotropic material are found by averaging the properties of
the single crystal grain which has been constrained by the surrounding
material. The averaging is done about the axis of the columnar grain.
Directionally
solidified
Transversly isotropic
FCC axis Z
Global
Matrix
A_mlltrix
I Global axis I FCC alxis -
I Iis.., ,,I/[ crystal _ I I
Known Known To be computed
Aamatrix = Dmatrix A_matrix -- A_'matrix
2r
1 /4
A_" (o(0), state (0)) dO
where A_'matd x = 2_" o
From the CONSTRAINED
cylindrical grain
2-68
EXTENSION OF THE SINGLE CRYSTAL CONSTITUTIVE
RELATIONS TO ISOTROPIC MATERIALS BY MEANS OF
A SELF CONSISTENT MODEL
Using a similar self consistent approach, it is possible to develop a
constitutive model for isotropic materials. In this case the averaging in
the constrained grain must be done with respect to all directions.
In both of these cases, transversley isotropic and isotropic, the slip
system formulation developed in this effort can serve as the base model for
the grains. Appropriate model constants for the particular material would
of course be required and the influence o£ grain boundary sliding may
require additional modeling.
Conventionally
cast
Where
FCC axis
A_matrix
crystal sphere
To be
Known Known computed
AOmatrix = Dmatrix A_matrix -- A_'matrix
2-69
WHAT IS THE CURRENT STATE OF OUR TECHNOLOGY?
The single crystal constitutive model developed in this effort represents
an advance in the state of our technology for durability prediction. A
structural analysis using the single crystal constitutive model can now be
used to obtain a more accurate stress - strain response at the fatigue
critical location of a component. These more accurate results can now be
used to develop more accurate life models. Global stress and strain
quantities as well as slip systems quantities are available from the model
for life prediction.
A similar, although more complicated, approach is envisioned for composite
materials. In these materials, the size scale of the material constituents
relative to the overall composite structure prohibits modeling individual
fibers and their surrounding matrix explicitly throughout the entire
structure. Instead, a homoginized or "smeared" material model could be
constructed for the bulk propreties of the composite by assuming some
periodicity of the fibers in the matrix and then volume averaging the
contltuent responses. This homoginlzed material model can then be used in
the structural analysis program to obtain the overall composite structure's
response. Damage models can be developed based on these "smeared" results
directly or these results can be used as boundary conditions on a composite
sub-element. The same constituent material models that were used to obtain
the smeared material model can now be used to obtain local flber/matrix
stresses and strains in the sub-element.
Fatigue
critical
location
Smeared
or volume
averaged
homogeneous J
CompOoSifi_erWith material model J[[
_i['_ r Im"[ [ CO:_utineJS__[S
Single crystal Strain-temperature
history at fatigue
critical location
Structural Analysis [
Program I
Smeared
or volume
averaged
damage
 onhomo,.n ous,1! 1  tre.- tr.,o-t.moer.,ureh,,to tutMicrostructural _ fiber/matrix structure
I con t,,ut,ve°o e,II" '
LeRC LST'88
2-70
References
Eshelby, J.
Inclusion,
p376, 1957.
D., "The Determination of the Elastic Field of an Ellipsoid
and Related Problems," Proc. Royal Society of London, A241,
Hill, R., "A Theory of the Yielding and Plastic Flow of Anisotropic
Metals," Proc. Royal Society of London, Set. A Vol. 193, pp. 281-297, 1948.
Jordan, E. H. and Walker, K. P., "Biaxial Constitutive Modeling and Testing
of a Single Crystal Superalloy at Elevated Temperature," presented at the
Second International Conference on Biaxial/Multiaxial Fatigue, Sheffield,
England, December 1985. To appear in Fatigue and Fracture of Engineering
Materials and Structures.
Lee, D., Zaverl, F. Jr., Shih, C. F., and German, M.D., "Plasticity
Theories and Structural Analysis of Anisotropic Metals," Report No.
77CRD285, General Electric Corporate Research and Development Center,
Schenectady, New York, 1977.
Swanson, G. A., I. Linask, D. M. Nissley, P. P. Norris, T. G. Meyer, and K.
P. Walker, "Life Prediction and Constitutive Models for Engine Hot Section
Anisotropic Materials Program, Annual Status Report, " Nasa CR-174952,
February, 1986.
Walker, K. P., "Research and Development for Nonlinear Structural Modeling
with Advanced Time-Temperature Dependent Constitutive Relationships," NASA
CR-165533, November 1981.
Walker, K.P. and Jordan, E. H., "Constitutive Modeling of Superalloy
Single Crystal and Directlonally Solidified Materials," NASA CP-2369, pp 65
- 81, 1984.
Walker, K.P. and Jordan, E. H., "First Annual Report on NASA Grant
NAG3-512," 1985.
2-71


Unified constitutive model for single crystal deformation behavior with applications

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