In developing the constitutive equations an interdisciplinary approach is being pursued. Specifically, both metallurgical and continuum mechanics considerations are recognized in the formulation. Experiments will be utilized to both explore general qualitative features of the material behavior that needs to be modeled and to provide a means of assessing the validity of the equations being developed. The model under development explicitly recognizes crystallographic slip on the individual slip systems. This makes possible direct representation of specific slip system phenomena. The present constitutive formulation takes the anisotropic creep theory and incorporates two state variables into the model to account for the effect of prior inelastic deformation history on the current rate-dependent response of the material

Jordan, E. H.

Walker, K. P.

English

NASA Technical Reports Server

N.87-I1217
Biaxial Constitutive Equation Development
Eric H. Jordan and Kevin P. Walker
University of Connecticut and Engineering Science Software, Inc.
INTRODUC_ON
Some current production gas turbine engines utilize large scale single crystal super-
alloy hot section components. Structural analysis of these components requires a valid
stress-strain-temperature relation, often referred to as a constitutive equation. At present
this behavior can only be represented by elastic constitutive equations or pure creep equa-
tions. The actual application involves viscoplastic strain cycles that are not represented
well by either of these types of equations. The goal of the work described in this report
is to develop and verify viscoplastic constitutive equations for superalloy single crystals and
directionally solidified alloys.
In developing the constitutive equations an interdisciplinary approach is being pursued.
Specifically, both metallurgical and continuum mechanics considerations are recognized in
the formulation. Experiments will be utilized to both explore general qualitative features
of the material behavior that need to be modeled and to provide a means of assessing the
validity of the equations being developed. The model under development explicitly recog-
nizes crystallographic slip on the individual slip systems. This makes possible direct repre-
sentation of specific slip system phenomena previously studied by metallurgists.
GENERAL FEATURES OF THE PROPOSED MODEL
Viscoplastic constitutive equations currently used for describing the high temperature
structural behavior of isotropic superalloys make use of a state variable concept. These
viscoplastic equations are basically rate-dependent creep equations in which the creep or
inelastic strain rate depends not only of the state of stress, but also on two state vari-
ables. These state variables represent the effect of prior inelastic deformation history on
the current creep response of the material.
Constitutive equations for describing the anisotropic creep behavior of superalloy single
crystals were developed by Paslay, Wells and Leverant [1] in 1971. The present constitu-
tive formulation takes the anisotropic creep theory developed in Reference [1] and incor-
porates two state variables into the model to account for the effect of prior inelastic de-
formation history on the current rate-dependent response of the material. This is done in
a manner analogous to the isotropic development of viscoplastic state variable models.
SIMULATIONS WITH THE MODEL
Figure 1 shows a numerical simulation of a single crystal superalloy which can exhibit
slip on its four octahedral 111 planes in twelve 110 type directions. The single crystal
bar specimen is pulled along its axis under load control with the direction of the load axis
inclined at various angles to the crystallographic axes. Curve 1 shows the response ob-
tained by loading the bar specimen along its axis under fully reversed load control to a
value of +207 MPa (-+30Ksi) at a constant load rate magnitude of 6.9 MPa (1 Ksi) per
second. Curves 2 to 6 show the effect of inclining the specimen axis and corresponding
load direction by angles of 0 = 9 ° , 18 ° , 27 ° , 36 ° and 45 ° to the crystallographic axes in a
plane containing two of the crystallographic axes. The increasing angles correspond to
moving along the [001] - [011] side of the stereographic triangle. Curve 7 corresponds
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https://ntrs.nasa.gov/search.jsp?R=19870001784 2020-03-23T15:06:07+00:00Z
to the [ill] vertex of the stereographic triangle.
According to Schmid's law [2] the yield stress decreases from its maximum values
at the [001] and [011] vertices of the stereographic triangle to a minimum value at
0=22.5 ° halfway between the [001] and [011] vertices. The increased plasticity in the
loop for curves 3 and 4 corresponding to 0=18 ° and 0=27 ° reflect this minimum behavior.
Symmetry between the loops equidistant from the minimum at 0=22.5 ° is not achieved with
the present formulation because Young's modulus for the single crystal bar specimen
increases from the [001] vertex to the [001] vertex and from the [011] vertex to the
[ill] vertex of the stereographic triangle. This increasing value of Young's modulus is
reflected in the increasing slope of the curves labeled 1 to 7 in Figure 1.
Figure 2 shows the effect of strain rate on a bar specimen oriented along the [001]
direction corresponding to one of the crystallographic axes. The specimen was pulled
along its axis under strain control at constant strain rate magnitudes of 10 -3 to 10 -6 per
second and shows the usual type of positive strain rate sensitivity in which the "yield"
stress increases with increasing strain rate.
Octahedral slip on the four 111 planes in the twelve [112] slip directions and cube
slip along the planes containing the crystallographic axes are also being coded into the
FORTRAN subroutine to provide a constitutive model which can exhibit different types
of slip behavior in different temperature ranges.
CONSIDERATIONS RELATING THEORY AND EXPERIMENT
In experiments on isotropic materials involving hollow tube biaxial specimens it is
relatively straight forward to calculate from the measured load and torque values using
"strength of materials" formulae. Unfortunately, these simple formulae cannot, in general,
be used with anisotropic materials. Instead, the specimen must be analyzed as a structure.
It is essential to make the analysis of the specimen computationally efficient if simulation
of the measured load-displacement histories is to be practical. The approach taken in this
investigation is to solve for the stress distribution in the specimen using a specially con-
structed finite element program written in FORTRAN on the IBM PC-XT computer. This
program represents the tubular specimen as a single high order finite element whose shape
function includes terms which reflect the spatial symmetry corresponding to the cubic
symmetry of the superalloy single srystal. The code for this finite element model has
been written and is currently undergoing numerical experiments.
Experiments performed on single crystal superalloys [3-5] have shown that there is a
tension-compression asymmetry in the flow stress. This has been accounted for by metal-
lurgists by assuming that the applied stress constricts the Shockley partials during cross
slip. Asymmetric flow stress behavior may be incorporated into the model in many ways.
The specific manner chosen to incorporate this asymmetry will be chosen in accordance
with the constriction stress explanation put forth by metallurgists. Specifically, the drag
stress state variable in the constitutive model will be assumed to depend on the constric-
tion stress component.
PROPOSED EXPERIMENTAL PROGRAM
Biaxial experiments with tubular specimens will be conducted at 1600°F on a
combined tension-torsion machine. The biaxial setup used to conduct these experiments
was developed previously with partial support from NASA grant NAG-3-160. The setup
336
includes a servo-hydraulic tension-torsion machine and a capacitance probe based extenso-
meter shown in Figure 3. This extensometer utilizes four high temperature non-contact-
ing capacitance probes that do not require cooling. Two probes per channel are used to
cancel out cross-talk between extension and rotation. In order to heat the specimen to
1600°F an audio frequency heater has been purchased and is undergoing operation trials.
Gripping a tubular tension-torsion specimen at high temperatures is problematical.
To accomplish this the single crystal tube has Hastelloy-X extensions which are vacuum
brased onto the specimen. These extensions are welded into disposable 304 stainless steel
flanges that bolt to the tension-torsion machine. This procedure eliminates both the need
to cool the grips and the need to machine complicated end shapes on the single crystals.
The first three specimenshave recently been receivedand are ready for testing.
The testing machine is controlled by a microcomputer. The microcomputer also
places the load-displacement results from the tension-torsion machine onto a diskette
which can be read by the IBM PC-XT computer and compared with the theoretical predic-
tions from the constitutive model without manual manipulation of the experimental data.
This is necessarydue to the large volume of data generated in multiaxial testing. In con-
trolling the test machine the digital-to-analog converter sends voltages to a special, highly
stable, analog integrator that generates straight line voltage time segments. Without the
integrator the command signal would only be stepwise continuous and would involve un-
defined strain rates between the steps.
The test program is divided into two broad classes of experimerts. Specific features
of the response which need modeling will be addressedin the first class of experiments,
while the second class of experiments is designed to produce target verification tasks for
the model. Early experiments will investigate the possibility of removing the effects of
prior cycling by stress-free high temperature hold times or by steady cycling. If this can
be demonstrated many independent experiments can be performed on one specimen.
Two types of experiments can be performed to assess the values of the two state
variables comprising the constitutive model. The value of the scalar drag stress state
variable can be determined from experiments in which the strain rate is changed very
rapidly, since the proposed equations are such that the stress increment or decrement
due to a sudden change in strain rate dependsonly on the drag stress and not on the
tensorial equilibrium stress state variable. To study the value of the tensorial equilibrium
stress state variable, creep tests can be performed at various points on a steady state
hysteresis loop, searching for the load that gives zero initial creep rate corresponding to
the equilibrium stress value. This procedure was used in Reference [6], while the tests
involving sudden changes in strain rate have previously been used by Krempl in Reference
[7].
REFERENCES
1. Paslay, P.R., Wells, C.H., and Leverant, G.R.: "An Analysis of Primary Creep of Nickel
Base Superalloy Single Crystals," J. Appl. Mech., pp. 759-764, 1970.
.
.
Schmid, E. and Boas, W.: "Plasticity of Crystals with Special Reference to Metals,"
Chapman and Hall, London, 1968.
Umakoshi, Y., Pope, D.P., and Vitek, V.: "The Asymmetry of Ni3(AI,Ta) Single
crystals," Acta Met., Vol. 32, No. 3, pp. 449-456, 1984.
337
4. Paidar, V., Pope, D.P., and Vitek, V.: "A Theory of the Anomalous Yield Behavior
of L12 Ordered Alloys," Acta Met., Vol. 32, No. 3, pp. 435-448, 1984.
5. Shah,D., and Lin, L.,LJ. Metals, Vol. 32, No. 8, p. 62, 1980.
6. Walker, K.P.: NASA CR-165533, 1981.
7. Krempl, E.: RPI Report No. RPI CS 78-5, 1978.
338
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STRAIN TIMES lO_3
Figure 1. OrJentotion O_p_ndence of Hysteresis Loops
339
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STRAIN TIMES I0"'3
F£gure 2. Strain Rote _ependence oR Hysteresis Loops
340
ORIGINAL PAGE IS 
OF POOR QUAkIW 
Figure 3: Capacitance Probe Based High Temperature B i  ax i  a1 Extensometer 
341 


Biaxial constitutive equation development

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