APPLICATION OF UNIFIED CONSTITUTIVE RELATIONS TO
HYBRID STRESS FINITE ELEMENT FORMULATION
by
RICHARD EUGENE KEHL
A.B. Dartmouth College (1981)
M.S.M.E. Stanford University (1982)
Submitted in partial fulfillment
of the requirements for the degree of
MASTER OF SCIENCE
in
AERONAUTICS AND ASTRONAUTICS
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 26, 1989
© Richard E. Kehl 1989
The author hereby grants to MIT permission to reproduce
and to distribute copies of this thesis in whole or in part
Signature of Author _
Llepartment ot Aeronautics and Astronautics
May 26, 1989
Certified By
,, I.
H.H. Pian
*s Supervisor
Accepted By
Professor Harold Y. Wachman
N(/
Chairmani•,t)eii5fin•hfiai~tommittee
nAMsV'ituSETjTS INSTITUTE
OF TECHNOLOGY
JUN V/ 11,389
I~"ke
S M.I.T.T.
LIBRAR1ES
Application of Unified Constitutive Relations to
Hybrid Stress Finite Element Formulation
by
Richard E. Kehl
Submitted to the Department of Aeronautics and Astronautics
on May 26, 1989 in partial fulfillment of the requirements
for the degree of Master of Science in
Aeronautics and Astronautics
ABSTRACT
An incremental finite element method of inelastic analysis is developed
based on the application of unified constitutive relations to an assumed stress hybrid
finite element formulation. The unified constitutive model of Bodner-Partom is used
to describe the inelastic behavior of a material. An eight node solid finite element is
derived using the assumed stress hybrid formulation. An initial strain approach is
used to incrementally compensate for inelastic deformation and stress relaxation.
The inelastic analysis methodology is verified for the uniaxial case using Rene95
material constants. The effects of element shape and mesh density on the inelastic
response are investigated by analysis of fundemental beam and disk problems. The
finite element inelastic analysis gives reasonable results in these test cases.
Thesis Supervisor: Dr. Theodore H.H. Pian
Title: Professor of Aeronautics and Astronautics
ACKNOWLEDGEMENTS
It is with love and deep appreciation that I acknowledge the support of my
family, and in particular the sacrifices made by my wife, Sally, on the late nights
long weekends, giving me the time required to complete this thesis. To my children,
Adam and Briana, for helping me to keep perspective on what is truly important in
life.
To Professor Theodore H.H. Pian, for his counsel and support, and more
importantly, his patience over the years that this thesis has progressed. His
inspiration in the pursuit of the fundamental formulation of a problem is an
experience that I'll carry with me.
To General Electric Aircraft Engines and my managers Bill Vogel, Han-Yu
Kuo, and Bill Chan, for providing me financial support and for providing the
opportunity to expand my professional and academic horizons and pursuing new
solutions to challenging problems.
To my colleagues in Life Analysis, particularly Ann Chambers, Kai Tang,
and Jim Zisson, whose constant query on the status of my thesis, although often
frustrating, expressed concern and support.
Thank you
CONTENTS
Abstract
2
Acknowledgements
3
List of Figures
5
List of Tables
5
1 Introduction
6
2 Constitutive Relations
7
2.1 Unified Constitutive Models
2.2 Bodner-Partom Unified Constitutive Model
2.3 Bodner Rene95 Material Constants
3 Finite Element Formulation
3.1 Variational Principle
3.2 Eight Node Solid Element
4 Inelastic Analysis Methodology
4.1 Incremental Initial Strain Approach
4.2 Numerical Integration
7
8
11
14
14
17
21
21
23
5 Computational Analysis Verification
5.1 Cantilever Beam Analysis
25
25
5.2 Flat Disk Analysis
27
6 Discussion
6.1 Finite Element Solution
6.2 Inelastic Solution Stability
6.3 Inelastic Material Property Definition
6.4 Gas Turbine Blade Application
6.5 Future Work
40
40
41
43
44
45
7 Conclusion
51
References
53
Appendix: Computer Program Output
56
LIST OF FIGURES
2.3.1 Bodner [4] Rene95 Experimental Results
13
3.2.1 Solid Element of Arbitrary Geometry
20
5.1.1
5.1.2
5.1.3
5.1.4
5.1.5
Cantilever Beam Element Models
Cantilever Beam Load Cases
Cantilever Beam Inelastic Strain Response
Cantilever Beam Deflection (Tension)
Cantilever Beam Deflection (Shear)
29
29
31
32
33
5.2.1
5.2.2
5.2.3
5.2.4
5.2.5
5.2.6
Flat Disk Element Models
Flat Disk Load Case
Flat Disk Elastic Radial Stress
Flat Disk Elastic Tangential Stress
Flat Disk Radial Deflection (Inner)
Flat Disk Radial Deflection (Outer)
34
35
36
37
38
39
6.3.1 Stouffer [14] Rene95 Experimental Results
6.3.2 Rene95 Experimental Inelastic Strain Rate
47
48
6.4.1 Uncooled Shrouded LPT Turbine Blade Model
6.4.2 Cooled LPT Turbine Blade Model
49
50
LIST OF TABLES
2.3.1 Rene95 Inelastic Material Constants
12
5.1.1 Cantilever Beam Elastic Results
30
1 INTRODUCTION
A unified constitutive model, one that does not artificially decouple time
independent plasticity and time dependant creep material responses, provides a more
correct representation of the material, and will perhaps allow better insight into the
parameters governing material behavior. One such set of unified constitutive
relations, the Bodner-Partom model, is appropriate for a constant load isothermal
analysis and is chosen for incorporation into an assumed stress hybrid finite element
formulation. The Bodner-Partom material properties for a high temperature nickelbased alloy, Rene95, available in the literature, are used for inelastic analysis.
The inelastic capabilities represented by the unified constitutive relations are
incorporated into a three dimensional solid assumed stress hybrid finite element.
The finite element is developed from the Hellinger-Reissner variational principle and
uses eighteen 3stress parameters to model the stress field. An inelastic analysis
capability is added to the formulation using an incremental initial strain approach.
Elastic solutions to cantilever beam and plane stress flat disk problems are used to
evaluate the accuracy of the finite element analysis. The effects of element shape and
density on the elastic, and then on the inelastic finite element solution, are then
investigated.
The accuracy and stability of the inelastic finite element analysis, dependant
on the numerical integration of the changing strain field, is discussed and algorithms
for improving the analysis are suggested. The finite element formulation may be
expanded to address centrifugal and thermal loading, providing a useful tool in the
design of gas turbine blades.
2 CONSTITUTIVE RELATIONS
2.1 Unified Constitutive Models
Historically, inelastic material deformation has been separated into a time
independent plastic component, and a time dependant creep component. In a unified
approach these components are not assumed to be decoupled; the inelastic behavior
description includes plastic flow, creep, and stress relaxation. The derivation of a
particular unified theory is dependant on the treatment of directional hardening, and
the choice of a plastic flow rule. Realistic treatments allow for material work
hardening or work softening. In most unified approaches, internal variables are
chosen to represent the current material resistance to inelastic flow in a deformed
solid. Typically, two internal variables are chosen in order to represent both
isotropic hardening and directional hardening. Isotropic hardening is represented by
a scalar "drag" stress, and directional (kinematic) hardening is represented by a
"back, "end", or "equilibrium" stress tensor.
In a survey of unified constitutive models, Chan et al [1] identifies the
similarities and differences of several theories and presents the evolution (growth)
equations for the internal variables in a generalized form. Inelastic deformation is
assumed to be the result of two simultaneously competing mechanisms, a hardening
process based on the material deformation, and a recovery process that is time
dependant. The coefficients of the growth governing differential equations are
material properties which must be determined experimentally, and are time and
temperature dependant.
Constitutive models reviewed by Haisler and Imbrie [2] are presented for
comparison purposes in a more convenient uniaxial form, assuming that the thermomechanical loading is proportional. This study includes evolution equations of
theories proposed by Bodner, Krieg, Miller, and Walker. Of these, the constitutive
model proposed by Bodner and Partom [3], although lacking a "back" stress term to
allow for changes in loading, is the most appropriate for an initial implementation of
unified constitutive theories into a finite element analysis, and is adequate for an
isothermal steady-state analysis.
2.2 Bodner-Partom Unified Constitutive Model
The approach of a unified constitutive model is to decompose the material
strain response into an elastic portion and an inelastic portion combining both the
plastic and the creep material response.
etotal
elastic
e=
+
,inelastic
(2.2.1)
Taking a time derivative and using tensor notation, the strain rate equation can be
written.
kij = tije
+
itij P
(2.2.2)
The elastic strain rate, Nij, is a function of stress expressed as a time derivative of
Hooke's Law. The inelastic strain rate, tijP, is written as a function of stress,
internal variables, and temperature. Material constants for Rene95 are evaluated by
Bodner [4] for an isothermal case, so for this derivation temperature dependency is
omitted. In the Bodner-Partom formulation, a single scalar internal parameter,
"drag" stress, accounts for isotropic hardening. Directional hardening which is
typically modeled with a "back" stress is neglected.
The inelastic strain rate conforms to the Prandtl-Reuss flow law, and so
may be written in terms of the stress deviator tensor, sij.
(2.2.3)
tijp = x sij
The second invarient of the strain rate tensor, and of the stress deviator tensor are by
definition written:
D
= ('/2) tijP iP =
(ijP
)2
J2 = (1/2) Sij Sij = (Sij) 2
(2.2.4)
(2.2.5)
These definitions allow the flow law to be rewritten in terms of the invarients.
D2P =
ij
=
(2.2.6)
2J2
'D2P/J
2
Sij
(2.2.7)
A form for the second invarient of the inelastic strain rate, D2P, is written in terms of
an initial value and the internal parameter. This form is chosen based on dislocation
dynamics.
D2
= D02 expI -(a 32/3 J2)n ( (n+l)/n)}
(2.2.8)
In this equation Do and n are material properties; Do is the limiting value of the
inelastic strain rate in shear, and n is a parameter controlling the strain rate
sensitivity. Given an initial value of the internal parameter, (a 3)0, the strain rate
invarient and hence the strain rate tensor may be determined. A differential equation
for the internal parameter in terms of the invarients and material properties is derived
from a plastic work equation. This work equation is a relative measure of plastic
work done in reference to an initial state, where the internal parameter, o 3, has an
initial value, Zi. The formulation also includes a "saturation" value for the.internal
parameter, Z1, at large values of plastic work.
(3/Z
= m { 1 - (a
%/Z)
) 2 -D2P/J
3
- A ( (a• - Zi)/ Z1 }r
(2.2.9)
This equation introduces material properties Z1, Zi, m, A, and r, m controls the rate
of isotropic hardening, A is a coefficient in the recovery term, and r is the exponent
of the recovery term. The equation for the second invarient of inelastic strain rate is
used to arrive at the differential equation governing the growth of the internal
parameter, Oa3.
(3
= 2 m Do
(Z1 - 0X3) exp{ -(032/ 3 J2 )n ( (n+l)/n)
•T2
- A ( (C3 - Zi) Z1 }r
(2.2.10)
This equation determines the transient response of the internal parameter and in turn
the inelastic strain response of the material.
The second invarient of the deviatoric stress tensor, J2, has been written as
a function of the stress tensor, sij (Equation 2.2.5).
J2 = J2(sij) = J2 (o)
(2.2.11)
The second invarient of the inelastic strain rate tensor, D2P, has been written as a
function of the internal parameter, c 3, and the second invarient of the stress deviator
tensor, J2 (Equation 2.2.7).
10
D2P
= D2P (a 3 , J2 ) = D2P (a 3 , )
(2.2.12)
The internal variable evolution equation may then be rewritten, giving the internal
parameter rate, iX3, as a function of the internal variable, a 3, and stress state only.
X3 = iX3 (X3 , D2P, J2 ) = tX3(a 3 , a)
(2.2.13)
The implementation of the internal parameter evolution equation into a computer code
using these relationships greatly simplifies the apparent complexity of Equation
2.2.10, and is used in the computation of the inelastic strain response of the Rene95
material in the subsequent analysis.
The inelastic material response of the Bodner-Partom constitutive model
unifies the plastic and creep response, however, only the primary and secondary
phases of creep are modeled, the tertiary phase of creep is neglected.
2.3 Bodner Rene95 Material Constants
Material constants of the Bodner-Partom unified constitutive model are
evaluated for Rene95 material at 12000 F by Bodner [4]. These constants are
presented in Table 2.3.1 converted into English units. The inelastic constants were
chosen by Bodner to fit experimental data over a range uniaxial tensile and
compressive loadings. Bodner's fit of the experimental data for tensile loadings of
163 ksi - 184 ksi is presented in Figure 2.3.1. For these high load cases, the
material constants in the internal variable evolution equation result in calculated
Rene95 inelastic strain rates that are slightly higher than those seen experimentally.
Table 2.3.1: RENE95 INELASTIC MATERIAL CONSTANTS
Rene95 Constants from Bodner [4]
E = 25.7 x 106 psi
v = 0.3
Rene95 Inelastic Constants for Equation 2.2.10 from Bodner [4]
6 3 = 2 m Do0 <
(Z, - a 3) exp{ -(a 32 / 3 J2) ( (n+l)/n)}
- A { (a 3 - Zi)/ Z }r
3= = t 3(
3,
(Y) psi / sec
n = 3.2
r = 1.5
m = 2.76x 10 psi -1
A = 400.0 x 10-6 sec-1
Do = 10.0 x 103 sec - 1
Z,
Zi
= 319.0x 103 psi
= 232.0 x 103 psi
(a 3 ) 0 = Zi
12
Figure 2.3.1.
[4R) EN_E9 5
B ODN, R
_ ER
L
I MEN.A
7
rTv
i.
C.l)
C11
0
0.1
25
3300
TfMW ("M
I
13
3 FINITE ELEMENT FORMULATION
3.1 Variational Principle
The assumed stress hybrid finite element formulation is derived using the
Hellinger-Reissner variational principle by Pian [5]. Assuming that the function is
stationary gives:
(3.1.1)
8 7R = 0
The sum over the entire finite element structure must satisfy this condition.
=
7CR
Y (7R)n
(3.1.2)
The Hellinger-Reissner principle will include both elastic and inelastic strain
contributions, however, the inelastic portion will be addressed in a subsequent
chapter. The elastic components are written:
-
R
1/2)Sijklijkl "
=
+
Tii i dS +
ij
ij
-
Fi i i dV
Ti(ui-ui)dS
(3.1.3)
where itij is the stress tensor, tij is the strain tensor; iti is the displacement vector,
Sijk the elastic compliance tensor, Fi the body force components; and Ti the surface
traction.
As pointed out by Fernandez [6] in the derivation of a specialized hybrid
solid element, the choice of the Hellinger Reissner variational principle, which
14
assumes a displacement field throughout the entire element domain, not just on the
element boundaries as in the modified complementary energy formulation, means
that one volume integral, rather than several area integrals, need be evaluated. This
greatly simplifies the element formulation. In addition, in the element stiffness
matrix derivation, the following assumptions are made:
ii = 0
(3.1.4)
(no body forces)
Ti = 0 on the boundary
(traction free)
ui = ui on the boundary
(3.1.5)
(3.1.6)
The Hellinger-Reissner variational principle may then be reduced to:
-
R
=
( 1 /2)Sijk1dijdk1 - dijtij dV
(3.1.7)
In the assumed stress hybrid finite element formulation the stress tensor is
defined as a function of stress parameters, P. In matrix notation the stress is written:
Y= P
(3.1.8)
The strain field is written as a derivative of the displacement field in the element
which is in turn interpolated from the nodal displacements, q. In matrix notation the
strain is written:
e = DNq
= Bq
(3.1.9)
15
where D is a differential operator, and N is the displacement interpolation shape
function. The variational principle may now be written in terms of the stress
parameters, 0, and nodal displacements, q.
- IR
=
(1/)
(pp)TS(p
)
- (P)ITB q dV
(3.1.10)
V
- R = (1/2
T
v
pTsPdV
-
PT
v
pTBdV q
(3.1.11)
The matrices H and G are now defined as:
H =
(3.1. 12)
PTSPdV
V
G =
(3.1. 13)
PTBdV
V
The variational principle written in terms of these matrices is:
-
R
= (1/2)
TH -
fTGq
(3.1. 14)
Taking a variation with respect to P results in the following solution for [:
0 = H 1 Gq
(3.1. 15)
The variational principle may be rewritten as:
16
(3.1.16)
- R = (1/2) qTGTH-'Gq
This for results in the definition of the finite element stiffness matrix, K, which is
written:
K = GTHIG
(3.1.17)
The individual element stiffness matrix, defined in terms of the assumed stress field
and displacement interpolation field, may be globally assembled to determine the
finite element solution to an applied load.
3.2 Eight Node Solid Element
An eight node hexahedral solid finite element is developed by Pian and
Tong [7] using an eighteen 0 assumed stress finite element formulation. This
element has been programmed in a FORTRAN code by Kang [8] whose subroutines
were the basis for the elastic portion of the subsequent analysis verification. For an
can be
element of arbitrary geometry as shown in Figure 3.2.1, the stress tensor ij
written in terms of the natural coordinates 4, i1and C.
Tll = P1
+ 0•i
122 =
+
02
=
03 +
T12
=
+
T2 3
=
'33
04
s1306
=
010ý
(3.2.1)
+ 09 C + P11 1C
+
12C +
+ 0 14 T1
+13
+
015
r
0161T"
017C
+ 018
+5
+ 071
These equations give the definition of the stress shape function matrix P.
17
'
= P3
(3.2.2)
The physical components of stress may be obtained using the Jacobian matrix to
transform from the natural coordinate system to the Cartesian coordinate system.The
stress tensor is written:
aYd
(3.2.3)
= Ai j rij
where ijis the Jacobian between x,y and z and the natural coordinates 4, 1l and 5.
This relationship may also be written in matrix notation as:
a = JT
(3.2.4)
J = JT (P) J
It should be noted that for this transformation, the Jacobian should be calculated at
the element centroid, J (0,0,0), and used for transformation over the entire element,
in order for the finite element to pass the "patch" test.
The displacement field is also assumed in the element domain. The
interpolation functions may be written in terms of the natural coordinates.
N1 = (1/8) (1 - ) (1 -T) (1 - )
N 2 = (1/8) (1 -)
(1 + 1) (1-
(3.2.5)
)
N3 = (1/8) (1 +)
N4
(1 +T1) (1-)
= (1/g) (1 + 5) (1 - 1) (1 - )
N5 = (1/8) (1 -)
(1 -)
(1 + )
N7 = (1/8) (1 + 5) (1 + 11) (1 + )
N7 = (1/8) (1 + ) (1 - 1) (1 +)
N8
=
(1/8) (1+•) (1-Ti) (1+•)
18
The derivatives of these interpolation functions determine the displacement shape
derivative matrix B. The Jacobian matrix is again used to transform from the natural
coordinate system to the Cartesian coordinate system in which the nodal
displacements, q, are calculated.
B = J-'DN
(3.2.6)
The matrices P and B have been expressed in terms of the stress parameters and the
displacement interpolation functions. The matrices H and G are determined by
numerical integration over the element domain using Gaussian quadrature, which in
turn determines the element stiffness matrix K.
19
Figure 3.2.1: SOLID ELEMENT OF ARBITRARY GEOMETRY
c:
I1
20
4 INELASTIC ANALYSIS METHODOLOGY
4.1 Incremental Initial Strain Approach
An incremental initial strain approach to creep and viscoplastic problems
using the hybrid stress finite element formulation is presented by Pian and Lee [9].
The Hellinger-Reissner variational principle discussed in chapter 3 may be extended
to include inelastic strains as initial strains. For a given time step, the stress rate, oij,
and the displacement rate, iii, can be written as stress and displacement increments,
Aoij, Aui. The variational principle may be written as:
(/2)SijklAoijAakl -AGij
- 7R =
i
jA ~i-AFiAu i dV
V
ATiAui dS +
+
(4.1.1)
ATi(Aui-Aui) dS
where Aaij is the stress increment tensor; Aeije is the elastic strain increment tensor;,
AeijP is the inelastic strain increment tensor, Aui is the displacement increments; Sijld
the elastic compliance tensor; AF i the body force increment; and ATi the surface
traction increment. Making the same assumptions as in chapter 3, the variational
principle reduces to:
-C R
f(/ 2)SijklAoijAo
f=
kl-AijAijeAijA
dV
(4.1.2)
V
The stress increment tensor, Ao, is defined as a function of the stress parameter
increment, AP3, and the elastic strain increment, Ase , is written in terms of the nodal
displacement increment, Aq.
21
Ao = PA3
(4.1.3)
Aee = B Aq
(4.1.4)
The variational principle may be rewritten in matrix notation as:
- IR =
V
(1/2) (PA)TS(PA3) - (PA3)TBAq
+ (PAP3)TAe
-
=R
pTSPdV
(1/2) T
T
T
f PTBdV q
V
V
+
(4.1.5)
dV
pTAePdV
V
(4.1.6)
The inelastic matrix GP is defined as:
f pTAedV
(4.1.7)
V
The variational principle written in terms of matrices H, G,and GP is:
- ER =
AfTGAq + A3TGP
(1/2) AfTH
(4.1.8)
Taking a variation with respect to A~3 results in the following solution for AP:
At
= H-'(GAq- GP)
The inelastic stress parameter increment is:
22
(4.1.9)
AP P = - H-'GP
(4.1.10)
The inelastic strain, modeled for each time step as an initial strain, is
represented in the finite element formulation by the application of a nodal load
increment, AQP.
(4.1.11)
AQP = GTH-1GP
For the finite element solution the complete equation for the displacements is:
Aq = K
1 (AQ
- AQ P)
(4.1.12)
The inelastic analysis procedure using a hybrid stress finite element and the
Bodner-Partom unified constitutive relations starts by calculating the elastic stress
tensor at the element centroid. This stress tensor, and the corresponding deviatoric
stress invarient, is used to determine the inelastic strain increment for the first time
step. The nodal forces required to treat the inelastic strain as an initial strain are
calculated, the applied loading is modified, and the centroidal stress for the
subsequent time step is calculated. The stability of the solution depends on the
choice of time step in the numerical integration.
4.2 Numerical Integration
The appropriate numerical integration technique and the time step size for
the solution of the differential equations governing the material inelastic response is
discussed by Haisler and Imbrie [2]. Four numerical integration schemes are
considered: explicit Euler forward integration, implicit trapezoidal method,
23
trapezoidal predictor-corrector method, and Runga-Kutta 4th order method. It is
concluded that in terms of accuracy, computational time, and ease of implementation,
a simple scheme such as the Euler method is preferable provided the appropriate time
step is chosen.
In the inelastic finite element analysis, the Euler forward integration method
is chosen for the numerical integration of the Bodner-Partom evolution equation,
determining the growth of the internal parameter, and hence the inelastic strain
response of the material. Analysis by Pian and Lee [9] indicates that the appropriate
time step increment for a creep problem under a strain hardening rule, such as the
Bodner-Partom formulation, is determined by the condition:
P At
AE
< a (Etotal)time=
t
(4.2.1)
P is the effective inelastic strain increment, etotal is the effective total strain
where Ag
at time t, and a is a constant determined by the stability of the problem solution. The
effective strain is written:
E = (2/3)
+ (2/3)
(1/2){ ((-11 22) +( 22 - 33 )2+(ll-33)2
3({e
2+-
12
2
2
13 1
23 +C8
(4.2.2)
The time step optimizer increases the time step as a function of the effective inelastic
strain and the effective total strain. For the uniaxial verification of the analysis the
value of the constant a was taken to be 0.001.
At = a (Et ot al/A P) = 0.001 (Etotal/AP)
24
(4.2.3)
5 COMPUTATIONAL ANALYSIS VERIFICATION
5.1 Cantilever Beam Analysis
An eight node hexahedral solid finite element using an eighteen 0 assumed
stress finite element formulation, as discussed in chapter 3, is used in an inelastic
analysis based on the Bodner-Partom unified constitutive relations. The element
stiffness matrix formulation is developed by Pian and Tong [7] and programmed in
FORTRAN code by Kang [8]. Additional FORTRAN subroutines developed at
MIT [10], "FEABL", are used to assemble the elements and solve the global finite
element problem.
A cantilever beam problem is solved using regular shaped elements (interior
angles of 900) with a 5 to 1 aspect ratio, and also using highly skewed elements with
minimum interior angles of 140. The finite element model meshes and dimensions
are shown in Figure 5.1.1. Uniaxial tensile and pure shear loadings are applied to
the models as shown in Figure 5.1.2. The locations of the elastic finite element
analysis stress and displacement results presented in Table 5.1.1 are also shown in
Figure 5.1.2. Under uniaxial tensile loading (Load A), the error in axial deflection
and tensile stress for both the regular and skewed element meshs is negligible
compared to the beam theory analytical solution presented by Lindeburg [11].
Under the pure shear loading (Load B), the error in tip deflection and bending stress
for the regular shaped element mesh is less 1%, again compared to the beam theory
analytical solution. Even in the case of the highly skewed element mesh, where
significant errors are expected, the error in tip deflection is less than 5%, and the
error in the bending stress is less than 13%. The eighteen [ assumed stress finite
element formulation gives satisfactory elastic results to the cantilever beam problem.
The subroutines used for the elastic finite element analysis, together with a
subroutine based on the Bodner-Partom unified constitutive relations previously
25
discussed in chapter 2, were modified and integrated into a FORTRAN code for
inelastic finite element analysis. The elastic and inelastic portions of the three
dimensional total strain tensor, ij, are calculated from the stress tensor, aij,
evaluated at the element centroid. Based on observations by Haisler and Imbrie [2] a
simplified explicit Euler forward integration scheme is used in the integration of the
evolution equation. The finite element analysis FORTRAN code is also modified to
include a time step optimizer which uses criteria based on the effective total strain,
it
P , as discussed in chapter 4.
, and the effective inelastic strain increment, AE
An inelastic finite element solution to the cantilever beam problem under
various uniaxial loadings is calculated using the regular shaped finite element mesh.
The Bodner-Partom unified constitutive relation material constants for Rene95 at
12000 F are used for the analysis. The inelastic strain response is presented in
Figure 5.3.1. Comparison with Bodner's published Rene95 results, also included
in Figure 5.3.1, shows good agreement.
The constant used to calculate the optimum time step increment (a = 0.001
in Equation 4.2.3) was determined by the observation of a stable solution in the
initial time steps, particularly for the higher stress loadings. The choice of time step
determines the stability of the inelastic strain solution. A more analytical approach to
time step optimization based on consideration of the differential equations driving the
possible instabilities, is discussed in chapter 6. The inelastic finite element analysis
of the uniaxial load case verifies that with the choice of the appropriate time step, or
series of time steps, a meaningful solution may be obtained. The computer output
for the 175 ksi load case, showing the format of the results presentation, is included
in the appendix.
An inelastic solution to the cantilever beam under a uniaxial tensile load of
175 ksi was also calculated using the skewed finite element mesh. A comparison of
the inelastic axial deflection of the regular element mesh and that of the skewed
26
element mesh, presented in Figure 5.1.4, shows a significantly different inelastic
response despite similar elastic centroidal stress solutions. The regular shaped
element mesh has a faster inelastic strain rate than that calculated using the skewed
element mesh, resulting in greater inelastic deflections.
A pure shear load, resulting in a bending stress of 175 ksi at outer layer
element centroids, was applied to both regular and skewed cantilever beam element
meshes. In this case, the skewed finite element mesh did introduce some error into
the centroidal stress solution, on the order of 13% compared to the beam solution. A
comparison of the inelastic tip deflection calculated using the regular shaped element
mesh to that calculated using the skewed element mesh is presented in Figure 5.1.5.
As in the uniaxial load case, the inelastic strain rate calculated using the skewed
element mesh has the slower inelastic strain rate.
5.2 Flat Disk Analysis
The inelastic finite element code, verified by comparison with the solution to
the cantilever beam problem, is used to calculate the solution to a flat disk under an
internal pressure loading. The analytical elastic solution to this plane stress problem
is presented by Den Hartog [12]. The flat disk radial and tangential stresses have
maximum values at the disk inner radius.
A mesh with five elements covering a 900 arc is chosen, which results in
elements that are only slightly skewed, and should eliminate the inaccuracies seen in
the shear solution to the cantilever beam problem for skewed elements. In order
evaluate the finite element solution, results are calculated for a series of element
meshes with increasing element density at the disk inner radius, the location of the
maximum stress values. Finite element meshes of 10, 20, 30 and 40 elements and
the flat disk dimensions are shown in Figure 5.2.1. The loading (internal pressure)
27
and finite element boundary conditions are shown in Figure 5.2.2. The elastic radial
stress results are presented in Figure 5.2.3, and the elastic tangential stress results
are presented in Figure 5.2.4, for the various finite element mesh densities. The
finite element results show good agreement with the theoretical solution.
An inelastic finite element analysis using the Bodner-Partom routines and
the Rene95 material properties as was used in the cantilever beam problem, is used
to calculate the solution to the flat disk problem for the various element mesh
densities. The inelastic radial deflection at the disk inner radius is presented in
Figure 5.2.5, and the inelastic radial deflection at the disk outer radius is presented in
Figure 5.2.6. As expected, the inelastic solution converges as the element mesh
density is increased. Since the material inelastic response in this formulation is
based on the calculated stress at the element centroid, the denser element meshes,
which use the high stresses near the disk inner radius, are needed to correctly
calculate the inelastic solution.
28
Figure 5.1.1: CANTILEVER BEAM ELEMENT MODELS
REGULAR ELEMENT MODEL
SKEWED ELEMENT MODEL
IVIIIN AINJLL. = YU
NVII
,t.J
"
I,+ .iJ.IU.,J
LENGTH = 20 IN
HEIGHT = 2 IN
WIDTH = 2 IN
Figure 5.1.2: CANTILEVER BEAM LOAD CASES
LOAD A
k
j
LOAD B
71k
29
Table 5.1.1: CANTILEVER BEAM ELASTIC RESULTS
Displacement at i
Load A (ux)
Load B (uy)
Bending Stress at j
Load A (Tx)
Load B (Yx)
Bending Stress at k
Load A (tx)
Load B (Yx)
THEORY*
REGULAR
FEM
SKEWED
FEM
0.1362 in
0.9076 in
0.1362 in
0.9096 in
0.1362 in
0.8681 in
175.0 ksi
175.0 ksi
175.0 ksi
-175.0 ksi
-174.4 ksi
-153.3 ksi
175.0 ksi
-175.0 ksi
175.0 ksi
-173.3 ksi
175.0 ksi
-192.3 ksi
* Lindeburg [11]
30
N
3_:CA
Figure 5.1
M
ER BEA
I
CI
)
120 1(K
100
/
i
@
31
31
120
Figure 5.1.4: CANTILEVER BEAM DEFLECTION (TENSION)
0.50 -
0.40-
0.30 -
0.20 -
0.10 - I
O
O
0.00 -1
REGULAR-175 KSI
SKEWED-175 KSI
.
20
40
60
TIME (MIN)
32
80
100
120
Figure 5.1.5: CANTILEVER BEAM DEFLECTION (SHEAR)
1.50
-
1.40
1.30 -
1.20 -
1.10
-
1.00
O REGULAR-175 KSI
0.90
O1 SKEWED-175 KSI
0.80
.
I
*
20
I
40
1
T
'
60
TIME (MIN)
33
T
T
80
1
100
120
Figure 5.2.1: FLAT DISK ELEMENT MODELS
10 ELEMENT MODEL
20 ELEMENT MODEL
30 ELEMENT MODEL
40 ELEMENT MODEL
INNER RADIUS = 2 IN
OUTER RADIUS = 10 IN
DEPTH = 2.5 IN
34
Figure 5.2.2: FLAT DISK LOAD CASE
P = 240 KSI
35
Figure 5.2.3: FLAT DISK ELASTIC RADIAL STRESS
-80
-100
-120
-140
-160
-180
-200
0.0
4.0
2.0
6.0
RADIUS (IN)
* Den Hartog [12]
36
8.0
10.0
Figure 5.2.4: FLAT DISK ELASTIC TANGENTIAL STRESS
240-
1
220
200 -
mV"P-
I
IT
ýT
T
TI
180
160140 120
10080 60O
40 20 0
A
-
0.0
-
2.0
4.0
6.0
RADIUS (IN)
* Den Hartog [12]
37
10.0
Figure 5.2.5: FLAT DISK RADIAL DEFLECTION (INNER)
q
0.100
0.080
0.060 -
A
A
A
+
+
+
Ah
0.040+
0.020
+
+
+
+
10 ELEMENT
A 20 ELEMENT
0.000-
O
30 ELEMENT
O 40 ELEMENT
-0.020
I.I.I.a.I.
0
20
I
60
40
80
100
TIME (MIN)
38
120
140
160
180
Figure 5.2.6: FLAT DISK RADIAL DEFLECTION (OUTER)
0.022-
0.020 0.018 -
o
0 0
0.016-
0]
0.014
0.012-
A
A
+
+
0.010 n nnQ
41 1 1
r
I
+
+
+
+
+
0.006 0.004 -
+
0.002
-
A 20 ELEMENT
0.000 -
o 30 ELEMENT
O 40 ELEMENT
-0.002
10 ELEMENT
-
I
0
1
20
'
I
'
40
1
60
·
1
80
'
I
100
TIME (MIN)
39
'
I
120
'
I
140
'
1
160
180
6 DISCUSSION
6.1 Finite Element Solution
In the finite element analysis of the cantilever beam problem is seen that,
although the elastic solution of stress and displacement has negligible error for both
the regular shaped element mesh and the skewed element mesh, there is a significant
difference in the inelastic solution results. The inelastic strain is treated as an initial
strain in the inelastic analysis formulation. This requires incrementally updating the
load applied at each time step to compensate for the inelastic deformation The
amount of the change in load is calculated by integration, Gaussian quadrature, of
P , over the element volume. For the cantilever beam
the change in inelastic strain, Ae
models the location of the element centroid is the same, however, due to the element
shape, the Gaussian point locations will be different with respect to the beam
geometry. This difference may account for the differences seen in the inelastic
results.
The flat disk problem may be used to comment on the finite element mesh
density required to calculate a valid stress field and a meaningful problem solution.
The four meshes of increasing element density accurately calculate the elastic radial
and tangential stresses. The accurate calculation of the centroidal stress is important;
under the present inelastic analysis formulation, the element inelastic strain response
is dependant on the centroidal stress only. For the flat disk problem the denser
element meshes are required to get the element centroid close enough to the peak
stress to allow for the correct inelastic material response.
An alternative to a fine mesh finite element analysis is the modification of
the inelastic analysis to use the Gaussian integration point stress results to determine
the inelastic response. This will require internal variable information for each
element Gaussian point to be carried through the analysis. This will increase the
40
accuracy of the sub-element solution and allow the use of coarser finite element
meshes.
6.2 Inelastic Solution Stability
The stability of the inelastic finite element solution is dependant on the
numerical integration of the time dependant differential equations, e.g. the internal
variable evolution equation (Eq. 2.2.10). In the inelastic finite element solution of
the cantilever beam under uniaxial loading, an instability was observed in the
inelastic strain response for the high stress load cases. This necessitated the choice
of a relatively small constant in the time step optimizer criteria in order to get a valid
solution (a = 0.001 for the unified inelastic formulation, compared to the constant
used by Pian and Lee [9], a = 0.05 for a power law creep formulation).
It was noted, however, that even with the much smaller constant, the finite
element solution becomes unstable. At a relatively high time, the inelastic strain
begins to increase without bound. This is not due to an instability in the numerical
integration of the internal variable evolution equation, however. At the time step the
the instability occurs, which can be identified in the cantilever beam problem by a
departure from the expected solution, the internal variable, %,
3, has already reached
the saturation value, Z1. The internal variable, %3,
becomes a constant and the
inelastic strain rate then becomes a function of stress only; a function that is
described by equations 2.2.7 and 2.2.8. The change in inelastic strain rate is
written:
kigp
= tijP(J2,sij)
(6.2.1)
For a time increment this becomes:
41
AEP
= AeP(Aa)
(6.2.2)
The inelastic analysis formulation treats the inelastic strain response of previous time
steps as an initial strain problem (the load vector Q, and the stress field parameter
vector 0, are modified). The stress field is a function of the stress field parameter
vector increment, A[ P.
AU = P AD = AG(A3 P)
(6.2.3)
The parameter vector increment, A[ P,is a function of the inelastic strain as
described by equations 4.1.7 and 4.1.9, equations which include the integration of
the inelastic strain rate over the time interval.
P
Ap P = A[ P(e
)
(6.2.4)
This series of relationships defines a differential equation for the inelastic strain.
0P
=
OP(O)
=
0P(f3P)
=
OP(e
P
)
(6.2.5)
An approach for the derivation of an appropriate numerical integration
scheme, and a corresponding rational selection of time step increment is given by
Zienkiewicz et al [13]. The algorithm presented includes an adaptation to nonlinear
problems, such as the problem presented by the finite element inelastic strain
calculation. The implementation requires the interpolation of the finite element
parameter values over the time step using a polynomial expansion and obtaining
higher order derivatives. The polynomial expansion allows an estimation of the
error in the numerical integration and in turn determines a criteria by which the size
42
of the time step, or time step increment may be set. The task of implementing this
approach into the inelastic finite element analysis, the numerical integration, and the
time step optimization is considered a subject for future work.
6.3 Inelastic Material Property Definition
One of the most difficult tasks in undertaking an inelastic analysis using the
unified constitutive model is the definition of the material properties. The BodnerPartom model uses an internal variable to represent material isotropic hardening. In
an evaluation of Rene95 material properties, Bodner [4] discusses the physical
interpretation of the parameters of his model, and arrives at the set of material
constants presented in chapter 2, and used in the finite element analysis. The
inelastic material behavior of Rene95 at 12000 F has also been investigated by
Stouffer et al [14]. The experimental data for the high stress loadings is compared to
Bodner's material constant calculated results in Figure 6.3.1. Experimental Rene95
minimum inelastic strain rates (secondary creep) for various loadings is presented in
Figure 6.3.2. It is noted that the data presented by Stouffer shows consistently
higher strain rates.
The Rene95 experimental data presents the opportunity to modify the
material constants of the Bodner-Partom unified constitutive model to reflect a
different set of data. The procedure would begin by fitting a set of isochronous
inelastic material response curves. Each of the material constants in the internal
parameter evolution equation affect this response as described in chapter 2. Several
of the constants, such as n, r, A and Do, can be assumed to be near the Rene95
values, as they affect the shape of the curves rather than the magnitude. Through the
appropriate choice of the internal parameter limits, Zi, the internal parameter at the
initial state, Z1, the internal parameter at the saturation value, and m, a coefficient of
the internal parameter rate equation, the isochronous curves may be fit. A similar
43
procedure would be used to fit experimental data of Rene95 at other temperatures,
and to fit the experimental data of other materials.
6.4 Gas Turbine Blade Application
The proper modeling of the inelastic behavior of a material is critical in the
analysis of aircraft engine hot section components, and in particular gas turbine
blades. The stress rupture, or creep, characteristics of a blade often determine the
functional life of the part. The finite element analysis presented in earlier chapters is
only the first step toward applying a unified constitutive relation approach to blade
life analysis. The inelastic analysis procedures used here need to be expanded to
address body forces, temperature dependant material properties, and thermal
stresses. There are, however, classes of turbine blade problems that may use
simplifying assumptions allowing analysis as the finite element formulation expands.
The incorporation of body forces into the finite element formulation would
allow analysis of uncooled LPT turbine blades. These blades are typically thin,
shrouded airfoils running at a relatively uniform temperature across a span. A finite
element model of a shrouded airfoil is shown in Figure 6.4.1. Because of a uniform
bulk temperature, the thermal stresses are small and may be neglected. For a
conservative analysis, the material temperature dependance may also be neglected as
the area of concern is the stress and inelastic response of the hotter upper portions of
the airfoil and of the shroud.
Material property temperature dependance and thermal stresses become
important in the analysis of cooled turbine blades. When cooled at all, LPT turbine
blades are lightly cooled, and the temperature for a particular span is still relatively
constant. An example of a cooled LPT blade finite element model is shown in
Figure 6.4.2. There may be a significant thermal gradient from the root to the tip,
44
and although the thermal stress may sometimes be neglected, the temperature
dependance of the material properties may not.
Finally, in the analysis of highly cooled HPT turbine blades, there are very
high thermal stresses that, although over time may relax out, can significantly affect
the inelastic behavior, and hence the creep life of the blade. It is this class of
problems that is historically the most important from the point of view of the engine
designer, and the most difficult to analyze. The incorporation of expanded
capabilities of body forces, temperature dependant material properties, and thermal
stresses into the inelastic finite element formulation would provide a useful tool in
the design of turbine blades.
6.4 Future Work
First, there are steps that may be taken to improve the accuracy and stability
of the present inelastic finite element analysis formulation. The numerical integration
scheme may be updated from the Euler forward method to the algorithm proposed by
Zienkiewicz [13]. This allows a measure of the error in the numerical integration,
and provides an analytical method for determining the appropriate time step
increment necessary to maintain solution stability.
The Bodner-Partom unified constitutive model is one of many unified
models currently proposed. Although satisfactory for a isothermal steady-state
analysis, models such as the one proposed by Walker [15] which include a "back
stress" term in the internal variable evolution equation, allowing kinematic
hardening, are necessary to model correctly the response to changing loads.
As mentioned in the discussion of the application to gas turbine blade
analysis, the incorporation of body forces to represent the centrifugal loading,
temperature dependant material properties to appropriately model the material
response in areas of temperature gradients, and thermal stress capability to calculate
45
the affect on creep life in areas of high thermal stress are features necessary for a
practical design tool.
46
Figure 6.3.1: STOUFFER [14] RENE95 EXPERIMENTAL RESULTS
5.0ItI
..........
4.0-
BODNER-184 KSI
BODNER-175 KSI
BODNER-163 KSI
O
EXP-184 KSI
O]
A
EXP-175 KSI
EXP-163 KSI
3.01
0
2.0 -
00
1.0-
[]r
enseme
0.0
a
i-
ememme1
I
I
mm1 ea
memesm
mesmam
iri
U
a
:
a
0
10
15
TIME (MIN)
47
memmes
I
I
20
I
.
"
ammesm
I
25
ememam
I
30
Figure 6.3.2: RENE95 EXPERIMENTAL INELASTIC STRAIN RATE
O
0O
0
SB
10 -6
0
0
0
0 0
10 -8
0 BODNER [4] EXPERIMENTAL
0 STOUFFER [14] EXPERIMENTAL
I
110
120
lI
130
'
l
140
I
Il
150
i
I*I
160
APPLIED STRESS (KSI)
48
lI
170
180
'
190
Figure 6.4.1 UNCOOLED SHROUDED LPT BLADE MODEL
49
Figure 6.4.2: COOLED LPT BLADE MODEL
50
7 CONCLUSION
Inelastic finite element analysis has been developed using eight node solid
assumed stress hybrid finite elements for definition of the stress field, the BodnerPartom constitutive relations for definition of the inelastic strain response, and an
incremental initial strain approach to calculate the inelastic stress and strain response
through time. Evaluation of fundamental cantilever beam problems using Rene95
inelastic material constants, verifies the elastic and inelastic finite element solutions.
Highly skewed finite elements are seen to affect the inelastic strain response in the
beam problems, reducing the inelastic strain and inelastic strain rate as compared
with a regular shaped finite element mesh.
The solution of a flat disk problem using finite element models of increasing
mesh densities used to evaluate a multi-axial load case. Although the elastic solution
of centroidal stress in satifactory for all the meshes, the inelastic solution is
dependant on the element centroidal stress only, so denser element meshes are
necessary to get the element centroid close enough to the inner radius peak stress to
allow a correct calculation of the inelastic response. The use of the stress state at the
element Gaussian integration points for the inelastic strain calculation is one possible
solution to this problem.
The stability of the inelastic solution is seen to be a function of numerical
integration technique and the time step increment chosen. The constant required in
the time step optimizer for the unified constitutive inelastic analysis required to
maintain a stable solution is seen to be more than an order of magnitude smaller than
the constant used in a traditional power-law creep finite element analysis. An
analytical approach to time step increment selection, such as that proposed by
Zeinkiewicz, would improve the stability of the solution.
51
The analysis of gas turbine blades requires an expansion of the current
formulation to address centrifugal and thermal loading, and temperature dependant
material properties. Incorporation of these capabilities, and the development of
unified constitutive material constants for a variety of other materials, would provide
a useful tool in the evaluation of turbine blade designs.
52
REFERENCES
[1] Chan, K.S., U.S. Lindholm, S.R.Bodner and K.P. Walker, "A Survey of
Unified Constitutive Theories", Second Symposium on NonlinearConstitutive
Relationsfor High Temperature Applications, NASA Lewis Research Center, 1984.
(NASA CP-2369)
[2] Haisler, W.E. and P.K. Imbrie, "Numerical Considerations in the Development
and Implementation of Constitutive Models", Second Symposium on Nonlinear
ConstitutiveRelationsfor High TemperatureApplications, NASA Lewis Research
Center, 1984. (NASA CP-2369)
[3] Bodner, S.R. and Y. Partom, "Constitutive Equations for Elastic-Viscoplastic
Strain Hardening Materials", Journalof Applied Mechanics, 42, 2, 385-389 (1975).
[4] Bodner, S.R., "Representation of Time Dependant Mechanical Behavior of
Rene95 by Constitutive Equations", Technical Report, Air Force Materials
Laboratory, 1979. (AFML TR-79-4116)
[5] Pian, T.H.H., "Nonlinear Creep Analysis by Assumed Stress Finite Element
Methods", AIAA Journal,12, 12, 1756-1758 (1974).
[6] Fernandez, J.A.G., "Hybrid Solid Elements with a Traction Free Cylindrical
Boundary", SM Thesis, Massachusetts Institute of Technology, 1986.
53
[7] Pian, T.H.H. and P. Tong, "Relations Between Incompatible Displacement
Model and Hybrid Stress Model", InternationalJournalfor NumericalMethods in
Engineering, 22, 173-181 (1986).
[8] Kang, D.S., "Solid Hybrid Element - 8 Node, 24 DOF", FORTRAN Code,
Massachusetts Institute of Technology, 1983.
[9] Pian, T.H.H. and S.W. Lee, "Creep and Viscoplastic Analysis by Assumed
Stress Hybrid Finite Elements", FiniteElements in NonlinearSolid andStructural
Mechanics, (Ed. P.G. Bergan et al), Norwegian Institute of Technology, 2, 807822 (1977).
[10] MIT, "FEABL", FORTRAN Code, Massachusetts Institute of Technology,
1983.
[ 11] Lindeburg, M.R., (Ed.), MechanicalEngineeringReview Manual (7th
Edition), Professional Publications Inc., Belmont, CA, 1984.
[12] Den Hartog, J.P., , Advanced Strength of Materials,McGraw-Hill, NY, 1952.
[13] Zienkiewicz, O.C., W.L. Wood, N.W. Hine and R.L.Taylor, "A Unified Set
Of Single Step Algorithms, Part 1: General Formulation and Applications",
InternationalJournalfor NumericalMethods in Engineering,20, 1529-1552 (1984).
[14] Stouffer, D.C., L. Papernik and H.L. Bernstein, "An Experimental Evaluation
of the Mechanical Response Characteristics of Rene95", Technical Report, Air Force
Wright Aeronautical Laboratories, 1980. (AFWAL TR-80-4136)
54
[15] Walker, K.P., "Research and Development Program for Nonlinear Structural
Modeling with Advanced Time-Temperature Dependant Constitutive Relationships",
Final Report, NASA Lewis Research Center, 1981. (NASA CR-165533)
55
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