A modification was formulated to Besseling's Subvolume Method to allow it to use multilinear stress-strain curves which are temperature dependent to perform cyclic thermoplasticity analyses. This method automotically reproduces certain aspects of real material behavior important in the analysis of Aircraft Gas Turbine Engine (AGTE) components. These include the Bauschinger effect, cross-hardening, and memory. This constitutive equation was implemented in a finite element computer program called CYANIDE. Subsequently, classical time dependent plasticity (creep) was added to the program. Since its inception, this program was assessed against laboratory and component testing and engine experience. The ability of this program to simulate AGTE material response characteristics was verified by this experience and its utility in providing data for life analyses was demonstrated. In this area of life analysis, the multiaxial thermoplasticity capabilities of the method have proved a match for the actual AGTE life experience

Mcknight, R. L.

English

NASA Technical Reports Server

Abstract 
MULTIAXIAL CYCLIC THERMOPLASTICITY ANALYSIS 
WITH BESSELING'S SUBVOLUME METHOD 
R. L. McKnight 
General Electric Co. 
Cincinnati, Ohio 45215 
In 1975, a modification was formulated to Besseling's Subvolume Method to 
allow it to use multilinear stress-strain curves which are temperature 
dependent to perform cyclic thermoplasticity analyses. This method 
automotically reproduces certain aspects of real material behavior important 
in the analysis of Aircraft Gas Turbine Engine (AGTE) components. These 
include the Bauschinger effect, cross-hardening, and memory. This 
constitutive equation has been implemented in a finite element computer 
program called CYANIDE which has been in production usage since 1977. 
Subsequently, classical time dependent plasticity (creep) was added to the 
program. Since its inception, this program has been assessed against 
laboratory and component testing and engine experience. The ability of this 
program to simulate AGTE material response characteristics has been verified 
by this experience and its utility in providing data for life analyses has 
been demonstrated. In this area of life analysis, the multiaxial 
thermoplasticity capabilities of the method have proved a match for the actual 
AGTE life experience. This paper will explore the multiaxial, 
vari?~le-temperature nature of the method and show examples demonstrating its 
utility. 
69 
https://ntrs.nasa.gov/search.jsp?R=19830026085 2020-03-21T03:03:20+00:00Z
BESSELING'S MATHEMATICAL MODEL 
The relation between the deviatoric stresses and the deviatoric 
strains is given by 
(1) 
where 
Sij is the deviatoric stress tensor 
eij is the total deviatoric strain tensor 
e" ij is the plastic strain tensor 
G is the shear modulus 
The yield strain, P, is given by the plastic potential function 
g ( II )( ") p2 0 • eij - eij eij - eij - • (2) 
The incremental plastic strains are given by 
(3) 
provided that 
(4) 
The incremental stress-strain relation is 
70 
(5) 
The new yield strain, eij + 6eij , is determined from 
(6) 
Besseling then introduced the concept of elastic-perfectly plastic 
subvolumes. The elastic potential, '1' of the subvolume of density p 
after prior plastic flow is given by 
where the eijl are the plastic strains due to ideal plastic yielding. 
If this subvolume constitutes the fraction '" of the volume element 
dV, its contribution to the total elastic potential of dV is 
(7) 
(8) 
If k subvolumes of the volume dV have exceeded their critical value of 
elastic potential and undergone plastic flow, the total elastic 
potential is given by 
(9) 
Now, the deviatoric stress tensor is given by 
(LO) 
71 
After yielding, the elasticity limit of 8ubvolume k is given by 
(11) 
The subvolume incremental plastic strains are given by 
(eij - eijk) (eaB - ea EDt) oeijk c: --..:=---....=..L..:.:...-p-=-2-=..:::...-~~~ oeaBk 
k 
(12) 
provided that 
(13) 
The incremental stress-strain relations are 
(14) 
DEVELOPMENT OF NONISOTHERHAL CAPABILITY 
The equation relating the stresses and the subvolume strains, 
Equation (10), can be rewritten to give 
(15) 
Now these stresses must be the same as the stresses given by Equation (1) • 
Therefore, the two right-hand s ides can be equated. When this 
is done, we get 
72 
e" ij (16) 
which gives a relationship between the subvolume plastic strains and 
the total plastic strains. 
Squaring both sides of Equation (16) and multiplying by 2/3, we 
get 
Now 
e" ij 
Therefore 
This gives a relationship between the total effective plastic strain 
and the subvolume effective plastic strains. 
The following ratio can be formed between a subvolume effective 
plastic strain and the total effective plastic strain: 
or 
(
£P£pn)2 _ 1 eijn eijn 
2 e" e" 3" ij ij 
73 
(17) 
(J 8) 
(19) 
(20) 
(21) 
By taking the square root of both sides, we obtain 
£ 
• ...E.!!. e" 
eijn £ ij 
P 
(22) 
This gives a means of determining the subvolume plastic strains from 
the total plastic strains if the effective plastic strains are known. 
This then provides the tools to convert Besseling's Isothermal 
theory into a nonisothermal theory. We note that for variable 
temperature problems 9 and 9k will be functions of both strain and 
tempe rat ure. 
g c 9 (elj,T) 
gk Ie gk (elj ,T) 
These functions can be specified by defining temperature dependent 
stress-strain curves. 
For tncremental loading tncludrng temperature changes, the 
change In the plastic potential function Is given by 
There are three posslbJe conditions that can occur due to this 
(23) 
(24) 
(25) 
load Increment and these are determined by the value of this differential. 
For loading beyond the present yield surface 
dg > 0 
!a de +.2E. dT > 0 
aetj tj aT 
74 
(26) 
(27) 
For the loading to place the point on the new yield 
surface 
dg = 0 
fa de.. + ~Tg dT = 0 Cle.. IJ CI IJ 
For the point to unload back into the elastic range 
dg <0 
~ de .. + ~ dT< 0 
IJ T ae. . a IJ 
These last two conditions are used to accomodate 
temperature variations. The solution to any load condition, 
(N-I), is arrived at when 
dg = ~ de.. ~ 0 
n-l ae ij IJ/T = Constant 
In proceeding to the next load step, (N), the temperature 
effects on the stress-strain curve are incorporated so as 
not to vialate this condition while holding the strains 
constant. 
- ~ dT dg(n-l),(N) - aT = 0 
Thus, we are requiring that the change of temperature alone 
not effect the inelastic condition of the material. We 
accomplish this by realizing that 
~dT = 2G e p 
at - T de ij de ij 
Therefore, by requiring that 
de~j = 0 
We force 
it dT = 0 
75 
(28) 
(29) 
(33) 
(35) 
(36) 
This then gives us the mechanism for positioning our new yield 
surfaces In step, N. The step, N, solution then proceeds by applying 
the loads and boundary conditions and iterating to obtain 
d 2L d ~O g .. e iJ• elj 
within your specified convergence tolerance. 
CREEP ANALYSIS 
(37) 
The creep analysis utilizes one of two possible creep representations. 
When tertiary creep is not considered to be of importance, the equation used 
is 
(38) 
where 
- /100000 = effective stress (Te - CI e ' O'e 
k, m, n, q, r = material-dependent and temperature-dependent creep coef-
tic-! ents. 
1~len the material exhibits a significant amount of tertiary creep capa-
bility, an alternate representation is used. Primary creep is represented by 
the 83i ley-Not-ton law. 
p - A2 A3 
e ::a Al O'e t 
c 
Sr.conJary creep is modeled with the expression proposed by Marin, Pao, and 
Cuff (Reference 19) 
E S 
c 
76 
(40) 
Tertiary creep is represented with an equation of the form 
(41) 
= 
AI, A2, ••• AIO c material-dependent and temperature dependent cr~1 ," 
coefficients. 
CYANIDE also contains an orthotropic creep formulation. The creep s~"l"ain 
rate is assumed to be given by 
where 
~ •• a strain rate tensor 
1J 
0kl - stress tensor 
(42) 
gijkl = Tensor whose components are functions of temperature, de, and 
hardening rule and are derivable from input creep curves. 
The user can select from time hardening, strain hardening, or life frac-
tion creep rule, depending upon the actual material characteristics. Strain 
hardening is ordinarily adequate for describing hardening behavior, providing 
that stress reversals do not occ~r. A stress reversal 1s considered to occur 
when 
c . 
e .. 0 •• < 0 1J 1J 
Where eijc is creep strain measured from its current origin. When a ~ev~rsal 
occurs, the origin is changed and the analysis proceeds (Reference20) . 
The combination of general creep equations and creep rule makes the pro-
gram very general In Its application to structures which undergo time-dependent 
plastic flow In which transient effects are not significant. 
77 
CYANIDE COMPUTER PROGRAM 
Many of the steps In the CYANIDE nonlinear finite element 
computer program are the same as those for a linear finite element 
analysis. The nonlinear effects are Introduced Into the system of 
finite element equations by adding vectors of pseudoforces to the 
right hand side. 
I KI {oJ = {F} + {F } + {F } p c 
where 
IKI is the clastic stiffness matrix. 
{oJ Is the vector of nodal displacements. 
{F} is the force vector including thermal terms. 
{F } p Is the plastic pseudoforce vector. 
{F } Is the creep pseudoforce vector. 
c 
(44) 
For each Increment of loading, the nonlinear pseudoforces are Iterated 
upon until the requirements of equilibrium, compatabillty, and the 
constitutive equations are met within user specified tolerances. Since 
this method does not require modification of the stiffness matrix during 
iterations it Is very economical. This economy is magnified during 
cyclic analysis. The stiffness matrix need only be regenerated If the 
material properties are revised by thermal variation or if elements have 
been added or removed. 
MULTIAXIAL, VARIABLE TEMPERATURE EXAMPLE 
In a previous NASA contract, we investigated one of the common thermal 
stress problems In AGTE' s : turbine blade tip cracking. In that case, the 
critical region was shown by analysis and confirmatory testing tb have the 
cyclic stress-strain behavior noted In Figure 1. High temperature, time 
dependent flow rapidly relaxes the compressive stress such that on 0001-
down high tensile stresses are genrated. This process shakes down very 
rapidly to an almost elastic hysteresis loop based on modulus changes. 
In that case the problem was almost totally uniaxial In nature. 
78 
A second type of thermal stress problem prevalent in AGTE's is the hot 
spot. In this case, the stress strain response is definitely multiaxial. We 
will investigate a hot spot on a combustor shingle as being typical of these 
problems. Figure 2 shows a shingle segment. Taking advantage of its large 
radius of curvature and thinness, it was modeled as a flat plate in a 
condition of plane stress. The model is shown in Figure 3. Figures 4, 5 and 
6 show the nature of the hot spot at peak temperature and Figure 7 shows the 
heat-up cool-down temperature cycle at the center of the hot spot. This cycle 
was analyzed assuming no time dependent effects occcurred during heat-up and 
cool-down but that a one hour hold time was associated with the peak of the 
hot spot. 
The stress-strain results of the first cycle are shown in Figures 8, 9 
and 10 for the center of the hot spot. Figure 8 shows effective stress versus 
effective strain and Figures 9 and 10 show the biaxial stresses versus 
strains. Once again the effect of plasticity and creep is to generate tensile 
stresses during the cool-down portion of the cycle. The next series of 
figures shows the shakedown stress-strain results for the center of the hot 
spot. Figure 11 shows the effective stress versus effective strain shakedown 
values and Figures 12 and 13 show the shakedown biaxial stress cycle at the 
center of the hot spot. Thus this multiaxial thermal stress case, just as the 
uniaxial case, shakes down to almost elastic cycling with a high tensile mean 
stress. In addition, the stresses are almost proportional. These types of 
analyses are important in indicating the types of response and life tests 
needed. 
79 
REFERENCES 
1. Besseling, J. F., "A Theory of Plastic Flow for Anisotropic Hardening in 
Plastic Deformation of an Initially Isotropic Material", Report S.4l0, 
National Aeronautical Research Institute, Amsterdam, 1953. 
2. Besseling, J. F., "A Theory of Elastic, Plastic, and Creep Deformations 
of an Initially Isotropic Material", SUDAER No. 78, Stanford University, 
April, 1958. 
3. Besseling, J. F., "A Theory of Elastic, 1'lastic, and Creep Deformations 
of an Initially Isotropic Material Showing Anisotropic Strain-Hardening, 
Creep Recovery, and Secondary Creep", J. of Applied Mechanics, pp. 
529-536, December, 1958. 
4. Duwez, P., "On the Plasticity of Crystals", Physical Review, Vol. 47, p. 
494, 1935. 
5. White, G. N., Jr., "Application of the Tbeory of Perfectly Plastic Solids 
to Stress Analysis of Strain Hardening Solids", Graduate Division of 
Applied Mathematics, Brown University, Tech. Rpt. 51, 1950. 
6. Turner, J. J., Clough, R. W., Martin, H. C., and Topp, L. J., "Stiffness 
and Deflection Analysis of Complex Structures, Journal of the 
Aeronautical Sciences, Vol. 23, No.9, p. 805, Sept., 1956. 
7. Armen, H. Jr., "Plastic Analysis", Structural Mechanics Computer 
Programs, edited by Pilkey, W., Saczalski, K., and Schaeffer, H., 
University Press of Virginia, 1974, pp. 37-79. 
8. Armen, H., Jr., Pifko, A., and Levine, H. S., "Finite Element Analysis of 
Structures in the Plastic Range", NASA CR-1649, February, -1971. 
9. Armen, H., Jr., Levine, H. S., and Pifko, A. B., "Plasticity - Theory and 
Finite Element Applications", Proc. of 2nd U.S.-Japan Seminar on Matrix 
Methods of Structural Analysis and Design, pp. 393-437, August 1972. 
10. Witmer, E. A., Kotanchik, J. J., "Progress Report on Discrete-Element 
Elastic and Elastic-Plastic Analyses of Shells of Revolution Subjected to 
Axisymmetric and Asymmetric Loading", Proc. of the Second Conference on 
Matrix Methods in Structural Mechanics, WPAFB, pp. 1341-1453, December, 
1969. 
11. Wilson, E. L., "Finite Element Analysis of Two-Dimensional Structures", 
University of California, Structures and Materials Research Department of 
Civil Engineering, Report No. 63-2, June, 1963. 
12. Salmon, M., Berke, L., and Sandhu, R., "An Application of the Finite 
Element Method to Elastic-Plastic Problems of Plane Stress", Air Force 
Flight Dynamics Laboratory, Tech. Rpt. AFFDL-TR-68-39, May, 1970. 
13. Zienkiewicz, D. C., The Finite Element Method in Engineering Science, 
McGraw-Hill, New York, 1971. 
14. Marin, J., Ulrich, B. H., and Hughes, W. P., "Plastic Stress-Strain 
Relations for 758-T6 Aluminum Alloy Subjected to Biaxial Tensile 
Stresses", NACA TN2425, National Advisory Committee for Aeronautics, 
Washington, D.C., August, 1951. 
15. Ueda, Y., and Yamakawa, T., "Thermal Stress Analysis of Metals with 
Temperature Dependent Mechanical Properties", Mechanical Behavior of 
Materials Vol. 3, edited by Taira, S., and Kunugi, M., The Society of 
Materials Science, Japan, pp. 10-20, 1972. 
16. Hunsaker, B., Jr., Vaughn, D. K., Stricklin, J. A., and Haisler, W. E., 
"A Comparison of Current Work Hardening Models Used in the Analysis of 
Plastic Deformations", Texas A&M University, Texas Engineering Experiment 
Station, TEES-RPT-2926-73-3, October, 1973. 
80 
17. McKnight, R. L., "Finite Element Cyclic Thermoplasticity Analysis by the 
Method of Subvolumes", Ph.D. dissertation, University of Cincinnati, 1975. 
18. McKnight, R. L. and Sobel, L. R., "Finite Element Cyclic Thermoplasticity 
Analysis by the Method of Subvolumes," Comput.Structures, Vol. 7, No.3, 
1977 • 
19. Marin, J., Pao, Y. R., and Cuff, G., "Creep Properties of Lucite and 
Plexiglass for Tension, Compression, Bending, and Torsion," Trans. ASME 
73, 705, 1951. 
20. Pugh, C. E., "Constitutive Equations for Creep Analysis of Liquid 
Moderated Fast Breeder Reactor (LMFBR) Components," in Advances in Design 
for Elevated Temperature Environment, S. Y. Zamrik and N. I. Jetter, Eds. 
AS ME , 1975. 
81 
lOAD 12.11W1M DlVIlION 
.ITRAIN O.01UM DIVIlIOO 
\1' i ':1 
-W"l-i" I ~~ I.j. .1 
"IYITtU; nRAIN COMTPIOllEI'IO 01 ..... ..,.. .• . I 
o I ! I C'i-: THf IIW.1. ITIWN 
I 1cyeLE~"; i ! 
r- --I -_.; TT r 
. ~ ..... ! -·1 ! ... ! '-1 . I 
~ '"1- ' L· LI----:--l-.;.-J-~_+_ 
.... 311000e 
12000" FI 
11251 .. 1 
• 0 
I'L 2 nooe-
111000 FI 
114 S .. I 
-
.... 11~OOCr i 
11Il000 F I 
175 ,.C! !,. 
i I 0'. 
Figure 1. Mechanical Strain Veraua Load - Test I. 
'Ilura 2. Shinll. S~laent. 
82 
. .£rIIDM1ION 
'1-"It" 
- . 
. I 
.+ .... '.1" 
1- _L i,' .;, .. 
lOl.D{n 
:, l:;:=! 
I1lIAIN 
•. I~ 
I 
! 
1000 
1100 
.. 
• 1100 
t ,. 
.. 
11400 
• .. 
1:100 
1000 
• Arrove Denote Fbi ty in Indicated Direction 
Figure 3. Boundary Conditions. 
C=---~ 
Figure It. Axial Te.perature Di.tribution W/o Hot Spot. 
83 
.. 
• 
.-
.. 
:J 
.. 
• .. 
!. 
I 
.. 
2000 
11100 
11100 
1400 
1200 
1000 
-------------
Solid Line. Diatribution Throu,h Center of not Spot 
Da.hed Line • Ko~.1 Distribution 
Figure 5. Axial Tr.perature Distribution Through Hot Spot. 
-'OfHrSik~ fo!OT roT PlJI'TlCITY MAL'ff3JIS B>.'\Tt GVJIVU HPlP~RATU~E OTMT U58.~ 1f1eR GO.GOt 
tVAHioE AHALYGIS 
x 
FIgure 6. Temperatur. Contours At !'uk Of Hot Spot 
84 
--III.~ u.._ 
.... -
.... -UN._ 
• 1=:= 
· .... -J lAe._ J .... _ 
1 U!M._ .-.-.-.-
· 1-.-~ =:: 1=:: 
f 
, 
• ! 
TE"PERATURE US. LOAD CASE (HSRSI2) CYAHIDE AHALYSIS 
-
" 
i i c l\.'~ • ... 
-
I 
I 
.... 
~ ~ 
-
/ ~ 
/ ~ 
II "\ .1 
a_ 
e 
e. a. •• 4. s. •. 1. •• t. ... II • II. 
..-ca. 
Figure 7. Heat-Up Cool-Down Cycle For Center Of Hot Spot 
EFF. STRESS US. EFF. ITRAI" (HSR8'2) CYAHIDE ANALYSJS 
.. 
C lUll • 
'II 
, 
~ .. .. , 
41 
'" -'" ~ 
--
~ ~ I / I 
V ~ ~ ~ ..I It 
/ -....::: ~ .. 
• 
II 
•. a_. _. _. _. _. _. _. ..... _. a_. 11_. ,_. 
err. TlTAI. l1Mall 
Figure 8. Effective Stress Versus EffectIve Streln At The Center of the 
Hot Spot For the First Cycle 
85 
• 
! 
I 
I 
, 
• I 
• 
R-STRESS us. R-STRAIH (HSRS02) CYAHIDE ANALVSIS 
.. 
C NIl 
" 
! ! i II 
~ I /' .. 
I!lP I I I 1/ I I It 
V I ! 
/' 1 • 
< 
v 
I 
i 1/ i 
"'19.... ~ I /1 JIft' I I ~ 
... I 
I i 
-It 
_. __ . _. -lIN. -_. -Gel. _. -1_. -1_. _. 
•• lIN. a_ . 
a .. 
t 
.. 
II 
• 
... 
-
• 'TDTAI. "..1_ 
Figure 9. Tr.nsverse Stress Versus Str.ln At The tenter of the Hot 
Spot For the FIrst Cycle 
Z-STRESS US. Z-STRAIH (HSRB02) CY~"JDE AHALYSI5 
t 1l.1li II 
/' 
~p 
/' ~ 
/ V 
V V 
~ /' B /" 
I-ts ~ V ~ ~ ~ 
~ 
-.t. _. ___ . __ . -<I,". _. -XH. -lON. -ISM. __ . -a_. -1_. _. 
Z TOTAl. "..1' 
FIgure 10. Longltudln.I Stress Versus Str.ln At the tenter of The Hot 
Spot For the First Cycle 
86 
J , 
, 
a 
I 
• 
a 
, 
a 
I 
• 
Eff. STRESS US. Eff. STRAIH (HSRSGS) CYAHIDE AHALYSIS I" C ILl" ., 
I 
lit 
I .. 
~P I ! . .. 
~ I 
.. '\ 
c, \. ~ • ~ ~ ~ ...... .. 
"s 
........ JI>. ~ 
"t:T ~ 
• 
_. ..... _. lOll. _. 1_. 1_. 11_. 11_. 1_. 1_. 1_. 1_. 
lit 
I .. 
.. 
.. 
.. 
It 
• 
-. 
m. tlT.~ .,..1" 
FIgure II. Shakedown Cycle of EffectIve Stress Versus EffectIve StraIn At 
the Centar of the Hot Spot 
R-STRESS US. R-STRAIH (HSRSGS) CYAHIDE AHAlYSIS 
tj ILlII .. 
I 
:---1\ I I 
--
~ 
~ 
,/' \ I r--~~ I 
~ 
,....... 
~ V I ~ ~ 
~ V ~ v-'" ~ L-a-
~ 
-
f--
i"" 
_. -_. -_. _. __ . __ • -_. -_. -14M. -lIM. -_. _. __ • 
a 'IOT.~ .,..1" 
FIgure 12. Shakedown Cycle For Tran.verse Stre •• Ver.us StraIn At the 
Canter of the Hot Spot 
87 
2-STRESS US. 2-STRAIH (HSR80S) CYAHIDE AHAlYSIS 
an 
t:l ILlJI n ~1 /"" 
! Vr 
V ! I 
'" 
I 
r; i I / ! 
v.V -1 I I 
/V V I i 
.~ ~ 
au 
.. 
.. 
.. 
II 
t 
-
.., 
__ u. _. .'IMI. ._. ..Itt. _. -I.... ...... ..,... ._. ._. ..... _. 
a" 
t:l ILiN 
au 
.. 
, u 
, .. 
• I 
" 
e 
-
.... 
.... 
-ceo 
r fOYAL .,...1" 
Figure 13. Shakedown Cy~le For Longitudinal Stress Versus Strain at the 
Center of the Hot Spot 
R-STRESS US. Z-STRESS CHSRS.S) CYAHIDE AHALYSIS 
.. 
/~ 
V 
--t V 
./ 
-~ /v 
l/ 
~ 
-. -ct. t. ... ... u. ... 
.... alt • 
I .nu. 
Figure I~. Shakedown BiaxIal Stress Cy~l. At the Center Of the 
Hot Spot 
88 
. ...
I 
. ...


Multiaxial Cyclic Thermoplasticity Analysis with Besseling's Subvolume Method

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