The Development of Validation Methodology and its Further Application
to Multi-axial Creep Damage Constitutive Equations for 0.5Cr0.5Mo0.25V
Ferritic Steel at 590°C
Q. X u
Department of Mechanical Engineering, School of Engineering The University of Huddersfield, England
A B S T R A C T
Research progress on the development of validation methodology for multi-axial creep damage constitutive equations and its further application to 0.5Cr0.5Mo0.25V ferritic steel at 590°C are presented including the fundamental consistency requirements, the two formulations, validation methodology, results and conclusions. This further reveals significant difference of creep curves between different set of constitutive equation and reveals the need to examine fully the coupling between damage and creep deformation.
I N T R O D U C T I O N
Material damage can have disastrous consequences on the human environment and examples range from the Liberty ships during World War II, the two Comet jet aircrafts in the mid 1950s, numerous bridges in Europe, Australia and USA, to nuclear reactors (Fermi, Three Mile Island and Cerobyl) [1 ].
Creep damage phenomena in material and structural members have been studied in recent years. Creep damage mechanics, originated by Kachanov in 1958, has been developed with the objective of dealing with creep damage problem in engineering design [1-7]. It is evident that significant progresses have been made in various aspects including theory (phenomenological approach or unified irreversible thermodynamics formulation, anisotropic damage theory), applications, experimental observation and experimental methodology, and even optimisation [7]. However, as it is rapidly developing, it is particularly important to distil the information to assess and refine the generic theory and/or methodology to ensure its general applicability.
The phenomenological approach can be broadly classified into weak coupling and strong coupling between damage and deformation. In case of weak coupling the effect of material damage in elastic properties is disregarded and a coupling is established by introducing the damage variables into the constitutive equation with the concept of effective states variables.
Within the weak coupling approach, a set of creep damage constitutive equations for uni-axial tension is generalised for multi-axial case. The success of this approach relies on the development of a set of appropriate constitutive equations capable to depicting the observed multi-axial material behaviour. This capability needs to be adequately assessed. That is the concern of this paper.
0.5Cr0.5Mo0.25V ferritic steel is an important material widely used in UK power generation stations. A set of conventional creep damage constitutive equations has been developed and been used in component design [8-13]. However, a critical review revealed its deficiency inherent from its generalisation method, namely: this method used lifetime (under plane stress condition) only and ignored creep deformation consistency [ 14]. Thus its general applicability is questionable. In fact, other researchers have reported a discrepancy between the predicted and experimentally observed damage development for a notched bar, though on different material [15]. Subsequently, a new set of creep damage constitutive equations for this material at 590°C was proposed in which two functions of states of stress were incorporated to better depict the damage influence on creep deformation and creep rupture [ 16]. For this set of constitutive equation, preliminary validation results can be found in [ 17]. Recently, a validation methodology was proposed [ 18].
This paper summarises research progress on the validation aspect including 1) the fundamental consistency requirements, 2) the two formulations, 3) validation methodology, 4) result, and 5) conclusion. It does not only demonstrate the capability of this new set of constitutive equations, but also reveals another deficiency in previous constitutive equation that was not discovered from previous validation. Thus, the author addresses that in the rapidly developing damage mechanics field, it is important and necessary to distil the information and conclusions cautiously prior to developing,
SMiRT 16, Washington DC, August 2001 Paper # 2042
accepting, and applying any theory and constitutive equation, either new or old (including this one), which supports a point addressed by Kemple [ 19].
T W O R E Q U I R M E N T S A N D F O R M U L A T I O N S
The two fundamental requirements for a set of creep damage constitutive equations to be developed are creep strain rate consistency and damage evolution consistency [14]. This means that the predicted creep strain rate and damage evolution by a set of constitutive equations under various states of stress and loading conditions should be consistent with the experimental observations. Therefore, any creep damage constitutive equation developed should be assessed through validation. It is important to include a wider range of stress states (multi-axil) and loading condition (proportional and non- proportional) in this validation process, which has been addressed by Kemple [19].
Uni-axial Form
The uni-axial form creep damage constitutive equations for 0.5Cr0.5Mo0.25V ferritic steel is written as [8]:
~o(1-H)
= Asinh( (1 - ~X 1 - m)) (1)
I~I = h ( 1 - ~ * * ) ~ c (2)
Ko
, = - - ~ - 0 - , ) 4 (3)
cb = C~ * (4)
where A, B, C, h, H* and IQ are material parameters. H represents the strain hardening that occurs during primary creep; initially H is zero and as strain is accumulated, increases to the value of H*. cI) describes the evolution of the spacing of the carbide precipitates which is known to lead a progressive loss in the creep resistance of particle hardened alloys as ferritic steels, o3 represents intergranular caviation damage and varies from zero, for the material in virgin state, to o3f = 1/3, when
all the grain boundaries normal to the applied stress have completely caviatated [8], at which time the material is considered to have failed.
K R H multi-axil Formulation
The multi-axial KRH f6rm is written as [8-9]:
• 3 S i j Asinh( B°e (1- H ) , (5)
g i j - ' 2 0 , e ~ ~ ~-~'- "g))
121= h (1-~**){: e (6)
13" e
Kc 4 (7)
(b - C g e * (o'1/O'e ) u (8)
where <> is heavy step function and v is stress state index defining the multi-axial stress rupture criterion. The adopted function of states of stress ( O l / o e ) ~ was originally proposed by Cane according to Perrin [8]. A value of 2.8 for v was obtained through calibration of isochronous rupture loci in plane stress condition and a couple of multi-axial test [8]. A length discussion on its determination is also reported in [8]. For completeness the material constants for this material at 590°C are given below [9]:
A - 2.1618x10 9 MPa h 1 B - 0.20524 MPa -I
C=1.8537 h = 2.4326x10 s MPa
H* = 0.5929 Kc = 9.2273X10SMPa 3 h -1
o = 2 . 8
New Formulation
The new form for constitutive equations is given as [ 16]"
3Sij
Asinh( B~e (1- H)~ij--2(~e ~ --~-~ _ O) d )) (9)
t:/= h (1- H ) ~
(10)
°-7
K c(~ ___~_ ~_ ¢)4 (11)
= CN~e"
f2
(12)(13)
iO d = iO" f l
where f~ and f2 are functions of stress states. The function f2 is introduced to depict the effect of states of stress on the damage evolution, which was employed in previous formulation. The additional function fl is introduced to better phenomenologically couple tertiary deformation, creep damage and creep rupture. The physical implication is that creep deformation and creep rupture are two different processes and two separate internal variables are needed [ 16].
Specific Form of fl and
f2
The Huddlestone's formulation was chosen for fl given as [20]:
f ~ ~,-~ ) exp b S s
-1 (14)where
S s
= ~ / ~ + ~ + ~ and S 1 = ~ 1 - ~ m 'The Spindler's relation of strain at failure is adopted for function
f2
[21 ]:f2
exp
p 1 ~1 1 3~ m (15)= - +q 2 2o e
which equals to 1 uni-axial condition and coincides with Rice & Tracey's equation if p = 0 and q = 1. Details for this equation can be found in [21 ]. Substituting equations (15-16) into equations (13-14) gives:
3Sij
Asinh( B~e (1- H) )) (16)~ij -- 2t~e 0 __ (~ )0 __ 0) d
I~I= h (1- H ) ~
ere ~ e (17)
Kc
(~ = - T - O - , ) 4 (18)
E ?)/-1
d~ Ci~ e exp
p 1 ~1 1 3~ m (19)=
-
+q 2 2o" e
d~ =d~*C2~ei"explbl gem
-11} (20)d
~3S1)
L [- Ss
The constants a and b should be determined by the experimentally determined isochronous rupture loci.
V A L I D A T I O N M E T H O D O L O G Y [18]
With clear understanding of the two fundamental consistency requirements addressed above, it is clear that an adequate validation should be designed and conducted considering:
1) The items: a) creep strain rate; and b) damage evolution;
2) The stress states: a) creep curves under uni-axial conditions; b) multi-axial stress states under proportional loading conditions; and c) multi-axial states of stress under non-proportional loading conditions.
O,)--Of
multi (21)
t = t f = ~nulti
where o8 is the critical value of damage and E is effective creep strain. These results will be used in validation. The following validation methodology is proposed here:
1) To check isochronous rupture loci under plane stress and plane strain (proportional) loading conditions; 2) To check strain at failure under plane stress and plane strain (proportional loading) conditions;
3) To check typical creep curves under plane stress and plane strain (proportional loading) conditions;
4) To check the damage development, creep strain developmem, strain at failure and lifetime for multi-axial stress states complex (or non-proportional) loading condition. One way to achieve that is by notched bar test.
In this approach, the plane stress and plane strain stress states (proportional loading) are selected to present a wider range of stress states (under proportional loading). It is suggested that all the material constants should be determined in the first two Steps. The Step 3 intends to further check the coupling of damage and creep strain. The Step 4 validates the constitutive equations under multi-axial non-proportional loading conditions. It is clear that previous practice is not adequate as it ignored the need to include strain at failure (under plane stress conditions) and did not consider plane stress conditions [14]. Further more, it also did not use damage developmem information. There was a reported discrepancy between predicted and observed damage development [ 15]. In this paper will presem validation results on the first three accounts.
R E S U L T S
These two sets of multi-axil creep damage constitutive equations are validated in terms of lifetime, ratios of strain at failure, and creep curves, which corresponds mainly to the first three Steps. Typical results are presented here, while extra results can be found in [ 18].
The isochronous rupture loci and ratios of strain at failure for previous formulation are presented in Fig.1 and Fig.2, respectively, while typical results for new formulation are presented in Fig.3 to Fig.4. Typical results of creep curves and damage evolution under plane stress condition (proportional loading) are shown in Fig.5 and Fig.6. Fig.7 gives a comparison of creep curves under pure share condition for both formulations.
The following observations can be drawn:
1) A significant creep strength increase under plane strain condition when the tri-axility is about the order of 1.5 to 2.8 as shown in Fig.1. This increase is not realistic according well-known creep strength theory. Thus, this formulation is unable to find a value for stress sensitivity that can satisfy the isochronous rupture loci under plane stress and plane strain conditions simultaneously. This deficiency was not revealed in previous constitutive equation development and/or validation. The difficulty of determining the value for stress sensitivity reported in [8] needs re-examined.
2) Further more, the ratios of strain at failure for previous formulation shown in Fig.2 are conjugated with the shape of isochronous rupture loci shown in Fig.1 through the common stress sensitivity parameter o. Thus, there is no freedom provided to produce strain at failure consistent with experimental observation. This ft~her demonstrates the need for modification of previous formulation though it has been addressed in [ 14, 16].
3) A wide range of ratios of strain at failure can be predicted by new formulation independent of the shape of isochronous rupture loci, as shown in Fig.3. In fact the explicit function of ratio of strain at failure is an empirical one consistent with experimental observation [21 ].
4) For a given ratio of strain at failure a wider rage of isochronous rupture loci can be produced, as shown in Fig.4. The isochronous rupture loci under plane stress condition, shown in Fig.4, are very close to experimental observations complied in [22]. However, there are not adequate results for plane strain conditions.
C O N C L U S I O N S
The overall conclusions can be made:
1) The proposed new validation methodology addressed the need to validate both creep strain rate and damage development under a wider rage of stress states (multi-axial) and loading (proportional and non-proportional) loading conditions.
2) Application of this validation method to previous constitutive equation revealed the impossibility to obtain a value for stress sensitivity simultaneously satisfying plane stress and plane strain conditions.
3) The new formulation is better constructed offering the capability to produce consistent results in lifetime and strain at failure under selected stress states under proportional loading conditions. On contrast, previous formulation was not properly constructed, as the strain at failure was not considered properly suffering an impossibility to obtain an appropriate value for stress sensitivity t~ even just considering lifetime only under plane stress condition and plane strain condition. No substantial experimental results of lifetime are available for plane strain condition.
4) There are not experimental results available for validate the coupling of damage and creep deformation under multi-axil stress states.
5) Further validation should include non-proportional loading under uni-axial and multi-axial loading conditions. It is anticipated that further research should investigate, in detail, the coupling of damage on creep deformation and rupture under complex stress states from which the accuracy and applicability of this new approach will be established or what improvement/modification is needed.
R E F E R E N C E S
(1) Krajcinovic, D. 'Damage mechanics', 1996 (Elsevier, The Netherlands)
(2) Kachanov, L. M. 'Introduction to continuum damage mechanic', 1986 (Martinus Nijhoff, Dordrecht) (1986). (3) Lemaitre, J. and Chaboche, J. L. 'Mechanics of solid materials', 1990 (Cambridge University Press).
(4) Lemaitre, J. 'A course on damage mechanics', 1992, (Springer-Verlag, Berlin)
(5) Boehler, J. P. (Eds.) 'Yielding, damage, and failure of anisotropic solid', 1990 (Mechanical Engineering Publication Ltd., London)
(6) Altenbach, H & Skrzypek, J (Eds). Creep and Damage Processes in Materials and Structures, Springer, New York (7) Skrzypek, J & Ganczarski, A. 'Modeling of material damage and failure of structures: theory and applications', Springer (8) Perrin, I.J. and Hayhurst, D.R. (1996). J. Strain Analysis, 31(4), 299
(9) Hayhurst, D.R. and Miller, D.R. (1997). In: IMechch Seminar: Remanent Life Prediction, IMech Publication, London (10) Hayhurst, D.R. and Xu, Q. "Summary of the preliminary results obtained from damage XX on crotch section
idealisation of branch vessel with internal pressure 4MPa", August, 1997, Research Report, Department of Mechanical Engineering, UMIST, Manchester, M60 1QD, U.K.
(11) Hayhurst, D.R. and Xu, Q. "Summary of the preliminary results obtained from damage XX on crotch section idealisation of branch vessel with internal pressure 4MPa: Enhancement", December, 1997, Research Report for a HSE funded project, Department of Mechanical Engineering, UMIST, Manchester, M60 1QD, U.K.
(12) Hayhurst, D.R. and Xu, Q. "Summary of the preliminary results obtained from damage XX on flank section idealisation of branch vessel with internal pressure 4MPa", October, 1997, Research Report for a HSE funded project, Department of Mechanical Engineering, UMIST, Manchester, M60 1QD, U.K.
(13) Hayhurst, D.R. and Xu, Q. "Summary of the preliminary results obtained from damage XX on crotch section idealisation of branch vessel with internal pressure 4MPa: Enhancement", December, 1997, Research Report for a HSE funded project, Department of Mechanical Engineering, UMIST, Manchester, M60 1QD, U.K.
(14) Xu, Q. "The development of constitutive equations for creep damage behaviour under multi-axial states of stress", in EDS. SELVADURAI, A.P.S. and BREBBIA, C.A. Damage and Fracture Mechanics VI, WIT Press, 2000.
(15) Kobayashi, K.K., Imada, H. and Majama, T. "Nucleation and growth of creep voids in circumferentially notched specimens", JSME Series A, 1998, 41,(2), 218-224.
(16) Xu, Q. "The development of creep damage constitutive equations under multi-axial states of stress", European Conference on Advances in Mechanical Behaviour, Plasticity and Damage, 7-9 November 2000, Tours, France. (17) Xu, Q. "The validation creep damage constitutive equations for multi-axial states of stress for 0.5Cr0.5Mo0.25V
stainless steel at 590°C '', submitted to Journal for publication.
(19) Kremp, E. 'Creep-Plasticity Interaction', in Altenbach, H & Skrzypek, J (Eds). Creep and Damage Processes in Materials and Structures, Springer, New York
(20) Huddleston. R.L. 'Assessment of an improved multi-axial strength theory based on creep-rupture data for type 316 stainless steel', JPress. Vessel Tech. 1993, 115, 177-183.
(21) Spindler, M.W., Hales, R. and Ainsworth, R.A., Multiaxial Creep-Fatigue Rules, IAEA/IWGFR Technical Committee Meeting on "Creep Fatigue Damage Rules to be Used in Fast Reactor Design", Altrincham UK, 11-13. June 1996, published by IAEA, as IAEA-TECDOC-933, ISSN 1011-4289, 1997
7
,.•,/01234
,/i /, . . . -1 -0.5 0 0.5 1 1.5Normalized stress
] , 1 2 3 4
/ . y ,
. . . Z , ~ , , . . .
-1 -0.5 0 0.5 1 1.5 2 2.5
Normalized stress
Fig.1 Isochronous rupture loci for previous formulation
(left for plane stress condition and right for plane strain condition)
lu
, L
9
8
!
0-1 -0.5 0 0.5 1
Stress Ratio
. . . I
0 1 2 3 4 -1
Fig.2 Ratios of strain at failure for previous formulation
(left for plane stress condition and right for plane strain condition)
5
4
-1 -0.5
~,..,..~=,,===~
0 0.5
Stress Ratio
P & q 2.5, 0 2.5, 0.25 2.5, 0.5 2.5, 0.75 2.5,1
2.5, 0.1 \ N N ~ \ 2.5, 0.25
2.5, 0.5 2.5, 0.75 2.5,1 ,,^,
0 0.5 1
Stress Ratio
Fig.3
Fig.4
Ratio of strain at failure for new formulation
(left for plane stress condition and right for plane strain condition)
.5
L5 I
0
).5 ~
-1 . . .
-1 -0.5 0 0.5
Normalized stress
-1 1.5
. . .
-0.5 0 0.5 1 Normalized stress
Isochronous rupture loci for new formulation with p = 2.5, q = 1, and a = 2 (left for plane stress condition and right for plane strain condition)
1.2
¢ - .
• ~ 0.8
~.. 0.6
0 0.4
0.2 A
B
C °
° o J
J
i
Time
Fig.5
0
g
(
4
~ A
3 ~ B
C
2
.1
0
!
I
J
0.00E+00 1.00E+04 2.00E+04 3.00E+04 4.00E+04 0.00E+00 1.00E+04 2.00E+04 3.00E+04 4.00E+04
Time
1.2
The creep curves and damage evolution under plane stress for new formulation (A: pure shear; B" uni-axial tension; and C: equal bi-axial tension)
¢ -
• ~ 0.8
~. 0.6
0 0.4
0.2
0
0.00E+00 1.00E+04 2.00E+04 3.00E+04 4.00E+04
Time
0.4
0.3
g
0.2
0.1
0.00E+00 1.00E+04 2.00E+04 3.00E+04 4.00E+04
Time
Fig.6 The creep curves and damage evolution under plane stress foe previous formulation. (G: pure shear; H: uni-axial tension; and I: equal bi-axial tension)
1.2
1
• ~ 0.8
~. 0.6
0 0.4
0.2
0 0.00
~ G
m A
1
A
"+00 1.00E+04 2.00E+04 3.00E+04 4.00E-','04
Time
