1. INTRODUCTION The long time duration of structures at elevated temperatures is governed by creep deformations. To design structures for an expected life of several years, it appears crucial to unde rst and how fra cture is gove rne d by pla sti c fl ow under nomina l stresses and temper atures tha t are mai nta ine d cons tant. In the ear ly fif tie s, cre ep experi ments on metallic and polymeric materials have demonstrated the complex dependence of fracture mec hanism wit h res pec t to the appl ied str ess. The experi mental obs ervati ons are summarized in Figure 1 drawn from the work of arson and !iller "1#$%&. ' thin bar ofstainless steel at elevated temperature is sub(ected to a constant tensile force. Figure 1 shows the dependence between the nominal stress and the time to rupture in log)log scale. *imilar results are given in the case of brittle low alloy steel at $++ o in the workof -ichard "1#$$& "reported by d/vist, 1#01&. Two linear branches are observed in these diagrams, with two markedly different slopes. The time to failure is strongly stress sensitive on the upper branch while a smaller stress sensitivity is observed for the lowerbranch. The change of slope in these log)log diagrams ap pears to be a salient feature ofthe cr eep experi ments. In addi ti on, two di ff erent modes of fr acture are obs er ved depending on the stress level "d/vist, 1#01& 1
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The long time duration of structures at elevated temperatures is governed by creepdeformations. To design structures for an expected life of several years, it appears crucial
to understand how fracture is governed by plastic flow under nominal stresses and
temperatures that are maintained constant. In the early fifties, creep experiments on
metallic and polymeric materials have demonstrated the complex dependence of fracture
mechanism with respect to the applied stress. The experimental observations are
summarized in Figure 1 drawn from the work of arson and !iller "1#$%&. ' thin bar of
stainless steel at elevated temperature is sub(ected to a constant tensile force. Figure 1
shows the dependence between the nominal stress and the time to rupture in log)log
scale. *imilar results are given in the case of brittle low alloy steel at $++ o in the work
of -ichard "1#$$& "reported by d/vist, 1#01&. Two linear branches are observed in these
diagrams, with two markedly different slopes. The time to failure is strongly stress
sensitive on the upper branch while a smaller stress sensitivity is observed for the lower
branch. The change of slope in these log)log diagrams appears to be a salient feature of
the creep experiments. In addition, two different modes of fracture are observed
large pressures, fracture of the tube by development of an aneurysm is observed "ductile
failure&. n the contrary, for low pressures the tube fails by slow crack growth with no
appearance of an aneurysm "3brittle4 failure&.
1.1 Initial and secondary creep
The common ground of all dislocation creep theories is the knowledge that the
material is hardened with the deformation and is softened with time "while heated&. These
two procedures take place simultaneously and define the strain. This idea was first
formulated by 6ailey and rowan and was successively developed by a large number of
scientists. 't a high temperature, usually about the one)third of the absolute melting
temperature, the dislocations ac/uire a new degree of freedom. xcept for gliding, there
is also climbing, and therefore the dislocations are not obliged to move only on their slip
planes. These results in the gradual release of dislocations previously created with the
strain. The structure of the dislocations is sub(ected to the so)called recovery. This means
that if a dislocation is held by an obstacle, the recovery procedure will release it, allow it
to slide down to the next obstacle, the glide step of the dislocation being responsible for
almost the total strain. The mechanisms which are based on this succession of glide climb
of dislocations are referred to as hardening recovery mechanisms. The characteristic
difference which distinguishes these mechanisms from those of plastic flow under lower
temperatures "which mechanisms may also be thermally activated& is that the procedure,
at an atomic level, is rather the diffusive movement of the atomic voids towards or from
the dislocation, which glides more, than the gliding of the dislocation as a whole . The
modern unified way of describing the gliding phenomena and of dislocation hardening
and recovery, is the theory of a dislocations network and of an internal back stress.
In accordance with the experimental observations, it has been assumed that the
dislocations are arranged to form a network. The creep procedure consists of continuousevents of recovery and hardening. The coherence of the network is ensured by the
repulsive and attractive forces among the dislocations. 's a result of the applied stress
and of the thermal fluctuation, some of the dislocations will escape from the network and
will slide a certain distance down to the point where they encounter some obstacle
"dislocation, second)phase particle, etc.&. 7uring their movement the strain and hardening
increase because the dislocations sub(ected to stress, increase their length and, therefore,
their density. The recovery procedure takes place simultaneously. The force for the creep
of the dislocations results from the linear stress of the dislocations which have been bent
by the obstacles. 'fter the climbing, the procedure is repeated. 't the beginning of the
loading "initial creep&, many of the dislocations loosely connected to the network will
move, and therefore the creep rate will, initially, be very high. 6ut the number of easily
escaping dislocations decreases, gradually, with time. This results in the fact that the
creep strain increases with a diminishing rate. 't this stage, the hardening is dominant
and the dislocation density increases. 8owever, the recovery trend increases with the
increase in the dislocation density, therefore, the recovery rate increases and finally, a
situation results where the two procedures balance one another while the dislocationdensity and the creep rate remain constant "secondary creep&. The time re/uired for a
dislocation to overcome an obstacle has to do with the flow of atomic voids at (ogs of its
length. 9hat is characteristic is that the higher the stress and the temperature, the faster is
the climb. The time re/uired for a dislocation to slide depends on the relationship
between the applied stress and the stress coming from the dislocations network "elastic
dislocation field&. 7issolved atoms "foreign atoms& which are attracted to the elastic
dislocation field and which attempt to be diffused while the dislocation slides create a
friction force with the lattice and can decelerate its movement. In a material that was
strengthened with second)phase particles and the ratio of the particles volume with
respect the total volume is large enough, the opposing stress consists of a component that
is due to the particles. The model of a dislocation sliding on the sliding plane is
represented by a load sliding on a plane, r: is the friction stress in the lattice during the
movement of the dislocation because of foreign "interstitial& atoms. is the internal
stress of particles coming from the elastic field of the distributed second)phase particles.
The stress that is applied on the dislocation and that is due to the network of the rest
dislocations, is called the internal back stress and is represented by the symbol . In
general, it can be assumed that when the atomic size of the interstitial atoms is not
considerably different from that of the lattice atoms. 9hile is taken into account only
the internal oxidation "mainly at the boundary grain& and the following creation of a
crack. <nder relatively medium stresses and high temperatures the metallic materials are
broken with a relatively low ductility. The decrease is due to the intercrystalline
development of cavities. Isolated cavities have been tracked at the second stage of creep
and, in some cases, at the first stage. 't the later stage, the
=oids start to become unified at the sides of the grain boundaries, forming small cracks
"micro cracks&. The unification of the micro cracks leads to the characteristic fibrous -
porous surface of the intercrystalline fracture. The general cause of the transition from the
intercrystalline failure "under low temperatures& to the intercrystalline failure is that the
atomic voids are rendered agile under high temperatures. The atomic voids that are
dispersed at the boundaries of grains can be concentrated to form cavity nuclei.
!oreover, the grain boundaries are active sources of voids so that they supply thecreation of cavities. In the alloys that usually contain second)phase particles at the limits
of grains, the cavities are, usually, created at the place of the particles. 5umerous studies
have shown that in the low alloyed steels, the austenitic steels as well as the nickel super
alloys, the cavities are related to carbides, as well as to sulphides, silicates and oxide
existing in the main material. The development of creep damage can be expressed in
terms of two mechanisms. The one mechanism is the creation of cavities and provides a
measure for the rate at which the number of voids increases and the other mechanism has
to do with the development of cavities and provides a measure of the magnification of
cavities with time. The rate of creation of voids is represented as and is measured
with the number of cavities created per unit of time on the unit area of the grain4s
boundary. et be the rate of creation of cavities in a time Therefore, in the time
interval the number of new cavities created is In the later time, t , these
cavities will be magnified and the assumed rate of development of the cross)sectional
area of the cavities created in time will
The total area, S , of the voids in time is therefore
where S + is the total cross)section. The above formulation is general but shows that, in
order to define a measure of damage, the effect of both functions M and 9 should
necessarily be known.
-abotnov and ?achanov simplifying the analysis at this point, considered the
ratio as a phenomenological "state& variable, and used the condition as
the failure condition. They assumed that the following relation is valid
>>>>>>>>>>>>>>>.."1.%&where k is a constant. !oreover, similar relations have been proposed for the
development of damage .It is important to emphasize that, using the effect of damage on
the tertiary creep, it is possible to measure the /uantity In fact, in other studies, amethod of measuring is suggested. !oreover, /uation is in perfect accordance with
the experimental data. Therefore, the variable concept is a fully defined macroscopic
The metallic materials creep behavior has been described and a complete model is
presented. The basic constitutive e/uation, as well as the structure parameters, has been
derived from a mathematical analysis that represents the dominant physical procedures
and mechanisms. The model is very general because it is referred to all stages of creep
and describes the creep behavior of all metallic materials, including those strengthened
by a dispersion of second)phase particles. ' creep function has been derived from the
constitutive e/uation describing all three stages of creep under constant loading. The
function has the minimum possible number of fitting, parameters. The dependence of the
fitting parameters on the loading conditions has been described using very simple
mathematical relations. 'pplications and predictions have been carried out in a widerange of metallic materials. @ood agreement has been shown by a comparison made also
between the creep curves determined experimentally, and those obtained from creep
function and determined fitting parameters. A1B
ife time and failure modes are predicted for metallic bars sustaining tensile
creep. xperimental results show that a ductile or a 3brittle4 mode of fracture occurs
depending respectively on whether the nominal applied stress is large or small. The
analysis is based on a modeling of void nucleation and growth in which damage
evolution is controlled by two mechanisms of plastic flow in the matrix material. Fracture
is supposed to occur when the porosity attains a critical value which depends on the mode
of fracture considered. xperimental results are explained and described in terms of the
proposed model. A%B
From the e/uations of motion within the field theory of defects, creep curves are
derived and a relationship between the applied stress and the time to rupture under
different deformation conditions is obtained. The creep duration as a function of the
applied stress and the initial strain rate, as well as the ultimate strain, specifying the
The basic e/uations of the problem are presented in this section. The material is
assumed to be porous and elastic)viscoplastic. 6ecause elastic effects will be neglected inthis work, the total strain rate is e/ual to the plastic strain rate d dp. Gorosity is
accounted for by using the formulation of Hu and 5eedleman "1##1&. They worked along
the lines developed by @urson "1#00& to analyze ductile fracture by void nucleation and
growth. The generalization of @urson4s approach to a rate)dependent matrix material was
made by Gan et al. "1# C&. There is no yield function in their approach. The viscoplastic
strain rate dp is given by the flow rule in terms of a viscoplastic potential J.
>>>>>>>.>>>>>>>>>>>."%.1&
9here is a positive scalar determined later. The viscoplastic potential has the
form
>>>>."%.%&
9here the mean stress and the effective stress are defined in terms of the
macroscopic auchy stress, and auchy stress deviator as
>>>.."%.C&
/uation "%& defines the effective matrix stress the porosity f is introduced In the
potential through the function f *(f) as proposed by Tvergaard and 5eedleman "1# D&
to model the loss of stress carrying capacity due to void coalescence;
The effect of porosity is excluded in this section by keeping f M + through all the
calculations. 9e (ust have to solve /uations "%.0& and "C.D&. volution of elasticdeformations during the process is neglected as stated in the beginning of the paper and
values of the material parameters are those of Table I. This analysis is first made for an
ideal specimen with uniform cross)section. The solution of this problem will be
designated as the fundamental uniform solution. In the next section, the development of
heterogeneous deformations due to an initial cross)section defect is analyzed. The
nominal stress is defined by
>>>>>>>>>>>>>>>"C.1&9here S + is the initial cross)section
Figure D.1. 5umerical simulations of the evolution of the axial strain with respect totime, in a creep test on a cylindrical bar.
fracture can be observed with grain boundary cavitations. This is a brittle damage
mechanism. Two different values of the critical porosity f D and f B are attributed to the
ductile and brittle failure respectively. In I F, voids are located at the grain boundaries.
Therefore at a given value of the overall porosity, interaction between voids is enhanced,
compared to T F where voids are spread uniformly in the material. *omehow in the case
of I F, the notion of an effective local porosity R f at the level of grain boundaries can beintroduced. 7ue to the oncentration of voids within thin layers, the effective porosity f
is much larger than the overall porosity f. For that reason, fracture is initiated in the case
of I F, at values of the global porosity significantly smaller than for T F. Therefore it is
assumed that f D > f B.
In a first step, the cross)section is assumed uniform. Two different characteristic
times are introduced. 's before, t + is the time needed for a uniform bar to have its cross)
section reduced to zero2 t R denotes the time for a uniform bar to develop the critical value
of the porosity f M f D, resp. f M f B, associated to a given fracture mechanism "ductile
fracture or T F, resp. I F&. In the following, the time to rupture t R is calculated in terms
of the nominal stress . The initial porosity is f + M +:+D.
model. This point is demonstrated in 'ppendix '. 9e can remark that the lower branch
of the curve in Figure is affected by the value of f R while the upper branch remains the
same for the two values of f R considered. 9e have to consider now that the value of f R
depends on the fracture mode. 9e have chosen f R M f D M +.1 "ductile fracture& and f R M f B
M +.+E "brittle fracture&.
(.2 )ail-re &ode Analysis
*o far, the discussion has dealt with the idealistic case where the flow remains
homogeneous along the bar. This case is referred to as the fundamental homogeneous
solution. The actual response observed in experiments is different. 7uctile fracture is
accompanied by the development of an inhomogeneous deformation or necking. 's presented in the Introduction, mechanical tests indicate that for large values of the
nominal stress , failure is preceded and accompanied by neck development "ductile
failure&. n the other hand, when is made smaller, the importance of necking is
reduced "sometimes a neck is hardly observed&, while damage plays a ma(or role in the
rupture process. 7amage growth triggers rupture by crack initiation and crack
propagation. These experimental observations can be interpreted using features related to
the homogeneous fundamental solution.
et us consider a bar with a cross)section defect. 7enote by S A and S B the largest
and smallest cross)section area. In Figure #, an example is shown of a model with two
zones ' and 6, the cross)section area being uniform in each zone. For a large value of ,
the material flow is governed by the small strain rate sensitivity m1, as discussed before.
'lternately, for small values of the dominant mechanism is controlled by the large)
strain rate sensitivity m%. It is known that plastic instability and neck development is
favored by a small)strain rate sensitivity "here m1&. This is schematically illustrated in
Figure # where the evolution of the axial deformation "zone 6& in terms of "zone
'& is reported for two values of the strain rate sensitivity " m1 < m %&. 't the beginning of
the process, the material flow is stable and the deformations and are almost
identical. ocalization of the deformation occurs later in zone 62 increases to large
values, while saturates at some critical value . It is known "8utchinson and 5eale,
1#00& that for a given geometrical defect depends strongly on the value of the strain
rate sensitivity. is smaller for a process controlled by the small strain rate sensitivity
"m1& than for a process governed by the large strain rate sensitivity " m%&, . The schematicevolution of the deformation in the thinner zone 6, in terms of the deformation in
zone ', is shown in Figure #, for two values of the nominal stress "continuous line&
and "dashed line&, such that >> . Therefore, we have
>>>>>>>>>>>>.."D.1&*train localization for the small stress occurs later than for the large stress .
Figure $.D. *chematic representation of strain localization in a cylindrical bar with a twozone model.
Two levels of the applied stress are considered with >> ...
7epending on whether the stress or is applied, mechanisms I or II of the plastic
flow are respectively activated. 5ote that the localization strain in zone ' depends on
the strain rate sensitivities of the deformation mechanisms.
et us denote by the strain at which the critical value fR of the porosity is
reached. It can be checked by numerical calculation that is weakly dependent on the
strain rate sensitivity. This can be related to the fact that the evolution of porosity is
driven by plastic deformation.
The value of fR depends on the fracture mode considered; ductile fracture fR M fD,
2 brittle fracture fR M fB , . The values of and
shown in Figure # correspond to the onset of rupture. It appears that for the small stress
, rupture occurs before strain localization. The damage failure mode is favored here
because of the large strain rate sensitivity m% which refrains strain localization and neck
formation. n the other hand, for , strain localization is initiated much earlier, due to
the small strain rate sensitivity m1. 5ot enough time is left for the porosity to reach the
critical value fR M fD before neck development. -ather, rupture is triggered during
necking. This description, although only /ualitative, is consistent with experiments.
Ruantitative investigations could be performed with a three)dimensional finite elementnumerical model. 8owever, the main results and trends are certainly described by the
. CREE# DURATION ANAL$ I IN TER& O) T%E )IELDT%EOR$
The operation of e/uipment under severe conditions "high stress andtemperature& has resulted in the discovery of the creep effect and the development of
creep theory. ngineers have centered on creep analysis, i.e., on the evaluation of the
time period within which the strain reaches the ultimate value. This problem remains of
practical importance nowadays. 't its incipient stage, creep theory was developed as an
engineering science. ater, it evolved into a branch of continuum mechanics.
*imultaneously, physical mechanisms responsible for the creeping effect were studied.
6ecause of their complexity, comprehensive physical description of creep is lacking.
*ome progress in creep physics has been achieved by invoking the concepts of the
dislocation theory. ' number of dislocation models describing different creep stages and
conditions have been constructed. It is argued that, at moderate temperatures, elementary
creep events in solids are attributed primarily to the motion of dislocations. Therefore,
creep mechanisms will be considered in terms of the field theory, which involves the
dynamics of translational defect. 5ote that the field theory of defects deals with defect
ensembles and, according to, describes a system on the mesoscopic scale. In contrast, the
classical dislocation theory has to do with individual defects and their interactions andthus, implies microscopic methods of description.
The dynamic e/uations in the field theory of defects have the form,
>>>>>>>..>>>>.."$.1&
From these e/uations, the e/uation relating the tensor of the dislocation flux density I to
8ere, is the dislocation density tensor, V is the elastic displacement rate, is the
material density, is the viscosity, B and S are the theoretical constants, and is the?ronecker delta. The symbols "H& and " & stand for the vector and scalar products,
respectively, and"H.& designates the vector product with respect for the first subscripts of
the dyad and the scalar product with respect for the second ones. /uation "%.%&, which is
helpful in studying the creep process, is written for the uniform distribution of defects,
i.e., for space)independent field strengths and I . It is believed that these assumptions
valid near the yield point, where defects are distributed randomly and do not form spatial
structures. ' great body of experimental data on the creep effect has been obtained from
tensile tests of rods. Therefore, in our previous study, we considered uniaxial
deformation, for which /. "%.%&, written in the dimensionless variables,
, and becomes,
>>>>>>>>.>>>>."$.C&9here v is the plastic strain rate
The special attention has been given to the analysis of the functions v "t&, which specifies
the
creep curves under constant stress. In the following, we will carefully
investigate the relationship between the applied stress and the time to rupture of a system.
It has been found that there are two creep modes depending on the applied stress *; stable
at S U 1V% and unstable atS W 1V%. The corresponding expressions for creep rates are
Detection o, li,e o, s tructural materials used in sodium cooled fastreactors (SFRs)
Creep dissipation energ concepts to predict creep rupture
life!"
5otations
M Koung4s !odulus in 5 Vmm%
M <niaxial e/uivalent creep strain rate "Vhr&
M <niaxial e/uivalent deviatoric stress or !ises e/uivalent stress "5Vmm%&
M !ises e/uivalent stress at time3t4 "5Vmm%&
tM Total time "hr&
', m, n !aterial parameters
*tructural materials used in sodium cooled fast reactors "*F-s& shall have good
high temperature low cycle fatigue and creep properties, ade/uate weldability to fabricate
large size components and shall be compatible with the li/uid sodium environment in
service. 'ustenitic stainless steels have been the natural choice for structural components
of *F-s worldwide. The creep design life of *F- component is very long and is of the
order of D+ years. These calls for robust creep life rediction models to convert short and
medium term laboratory rupture data to design life. This paper discusses the application
of creep dissipation energy concepts to predict creep rupture life of four nitrogen alloyed
grades of C1E 5 **. 'ustenitic stainless steels have been chosen world wide as the
structural material for high temperature components of *F-s. The choice has been basedon their good high temperature mechanical properties, compatibility with the coolant
sodium and ade/uate weldability. The material chosen for present study was C1E" & 5
stainless steel with different nitrogen contents "+.+0 9t. S 5, +.11 9t. S 5, +.1D 9t. S
5 and +.%% 9t. S 5&. The creep design life of *F- components is very long and is of the
1"+.+0 S 5& $. D10 H 1+N)1D D.$E+0 )+.C1$%%"+.11 S 5& $.#0% H 1+N)10 $.0E+E )+.CC$1C"+.1D S 5& $. # H 1+N)10 $.0$+1 )+.C%$1D"+.%%S 5& $.$D0 H 1+N)%1 0.1D1 )+.C$$
Table E.1; reep constants for C1E" & 5 **
reep dissipation energy can be found using the formula;
>>>.>>>>>>>>>>>>>>> "E.10&
9here ^ o is the 'pplied stress, Zcr is found from the graph for experimental value of time
versus strain, = is total volume M '. and3t4 is the experimental creep life.
reep dissipation energy for C1E" & ** with different nitrogen content "+.+0S 5, +.11S
5, +.1DS 5 and +.%%S 5& have been tabulated in Table D. These values are used to find
out simulated creep life from the creep dissipation curve generated from analysis result.
reep dissipation energy for different specimens was calculated from the experimental
data using /. "D.1&. reep dissipation energy vs time has been generated for differentcases and creep life in terms no. of hours were given in Table D.