Mechanical seminar

7/21/2019 Mechanical seminar

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1. INTRODUCTION

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

depending on the stress level " d/vist, 1#01&

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Figure 1. xperimental results of arson and !iller "1#$%& concerning a cylindrical bar under Tensile creep at elevated temperatures.

The nominal applied stress is represented in terms of the time to rupture in a log)

log diagram. The material is a stainless steel. -upture time is in hours.

"a& ' ductile mode of failure is associated to the large stresses on the upper branch2 then,

fracture occurs by necking accompanied in general by an important damage.

"b& ' 3brittle4 failure mode corresponds to the small values of the stress on the lower

branch. 5o necking develops2 rather the fracture is initiated by crack propagation

preceded by an important damage development. The denomination 3brittle4 used in the

literature may be considered as misleading. The failure is actually accompanied by a

growth of damage which is controlled generally by plastic flow. *imilar observations are

reported from creep tests on polymeric materials. For example, the work of 6arton and

herry "1#0#&, experimental results are obtained concerning the time to failure of a

polyethylene pipe sustaining constant internal pressures2 see also -ichard "1#$#&.

7ifferent failure modes are again observed depending on which branch is considered. For

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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

3



"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

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in metallic materials heavily hardened with particles. In order for strain to take place, the

dislocation has to glide under the applied external stress which is herein represented by

I , glide will not take place for a limited period of time until the internal back stress

is self ad(usted to new values. The above formulation does not include existing inelastic

phenomena related to the grain boundary. 6ut as shown in a previous study, these

phenomena can be neglected. 't the microscopic scale, the hardening is controlled by the

rate at which the dislocations approach each other, while the recovery takes place through

rearrangement "and counterbalancing& of the dislocations. Therefore, the applied stress is

composed of two terms; the mean internal back stress, which is related to the recovery

hardening e/uilibrium, and the mean effective stress that is responsible for the glide. The

concept of the mean internal back stress "and, respectively, of the mean effective stress&

has been proven to be a very important phenomenological structure parameter because it

is directly related to the dominant procedure of dislocations movement.

1.2 Tertiary creep

The creep acceleration at the third stage is due to the creation and (oining of

cavities at the boundaries of grains, that is, the fracture in creep is, generally,

intercrystalline The cavities may be created at the beginning of creep, even in the first

stage. Initially, their effect on the strain rate is negligible but, as their number and size

increase, this effect becomes definitive. The accelerating rate of creep can also be caused

by the collapse of the materials microstructure. !any metallic materials contain second)

phase particles that act as obstacles in the movement of dislocations and improve the

strength to creep. <nder a straining over a long period of time the possible growth of the

larger of these particles and the disappearance of the smaller, leads to increasing rates of

creep. 6ut 7yson and !c ean showed that, even in nickel super alloys "with a large

number of particles&, the tertiary creep cannot be explained by the development of

particles. The main part of the accelerating rate of creep should be attributed to the

cavities of the grain boundary. 'nother possible cause "not definite& for the accelerating

rates of the third stage of creep is the corrosion on or below the surface, as, for example,

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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

>>>>>>>>>>>"1.1&

6



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

structure parameter.

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2. LITERATURE REVIEW

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

material rupture, are found. ACB

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*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. This call for robust creep life rediction models to convert short and

medium term laboratory rupture data to design life. This seminar discusses the

application of creep dissipation energy concepts to predict creep rupture life of four

nitrogen alloyed grades of C1E 5 **. ADB

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3. CON TITUTIVE E!UATION

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;

>>>>>"%.D&

10



9here differs from f, when the porosity f exceeds a certain critical value

fc. f * is e/ual to when in that case the material suffers a complete loss of

stress)carrying capacity as can be seen from /uation "%.%& where

The positive factor in the flow rule "1& is determined from

the e/uivalence between the macroscopic dissipation and the dissipation in the matrix

material "see Hu and 5eedleman, 1##1&;

>>>>>..>>>>>...>>>"%.$&

9here is the matrix effective plastic strain rate defined by

>>>>>>>>.>.>>>>>.>>..."%.E&

To complete the set of e/uations, the viscoplastic flow of the matrix remains to be given.

For metals at elevated temperatures, and for the level of stresses explored here, two basic

deformation mechanisms will be considered;

"I& thermally activated dislocation glide at large stresses, and

"II& 7iffusion controlled dislocation creep at lower stresses.

To account for these two mechanisms, a phenomenological formulation of the !atrix)material

flow law is used;

>>>>>.."%.0&In this relationship the effective plastic strain rate in the matrix is a function of the

effective matrix stress . Two strain rate sensitivity parameters m1 and m% are introduced.

They are respectively related to mechanisms I and II discussed above. 'n important fact

is that the strain rate sensitivities associated with these mechanisms differ by an order of

magnitude. !echanism I, associated with thermally activated dislocation glide, is

assumed to be sensitive to strain hardening. The strain dependence of the material flow is

11



described by the function introduced in /uation "%.0&, with being the cumulated

plastic strain obtained by time integration of. The following form is taken;

>>>>>>>>>>>>>>>>.

"%. &where is a reference stress, is the strain hardening exponent and is

the Koung modulus.

The evolution of the porosity f is due to void nucleation and growth "see Hu and

5eedleman, 1##1&;

>>>>>>>>>>>>>>>."%.#&

9here

>. >>>>>>>>>>>>.>."%.1+&9ith

>>>"%.11&

Table %.1. =alues of the material parameters typical of an alloy steel at $++ degree elsius

12





taken as m1 M +:+% andm% M +:%$. For the parametric analysis, variations with respect to

the values given in Table I will be considered. The stress vs. strain)rate relationship "%.0&

for the matrix material is illustrated, in Figure C, using values of Table I. The effective

matrix stress is represented in terms of on a logarithmic scale. Figure C shows the

activation of mechanisms I and II, respectively, for large and small values of the stress

5ote the slopes m1 and m% corresponding to these two mechanisms. 5ote the two inear

branches corresponding to mechanisms I and II of viscoplastic flow with strain rate *ensitivity

m1 and m%, respectively.

Figure C.1. *tress)strain rate response of the matrix material "log)log diagram&.

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". CREE# ANAL$ I W%EN DA&A'E I NE'LECTED

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.

15



8ere porosity is not taken into account; f M+. =alues of the material parameters

are defined in Table I. These values are typical of steels at elevated temperatures. For the

applied nominal stress , it appears that the early stages of the process are

dominated by mechanism II of diffusion controlled dislocation creep. 't the later stages,

mechanism I of thermally activated dislocation glide is activated, leading to an explosive

growth of the deformation. The dashed line corresponds to a simulation where

mechanism I is not considered.

The evolution with respect to time of the axial deformation is shown in Figure D

"continuous line& for . 6y we designate the time at which an

explosive growth of the deformation is observed. <ntil times close to M 1 .E H 1+NE

s, the plastic flow appears to be governed by mechanism II " m% M +.%$&. This point is

made clear in Figure D by considering the evolution of , when mechanism I is neglected

" 2 dashed line&. 9hile mechanism I appears to be negligible for

most times t < , it becomes important near and leads finally to the explosion of .

Thus the critical time seems to be governed in general by the interplay of both

mechanisms I and II. These results can be related to the contribution of the two terms of

the constitutive flow law. *ince f M +, it follows from that . Then from "E& we

have where is defined. Thus the flow law has the form

>>>>>>"C.%&

to 6ecause f M +, the material is incompressible, and therefore the cross)section area is

given by

S = S O exp. ") &. The true stress is then related to the nominal stress by

It , leads to

>>>>."C.C&

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To evaluate the respective contributions of the two mechanisms controlling the material

flow, the ratio of the two last terms of /uation "%D& is considered;

>>>>>>>.."C.D&7ue to the condition n1 > n % "m1 < m %&, the evolution of in terms of has a maximum

. For large values of vanishes. If , mechanism II is

dominant in some range of deformation. 8owever, because ,

mechanism I shall eventually overcome mechanism II as observed in the calculations

reported in Figure D. 'n interesting point concerns the role of the nominal stress. 9hen

is increased, is made smaller, meaning that the mechanism I is more prominent.

'ssume now that is large enough so that . Then the contribution of

mechanism II is always negligible. n the other hand, when is small enough,

mechanism II is dominant during the process. ' relationship between time and

deformation can easily be obtained from /uation "C.$&. 'fter separation of variables and

integration, it yields

>>>>..."C.$&The time t O as defined in Figure D, for which S P +, is obtained by making in

"C.$&.

et us now consider the case of large values of . Then from the preceding discussion,

the contribution of mechanism II can be neglected, and we obtain from "C.E&.

>>>>>>>>>>>>>> ."C.E&

9ith,

17



>>>>>>>"C.0&

Figure D.%. 5ominal stress vs. critical time t o.

This time corresponds to the moment where an explosive growth of the

deformation is observed in the fundamental homogeneous solution. 5ote that the two

slopes "Qm1 M Q+:+%& and "Qm% M Q+:%$& are associated to mechanisms I and II of the

plastic flow. Therefore, a linear relationship, with slope Q m1, exists between ln and ln

t O.

>>>>>>>>>>>>>>>."C. &

For small values of , a similar relationship is obtained with m1 replaced by m%.

These results have to be put in perspective with experimental data where the same type of

18





(. DA&A'E AND )AILURE &ODE ANAL$I

(.1. Da*a+e In,l-ence

The influence of damage on fracture is now examined. Gorosity evolution is taken

into account by /uation "C.1&. avities may interact when their size becomes large or

when their number is multiplied. 's a result, cracks are initiated and propagate across the

section, leading to failure. ' simple criterion is used here to account for the interaction

between cavities. 9e shall consider that fracture is initiated when a critical value f R of the

porosity is reached.

7ifferent values of f R will be introduced, depending on the fracture mode that is activated.

't large stresses, fracture is the result of void nucleation and growth within grains

"ductile fracture or transgranular creep fracture&. 't lower stresses, intergranular creep

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.

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In Figure $.1, we report the evolution with respect to time of the porosity and of

the cross)section area for an ideal uniform bar. ' small nominal stress is applied, M

1++ !Ga. 'n explosive growth of the porosity and a drastic reduction of the cross)

section area are reached at the end of the process.

Figure $.1. volution of; "a& porosity "b& cross)section area2 with respect to time for anominal stress M 1++ !Ga.

21



5ote that the critical porosity fR= +.+E, at which rupture is initiated, is reached at the

time tR. In the present case, this time is smaller than the time t + for which the cross)

section area vanishes

Figure $.%. ffect of the nominal stress on the time to rupture tR and on t +.

5ote that tR = t + when is large, and that tR < t + when is small. The transition

from the deformation mechanism II to mechanism I is indicated by arrows.

For discussion, a given value f R = +.+E is considered. It is of interest to note that

this critical value of the porosity "indicative of the onset of rupture& is attained before the

vanishing of the cross section area " t R < t +&. 's discussed in the previous section, the

process is controlled by mechanism II, providing that the nominal stress is small enough.

In fact no change in the result is found when calculations are made by switching off

22



mechanism I " M +&. The protuberance that appears in the curve of Figure D.Ca at the

time t M x 1+N0 s, results from the nucleation of voids activated at the strain

introduced in the relationship.

7ifferent values of the nominal stress are considered in Figure 0, where the

evolution of porosity with respect to time is shown. The values chosen for the critical

porosity are f D M +.1 for ductile fracture at large stresses "here > C++ !Ga& and f B M

+.+E for brittle fracture at low stresses " < C++ !Ga&. It is found, as discussed in *ection

D, that for increasing values of , the deformation mechanism I is taking a more

prominent role. In the early stage, the process is controlled by the deformation

mechanism II. 't a certain moment L marked by an arrow in Figure 0 L mechanism I is

activated. The absence of an arrow for the small stress M 1++ !Ga, means that the

process is entirely controlled by mechanism II. The porosity at which the mechanism I is

activated, is larger when is decreased "see arrows in Figure 0&. 's a conse/uence, the

time difference t +Q t R tends to zero when the nominal stress is increased. Indeed for M

D++ !Ga, we have t R Mt + since fP f R and S P + simultaneously.

23



Figure $.C . Influence of the damage evolution on the time to rupture

5ote that by choosing smaller values of f B, the deformation at fracture in the

brittle mode would be reduced. In that way, brittle fracture at strains less than $S

observed in certain engineering alloys would be described. In Figure , the nominal stress

is given in terms of the time to rupture on a logarithmic scale. The continuous line

refers to the situation where damage evolution is disregarded. Then, the time to rupture is

considered as being the time at which the section is reduced to zero " t rup M t+&. The case

where damage evolution is accounted for corresponds to the dashed lines. Two values of

the critical porosity are considered f R M +.1 and f R M +.+E. The time to rupture is now the

moment where the porosity reaches the value f R "t rup Mt R&. Two different slopes are

associated with the mechanisms of deformation I and II. 'ctually, scaling laws similar to

"%. & are still valid for t + and t R when the evolution of porosity is controlled by the @urson

24



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

25



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 >> ...

26



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

simple model presented here.

27



Figure $.$ . Influence of the damage evolution on the time to rupture

28



. 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

the applied stress was obtained;

29



>>>>>>>>>>.>"$.%&

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

>>>>>>>>>>>."$.D&at S < 1V%,

>>>>>>>>>>>"$.$&

30



at S > 1V%,

Then, the creep curves are given by,

>>>>>>>>>.."$.E&9here,

is the initial creep rate, and p, q= are the steady)state rates given by

d v Vd τ M +.

The critical applied stress S M 1V% corresponds to the bifurcation point where the

creep behavior changes. 9ith this parameter introduced, one can define the stable creep

limit

σ M η ² V% B, which depends on the material parameters. xpressions "%.D&L"%.0& yield the

condition

>>>>>>>>>..>>>."$.0&

From which the creep life of a real system at S WS is obtained. <nder condition "%. &,

creep rate "%.$& becomes infinite and the time to rupture is given by

>>>>>>>>.>"$. &

urves plotted in Fig. 1 relate the time to rupture of the system to the applied stress at S W

S and v + M "1& +.$ and " & +.#.

'ccording to, when S U S , solutions of /. "$.0& are strongly dependent on the

initial value of v+. The range of v+ can be divided into the intervals + U v + U q, q U v+ U p,

and v + W p. 't v + W p, the creep rate becomes infinite when the denominator of "D&

vanishes2 hence, the time to rupture is

>>>>>>>>..>>>>>.."$.#&

xpression "%.1+& is illustrated in Fig. 1 by curves ! and " for v + M 1.E and 1.#,

respectively. If the initial creep rate lies in the range q U v+ U p, v "X& P q at X PY2 i.e.,

the stationary creep mode is observed. In this situation, creep duration analysis turns to

31



the conventional problem "see the first paragraph&. If the critical strain Z at which the

material fails is known, /. "$.#& with X PY yields the relation of interest between the

stress and the time to rupture ;

This dependence is demonstrated in Fig. % for Z M " 1& +.E$ and " & +.C$. In both

cases, the initial strain Z + is taken to be e/ual to +.+1S. This value is in /ualitative

agreement with Figs. 1 and C. The latter depicts experimental data for a number of

materials tested at various temperatures. The test duration was up to 1++ days. urves 1

and " were obtained for carbon steel "+.$S and +.%DS & at temperatures of C++ and

DC%[ , respectively2 curve , for nickel steel at D++[ 2 curves ! and #, for high)alloy

nickelLchromium steel at E++ and 0++[ , respectively2 $ and 11, for high)speed steel at

$#C and 0C%[ , respectively2 % , for cast heat resistant steel at ++[ 2 &, for lead at room

32



temperature2 ' , for nickelLcopper alloy at E++[ 2 and "%. 1 &, for magnesium alloy at

1$+[ . Following the conclusions made .we leave without consideration the initial range

+ U v+ U q, where attempts to describe the early portion of the creep curve with decreasing

rate have failed.

33



7. CASE STUDY

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

34



order of D+ years. *tructural analysis under creep conditions re/uires reliable constitutive

models which reflect time dependent creep deformation. In the present investigation, an

attempt has been made to predict creep curves and creep rupture life on the basis of creep

dissipation energy. The 6ailey)5orton form of creep law is derived from the experimental

data from uniaxial creep tests for C1E " & 5 stainless steel with different nitrogen

contents. ' strain hardening form of creep law is used in '6'R<* F ! code to evaluate

the e/uivalent creep constants.

The data4s for C1E" & 5 ** with +.+0S 9t. S 5, +.11S 9t. S 5, +.1D 9t. S 5

and +.%% 9t. S 5 at 10$ !Ga and %++ !Ga at #%C ? from the uniaxial creep test are

fitted to the 6ailey)5orton form of creep law and e/uivalent strain hardening '6'R<*

creep constants were evaluated accordingly .

The 6ailey L 5orton law,

>>>>>>>>>>>>>>>>>>>.

"E.1&

Taking log on both the side of the e/uation

>>>>>>>>>>>>>.."E.%&

/uation "%.%& can be compared to the standard plane e/uation of

>>>>>>>>>>..................>>>."E.C&

The best fit plane for the above mentioned plane is found using the least L s/uare method.

The e/uation of plane can be written as follows.

\ "x, y2 a, b, c& M c ] ax ] by>>>>>.>>>>>>>.>."E.D&

!inimizing the sum of the s/uares of the vertical distances between the data and the

plane and which is called as

For extreme values the gradients must be zero,

35



>>>>>>>>>>

"E.$&

*olving the above set of linear e/uation we get the values of the constants a, b and c

which are e/uivalent to the values of p, / and log "?& respectively of 6ailey)5orton

e/uation. The value of ? is obtained by taking antilog "exponential&.

7.# Deri$ation of e%ui$alent creep constants of strain &ardening!"

*train hardening creep is modeled using '6'R<* software. The available time

hardening creep subroutine in '6'R<* can be used for simulating strain hardening

creep by deriving its e/uivalent constants. The '6'R<* time hardening is calculated

using the formula,

>>>>>>>>>>>>>..>>>>>>>>"E.E&

Integrating /. "%.1C& w.r.to3t4, creep strain can be written as

>>>>>>>>>>>..>>>>>>>.."E.0&From the above /. "%.1D&.,.

>>>>>>>>>.>>>>>>>>>"E. &

36



*ubstituting3t4 in /. "%.1C& we get the '6'R<* e/uivalent strain hardening form of

power law and is given by AD)$B

>>>>>>>>..>>>>>>> "E.#&

The 6ailey L 5orton law given in /. "%.1&

>>>>>>>>>>>>>>>>>>>>> "E.1+&

7ifferentiating /. "%.1& with respect to3t4 we get

>>>>>>>..>>>..>>>>>>>..>> "E.11&

*olving for3t4 from /. "%.1&

>>>>>>>>>>>>>.>>>>> >.. "E.1%&

*ubstituting /. "C.D& in /. "C.C& we get

>>>>..>>>>.>>>>.>>>> "E.1C&

omparing /. "C.$& with the e/uivalent strain hardening form used in '6'R<*) /.

"1E& we get

>>>>>>>>>>>>>>>>> "E.1D&

>>>>>..>.>>>>>>>>>>>>>> "E.1$&

>>>>>>>>>>>..>>>>>>>>. "E.1E&

The creep constants for C1E " & 5 ** for the /. "%.1C& or for /. "C.%& are given in Table

1 for different loads which were calculated from experimental data fitting in the /. 1+)

1%.

37



*pecimen. 5o ' 5 m'pplied load 10$ !Ga at #%C ?

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.

38



Fig. 1; reep dissipation energy for the specimen 1 "+.+0S 5& C1E" & 5 ** of load 10$!Ga

*pecimen

5o.

*tress"^&

!Ga

*train rate

"Ger hr&

=olume"mmNC& Time"hrs& reep

dissipation

nergy"5)

mm&1"+.+0 S 5& 10$ +.+++D$ C$DD.1+ 0C %+$#0D%"+.11 S 5& 10$ +.+++C1 C$DD.1+ 1D D % $C%$C"+.1D S 5& 10$ +.+++%% C$DD.1+ 1D%% 1#D+%#D"+.%%S 5& 10$ +.+++1E C$DD.1+ DD1% DC0 %D

Table D; xperimental results for C1E" & 5 **

RE ULT/0

39



8ence the simulation results can help to find out the creep life for different loads

using the less number of specimen data.

From the results, it shows that increase in creep life was found for C1E" & 5 **

by increasing the level of nitrogen content "/uantitatively estimated and tabulated for

different specimens of +.+0S 5, +.11S 5, +.1DS 5 and +.%%S 5&. omparing specimen

of +.%%S 5 with different loading condition of 10$ !Ga load is increased by 1.1D times

and respective creep life is decreased by +.01 times.

40



RE)ERENCE

1 _-eview !athematical description of the mechanical behavior of metallic materials

under creep conditions`, by 5. 7. 6atsoulas. ournal of material sciences C%"1##0& %$11 ) %$%0.

2 _ reep Failure 'nalysis of 6ars *ustaining onstant Tensile oads` by lf 'tochem,

erdato. !echanics of Time)7ependent !aterials D; $0L0#, %+++

3 _ reep 7uration 'nalysis in Terms of the Field Theory of 7efects` by 5. =. hertova

and Ku. =. @rinyaev. Technical Ghysics, =ol. DE, 5o. 0, %++1, pp. DDL DE.

" _ reep rupture life prediction based on creep dissipation energy analysis` by =ela

!urali1,!.7. !athew%, ?. 6hanu *ankara -ao%, =. @anesan%, *. -avi%, *.6askar1 and 5. 'ngara(1. =ol EC, issues %)C , 'pril) une %+1+ , pp EC$)EC#.

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