boundaries N. Tada, T. Kitamura, R. Ohtani 606-07, · Relationship between creep damage parameters...

Relationship between creep damage parameters

and distribution of creep cavities on grain

boundaries

N. Tada, T. Kitamura, R. OhtaniM?c Wzz'ca,

606-07,

Abstract

Several kinds of creep damage parameters, such as area fraction of cavities,fraction of cavities on grain boundary lines, areal cavity density, and A-parameter, have been proposed and examined for the lifetime assessment ofhigh-temperature components. In this study, physical meaning of damageparameters is analyzed on the basis of a probabilistic model of creep cavitieswhere spherical cavities with the same radius are randomly distributed on everygrain boundary plane nearly perpendicular to the stress axis direction.

1 Introduction

Under creep dominant conditions, cavities initiate along grain boundaries andtheir linkage forms intergranular cracks the size of which is about one grainboundary. Then, these intergranular cracks begin to grow and coalesce witheach other, and a large crack brings about the final failure of the material.Therefore, it is crucial to detect and to evaluate the distribution of cavities forassessing the creep damage of material at the early stage.

Several creep damage parameters, such as area fraction of cavities, fractionof cavities on grain boundary lines, areal cavity density, and A-parameter, havebeen proposed and used to evaluate the creep damage; e.g. Ejima et al.[l],Shamas[2], Murakami et al.[3], Kadoya & Goto[4], Miura & Ogata[5].However, their physical meaning is still unknown because cavities are usuallydistributed on grain boundary planes which are randomly distributed in threedimensions, while the damage parameters are evaluated from a two-dimensional observation of cavities at a surface or on a cross-section.

In this study, four kinds of creep damage parameters are investigated on the

Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533

536 Localized Damage

sound grainboundary lines

Figure 1: Creep damage parameters; (a) Area fraction of cavities, (b) Fractionof cavities on grain boundary lines, (c) Areal cavity density, (d) A-parameter.

basis of a probabilistic model of distributed creep cavities.

2 Definition of creep damage parameters

Figure 1 schematically shows creep damage parameters discussed in this paper.All of them are evaluated from the two-dimensional observation of creepcavities on a cross-section. Definition of the damage parameters is given asfollows.

(1) Area fraction of cavities, P: fraction of the cavitated area.(2) Fraction of cavities on grain boundary lines, L: fraction of the total

length of cavities, Y 2c., to the length of each grain boundary line, 21.

(3) Areal cavity density, M: the number of cavities in a unit area.(4) A-parameter, A: number fraction of cavitated grain boundary lines (i.e.

grain boundary lines with one or more than one cavities) to the total number ofgrain boundary lines. A-parameter focuses only on the normal grain boundarieswhich are defined as those intersected by a straight line parallel to the stress (orprincipal stress) axis direction as shown in Fig.l(d).


Localized Damage 537

3 Formulation of creep damage parameters

3.1 A model of creep cavities

The distribution of creep cavities is modeled as shown in Fig.2.(1) All grain boundary planes are circular of the same radius, r, and they are

randomly distributed in three dimensions.(2) Cavities exist only on the grain boundary planes nearly perpendicular to

the stress axis direction. In other words, angles of cavitated grain boundaryplanes, which are defined as the angles against the plane perpendicular to thestress axis as shown in Fig.3, are less than a critical angle, 0̂ .

(3) N cavities exist on every normal grain boundary plane, and cavities arerandomly distributed on it.

(4) All cavities are spherical with the same radius, a, and the radius a ismuch smaller than that of grain boundary planes, r.

It should be noted that a grain boundary line is clearly distinct from a grainboundary plane in this paper. Circular planes which are distributed inside thematerial are called 'grain boundary planes,' and the intersected lines on a cross-section are called 'grain boundary lines,' as shown in Fig.2.

3.2 Distribution of angles of grain boundary planes

3.2.1 All grain boundary planesDensity function of angles of all grain boundary planes is given by

Figure 2: A model of creep cavities distributed on grain boundary planes.



Grain boundary plane

Figure 3: An angle of a grain boundary plane.

sin0 (1)

The above equation is easily derived on the assumption that the directions ofgrain boundary planes (i.e. normal vectors of the planes) are randomlydistributed in three dimensions. As it is assumed that creep cavities initiate onlyon the grain boundary planes the angles of which are less than 0̂ , the

proportion of cavitated grain boundary planes, #,, is given by

(2)

where F(0) is the distribution function of /(0).

3.2.2 Grain boundary planes revealed on cross-sectionDensity function of angles of the grain boundary planes which are revealed on across-section is approximately given by

g(0)s--sin0. (3)

when 0 is small comparing with jt/2; Tada et al.[6]. The proportion of cavitatedgrain boundary planes, p̂ is calculated from

(-<?(*.))

(4)

where G(d) is the distribution function of g(0).



3.2 3 Grain boundary planes intersected by inspection lineConsider an inspection line parallel to the stress axis direction. Densityfunction of angles of the grain boundary planes which are intersected by aninspection line is given by; Tada et al.[6]

h(6) = sin20. (Q*e*Jt/2). (5)

In the same manner, the proportion of cavitated grain boundary planes, #, is

given by

-sin*0<, (6)

using the distribution function, //(#).

33 Formulation of each creep damage parameter

3 J.I Area fraction of cavities on cross-sectionIt is proved by geometrical analysis that area fraction of cavities on a cross-section, P, shown in Fig.l(a), is equal to volume fraction of cavities. P is thenformulated as

(7)

where /̂ is volumetric density of grain boundary planes (i.e. the number ofgrain boundary planes in a unit volume). As is easily understood, the right sideof eqn(7) represents the total volume of cavities in a unit volume.

33.2 Fraction of cavities on grain boundary linesSince the fraction of total cavity length to the length of each grain boundaryline fluctuates on every grain boundary line, the fraction is evaluated by itsaveraged value. Then, the fraction of cavities on grain boundary lines, L,shown in Fig.l(b) is formulated as

"{7)'

where [ ]^ in the equation above means to take an average over all grain

boundary lines. L is approximately equal to the mean fraction of cavitated area



to the total area on each grain boundary plane, N-(a/r) •#, because #. is

approximately equal to #,. Thus, L corresponds to the normalized damagedarea which is usually adopted as a damage parameter in Continuum DamageMechanics (CDM).

3.3.3 Areal cavity densityAreal cavity density, M, is given by

(9)

which is derived from the relationship between the number of inner defects inthe material and the number of defects which can be observed on (i.e. cut by) across-section; Tada et al[7]. Eqn(9) points out that the value of M does notdirectly represent the actual number of cavities in the material because M isproportional not only to N but also to the cavity size, a.

33.4 A-parameterAs A-parameter, A, is defined as the number fraction of cavitated grainboundary lines to all grain boundary lines intersected by an inspection line, Ais given by

(f r-i_2.r^Jo^ jt-r •Pi (10)

(11)

where x is a distance between the center of a grain boundary plane and a cross-section, as shown in Fig.4. Eqn(lO) is constructed by the product of two

Cutting

Grainboundaryplane

Cavities

Figure 4: A grain boundary plane with N spherical cavities.



probabilities; the probability that the grain boundary line is made by a normalgrain boundary plane (i.e. a cavitated grain boundary plane) which is expressedby PI, and the one that at least one cavity out of N is revealed on the grainboundary line which is given by the other parts preceding /?, in eqn(10). It canbe found from eqn(ll) that A-parameter does not have a clear physicalmeaning.

4 Discussion

4.1 Dependence of parameters on radius of creep cavities

From eqns (7) to (11), area fraction of cavities, f, fraction of cavities on grainboundary lines, L, and areal cavity density, M, are in proportion to the third,second and first powers of 0, respectively. Regarding the A-parameter, thevalue of A increases in proportion to a when a is small. As a becomes large,the value of A converges on /?,.

4.2 Dependence of parameters on the number of creep cavities on eachgrain boundary plane

Every damage parameter increases in proportion to the value of N except thatA-parameter becomes constant when N is large.

43 Dependence of parameters on grain size

In this analysis, it is assumed that the radius of cavities, a, and the number ofcavities in a unit area on each grain boundary plane, N / nr*, are constant.These assumptions mean that the distribution of creep cavities on grainboundary planes is independent of the grain size, r. Moreover, volumetric

Table 1 Characteristics of creep damage parameters.

Cavity radius a

Number of cavities on grain boundaryplane #

Radius of grain boundary plane r(Grain size)

Physical meaning

Area fraction ofcavitiesP

OCg3

oc#

al/r

Volume fractionof cavities

Fraction ofcavities on grainboundary lines

Loca'

oc#

const.

Area fraction ofcavities on grainboundary planes

Areal cavitydensityM

oca

octf

ocl/r

(unknown)

A-parameterA

oca -» const.

oc# -» const.

ocr-» const.

(unknown)

It is assumed that a and N/nr* are constant.



density of grain boundary planes, n̂ , is in inversely proportion to the thirdpower of r. Consequently, the values of P and M are in inversely proportionto the radius of grain boundary planes, r, while fraction of cavities on grainboundary lines, L, is independent of the grain size. The A-parameter, A,increases in proportion to r when r is small while it converges on the value ofPi as r becomes large.

5 Conclusions

Four kinds of creep damage parameters, area fraction of cavities, P, fraction ofcavities on grain boundary lines, 1, areal cavity density, M, and A-parameter,A, are examined on the basis of a model of distributed cavities. In the model,grain boundary planes are assumed to be circular with the same radius, r, andN cavities with the same radius, a, exist at random locations on every grainboundary plane which is perpendicular to the stress axis. Dependence of eachdamage parameter on the distribution of cavities is clarified and the physicalmeaning of parameters is discussed. The results are tabulated in Table 1.

References

1. Ejima, T., Zhow, W., Ohtani, R., Kitamura, T & Tada, N. Initiation ofinner small crack in high-temperature creep-fatigue in Type 304 stainlesssteel, Proceedings of the 32nd Symposium on Strength of Materials atHigh Temperatures, pp.94-98, The Society of Materials Science, Japan,1994.

2. Shamas, M.S. Predicting the remanent life oflCrl/2Mo coarse-grainedheat affected zone material by quantitative cavitation measurements,CEGB Report, TPRD/IV3199/R87,1987.

3. Murakami, S., Liu, Y. & Sugita, Y. Interrelation between damagevariables of continuum damage mechanics and metallographicalparameters in creep damage, International}our nal of Damage Mechanics,1992,1,172-190.

4. Kadoya, Y. & Goto, T. Creep damage evaluation based on damagemechanics of Cr-Mo-V steel forging, Journal of the Society of MaterialsScience, Japan, 1992, 41,1736-1742.

5. Miura, N. & Ogata, T. Study on highly accurate evaluation of creep voidarea fraction rate by use of image processing technique, Transactions ofJSME, 1993, A-59, 2466-2471.

6. Tada, N., Kitamura, T. & Ohtani, R. Probabilistic analysis of therelationships between the distribution of creep cavities and creep damageparameters, Transactions of JSME, 1993, A-59, 2074-2080.

7. Tada, N., Ohtani, R., Kitamura, T. & Yamada, M. Inverse analysis ofdistribution of internal small defects, JSME International Journal, 1994,A-37, 450-455.


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