Carbon Materials for Advanced Technologies Episode 14 ppt

Although the design is based on a maximum principal stress failure criterion, the need for applying fracture mechanics to component strength assessment becomes obvious when one considers

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The difference between the work input and the elastic stored energy is the crack growth resistance, R,

As with the purely elastic case, the energy values associated with elastic-plastic

fracture may be ascertained fiom the load versus load-point deflection diagram for

a cracked body as shown in Fig 5

U Fig 5 Diagram of load versus load-point displacement for an elastic-plastic body

experiencing stable crack extension [48]

For an increment of crack extension, AA is represented by a movement from B to

D on the load versus deflection curve, the energy consumed may be represented by the area AnR Hence the crack growth resistance may be expressed as

and similarly

where cf+, = XJJdA is the plastic energy dissipation rate

The J value is defined as the elastic potential difference between the linear and

nonlinear elastic bodies with the same geometric variables [52,53] The elastic potential energy for a nonlinear elastic body is expressed by:

The J integral can be written as

The graphical evaluation of J, is presented in the load versus load point

displacement diagram of Fig 6

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P

U Fig 6 Diagram of load versus load-point displacement illustrating a graphical

determination of the J-integral[48]

Sakai et al [48] investigated the fracture of an isotropic nuclear graphte, IG-11, using the chevron notched short bar specimen and found it to exhibit significant elastic-plastic behavior Nonlinear fracture parameters y, R, J,, and @p were empirically determined from the load versus displacement diagrams These nonlinear fracture parameters were found to be increasing functions of aP, and to converge to a constant value of G, as @p - 0 The value of K, calculated using the Irwin relation, Eq 12, was only about one fourth the value measured using the

ASTM E399 test method This lfference was attributed to significant inelastic deformation which occurred during fracture It was estimated that approximately 38% of the total fracture energy was consumed as plastic energy dissipation

IG-11 using compact tension specimens and found crack growth resistance to increase with increasing crack extension for a range of d w values This rising R- curve behavior was attributed to inelastic deformation and fiacture in the process zone and crack wake region In the crack tip region, significant inelastic material response was identified with basal slip and extension of pre-existing microcracks Grain bridging tractions having a chctexistic dimension of one filler coke grain,

and compressive strains in the microcracking wake, contributed significantly to rising R-curve behavior

As noted in the above discussion on fracture mechanics characterization of graphites, EPFM has many advantages over LEFM in accounting for the nonlinear deformation and fracture behavior of graphite This is particularly true for laboratory specimens containing a single macroscopic, artificial flaw for which the

measurement of load-point displacements is rather straight forward Without a knowledge of local displacements, the work and energy terms used to calculate nonlinear fracture parameters can only be estimated The application of EPFM to graphite is further complicated by a number of factors Multiple flaws may act in unison as the critical flaw at fracture Furthermore, the critical flaw may have a size which approaches microstructural features with a location and orientation unknown Hence, for most fracture mechanics assessments of graphite, many investigators have employed a LEFM failure criterion as a workable solution to quantifjmg graphite fracture The authors have addressed critical issues in graphite fracture by exploring the lunits of LEFM while recognizing the nonlinear behavior

of this unique material

Graphite is used as a moderator and a structural material in the core of the Gas Turbine - Modular Helium Reactor (GT-MHR) Although the design is based on

a maximum principal stress failure criterion, the need for applying fracture mechanics to component strength assessment becomes obvious when one considers the possible presence of significant flaws in nuclear graphites An investigation was undertaken to determine the flaw sizes relevant to the fracture of graphite and

to assess the applicability of fracture mechanics to d e f i i g an appropriate failure criterion for small macroscopic flaws in two grades of nuclear graphite, H-45 1 and IG-11

5 I Material

In our fracture mechanics studies and fracture model development we have utilized several grades of graphite Three of the graphites examined are shown in Fig 7 Grade H-451 graphite is an extruded, medium-grained, near isotropic, nuclear

graphite which has been selected for use as the moderator and core structural material in the GT-MHR Grade H-45 1 was manufactured by Sigri-Great Lakes

Carbon Company in the USA In addition to macroscopic pores and cracks, H-45 i contains arrays of associated pores with trace lengths in the 1 to 5-mm range which are clearly visible on a machined surface of H-451 graphite Individual spherical

pores with diameters as large as 3 mm are common Grade IG- 110 graphite is an isostatically molded, fine-grained, isotropic, nuclear graphite which is used as the moderator and core structural material for the High Temperature Test Reactor (HTTR) which is currently under construction by the Japan Atomic Energy Research Institute Grade IG-110 graphite is manufactured by Toyo-Tanso in

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Japan and is a purified version of IG-11 Grade AXF-5Q is an isostatically molded,

ultrafine-grained, isotropic, high strength graphite Grade AXF-5Q is used for a variety of applications such as throats, nozzles, etc., in the aerospace industry, boats, crucibles and fumace parts in the semi-conductor industry and for other applications such as electrode discharge machining tools Grade AXF-5Q is manufactured in the USA by the POCO Graphite Company Grade AGX, (not shown in Fig 7) is manufactured by UCAR Carbon Company, and is used for the production of arc-furnace electrodes for the steel industry

The widely different textures of three of the graphites initially studied here are clearly shown in the photomicrographs in Fig 7 Grade H-45 1 [Fig 7(a)] contains relatively large filler coke particles, [F] in Fig 7(a), and pores, [PI in Fig 7(a) The microsiructure shown in Fig 7(a) reveals the presence of a large crack, [C] in Fig 7(a), that has propagated between pores in the graphite Microstmctural evidence

such as this supports the graphite fracture mechanism adopted in the Burchell

fracture model (section 6) In contrast to H-451, grade IG-110 pig 7@)] contains

much smaller filler coke particles, [F] in Fig 7(b) The particles are typically 10-

150 pm in length compared to the 0.5-1.5 mm filler particles found in H-451 Moreover, pores in IG-110, [PI in Fig 7(b), are considerably smaller than in H-

451, being in the range 10-250 pm Grade A m - 5 Q graphite [Fig 7(c)] has an extremely fine texture The filler particles are difficult to resolve in the photomicrograph Typical pore sizes are <10 pm and particle sizes appear to be

C 5 pm

5.2 Test procedure

Four-point bend specimens of square cross-section measuring 25 mm x 25 mm and

50 nun x 50 rnm were used in this investigation Specimen comers were chamferred to minimize failure initiation there A single artificial flaw was

machmed in the center of the tensile surface of most specimens The flaw geometry perpen&cular to the tensile axis was a circular section with a slot thickness of 0.25 mm and notch root angle of 45" Crack depth ranged from a = 0.025 to 10 mm and the surface length ranged from 2c = 0.55 to 22 mm The surface flaw geometry is shown in Fig 8 A number of specimens contained no artificial flaw, i.e., contained only intrinsic flaws Six specimens not containing artificial flaws were infiitrated under vacuum with a polyurethane bearing fluorescent dye to delineate surface connected porosity Four-point bend tests were conducted in strict accordance with ASTM standard C651-91 [55 1 All specimens were loaded in four-point bendmg to failure under displacement control The peak

load was measured and the fracture origin was noted as being at the artificial flaw

or away from the flaw Subsequent to testing, the flaw depth, a, and surface length, 2c, were precisely measured in the fracture plane using an optical comparator

Fig 7 Microstructures of the three primary graphites used in this work: (a) H-45 1, (b)

IG-11, and (c) AXF-5Q [F]-filler particles, [PI-pores and [C] cracks

crack tip geometry

Fig 8 Schematic of circular section flaws introduced in the tensile surface of IG-11 and

H-45 1 graphites

5.4 Results and discussion

Fracture at artificial flaws occurred by extending the plane of the flaw perpendicular to the tensile axis of the specimen A typical fracture surface for

specimens that failed at artificial flaws is shown in Fig 9 Failure away from

artificial flaws, i.e., at intrinsic flaws, always occurred between the loading points

in the region of constant tensile stress Typical fracture surfaces for failure at intrinsic flaws are shown in Fig 10 for an H-451 graphite specimen which had been infiltrated with fluorescent dye before testing Near surface flaws with maximum dimensions around 1 rnm appear to be the origins of failure Origins of failure at intrinsic flaws in IG-11 graphite could not be identified on fracture surfaces even for specimens treated with fluorescent dye

Test results for IG- 1 1 graphite are given in Fig 1 1 as a plot of fracture stress versus crack depth for specimens with and without artificial flaws At longer crack lengths, the fracture stress is proportional to the square root of crack depth The slope of fracture stress versus crack depth is approximately - % on the logarithmic

scale As the artificial flaw size is reduced, failure occurs at higher stress levels

until the fracture strength is equivalent to the unflawed specimens, i e., the mean

flexural strength At the transition crack depth, half of the specimens failed at the

artificial flaw and half failed at intrinsic flaws The transition crack depth is 0.050

mm for IG- 1 1 graphite It is notable that this value is comparable to the mean filler coke particle s u e for this graphite

versus crack depth for specimens with and without artificial flaws The transition from artificial flaws controlling fracture strength to intrinsic flaws controlling strength occurs at 1 mm for this graphite Although the mean filler coke particle size is around 0.5 111111, filler coke particles as large as 1 mm are common Here again, at the transition crack depth, half of the specimens failed at the artificial flaw and half failed at intrinsic flaws

The fracture stress for H-45 1 graphite exhibited greater variability than for IG- 1 1 graphite in all tests, regardless of whether failure occurred at or away from artificial flaws This greater variability may be attributed to a coarser microstructure for H-

45 1 graphite In the absence of artificial flaws, H-45 1 graphite presents a broader distribution of intrinsic flaw sizes from which failure may initiate When artificial flaws are large enough to control strength, the microstructure along the front of an incipient crack was more variable for H-451 graphite, thus offering more varied resistance to crack extension

4 , 5 m m ,

TENSILE SURFACE

big 9 Photosraph of fracture surface of H-451 graphite bend specimen illustrating

fulurc at artificial flaw

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,5 m m ,

TENSILE SURFACE

Fig 10 Photograph of fracture surface of H-451 graphite bend specimen illustrating

failure originating at natural flaws at the tensile surface

ED AT INTRINSIC FLA

CRACK DEPTH (mm) Fig 11 Fracture stress versus crack depth for small flaw fracture tests in IG-11 graphite

Irwin [23] developed an expression for the mode I stress intensity factor around an

elliptical crack embedded in an infinite elastic solid subjected to uniform tension The most general formulation is given by:

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factors to account for: the free surface - M, the flaw shape - H, and the relative flaw

size -S Under bending with the nominal outer-fiber stress ob, the stress intensity factor at the deepest point on the crack periphery is given by

and at the intersection of the crack with the surface

Empirical expressions for M, S, HI, and H2 based on fitting results of finite element analysis are given elsewhere [55, 561 The stress intensity factors, KA and KR, are

plotted in Figs 13 and 14 for small flaw tests on IG-I1 graphite and H-451 graphite, respectively Two regimes of behavior are observed for both graphites Fracture behavior at longer crack lengths, greater than 1 111111, is characterized by

a nearly constant value of stress intensity factor at failure The measured fracture toughness in this regime is approximately 1 O M P a 6 for IG-11 graphite and

1 2 M P a 6 for H-451 graphite The fracture condition is likely the larger of K,

or K, At smaller crack depths, fracture occurs at decreasing values of K, and consequently the fracture stress begins to fall below that expected from extrapolating the longer crack regime, as can be seen in Figs 11 and 12 Fracture

at intrinsic flaws is presumed to occur at flaw sizes smaller than or equal to the transition crack lengths and at stress intensity values equivalent to those shown in Figs 13 and 14 for an aMicial flaw of comparable dimensions

The value of K at fracture for IG-11 graphite decreased from 1 O M P a G to 0.5 M P a 6 as the artificial flaw depth decreased more than an order of magnitude from 1 mm to 50 pm The value of K at fracture for H-451 graphite decreased from 1.2MPafi to 0 8 M P a f i as the artificial flaw depth decreased to 1 mm, where intrinsic flaws comparable to this size are presumed to control strength This

observation is consistent with measurements of artificial flaw sizes for H-451 graphite specimens which had been infiltrated with flourescent dye Variability in

the maximum intrinsic flaw size results in a corresponding variable strength for

specimens without artificial flaws

The variability in fracture stress when small artificial flaws were controlling

strength was particularly pronounced for H-451 graphite, as can be seen in Fig 12

Here the crack dimensions and crack trip process zone dimensions are comparable

to the microstructural dimensions Consequently, local variations in microstructure

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present significant variation in crack growth resistance with attendant variation in fracture stress

The decrease in Knt with crack depth for fracture of IG-11 graphite presents an

interesting d i l e m The utility of fracture mechanics is that equivalent values of

K should represent an equivalent crack tip mechanical state and a singular critical

value of K should define the failure criterion Recall Eq 2 where K is defined as the first term of the series solution for the crack tip stress field, uy, normal to the crack plane It was noted that this solution must be modified at the crack tip and

at the far field The maximum value of oy should be limited to oms and that the far field stress should decrease only to the applied stress at increasing distance from the crack tip The nominal fracture stress for IG- 1 1 specimens with artificial flaws ranged from 28 to 100% of oms

The decreasing value of K with decreasing flaw size for fracture of IG- 1 1 graphite suggests that the near crack tip failure criterion would be better expressed by combining the near crack tip stress intensification estimated by K with the far field applied stress, Sa(Eq 28) Using simple superposition of the stress perpendicular

to the crack plane, it can be shown that ucnt is the largest principal component of the local stress in the highly constrained volume near the crack tip Here we may presume that onit is a material constant and defines the near crack tip failure criterion In applying Eq 28 to the IG-11 graphite fracture data of Fig 13, a value for r is found for which r,, = a constant such that r = r,, defines the critical crack

tip process zone dimension A simple iteration procedure was used to determine

that r = 90pm resulted in the minimum variance in ucrit The corresponding value

o f critical fracture stress,uCfit = 60.2 MPa (Std Dev = 2.45 MPa) was determined

for this data set Figure 15 shows onit ploted versus crack depth for IG- 1 1 graphte

data on the same scale as was used for Fig 1 1 Note that the fracture condition can now be defined by a single parameter It may seem physically inconsistent to define failure at a stress level nearly twice the nominal flexural strength of the bulk

material However, this high fracture stress may be related to the very small

volume of the near crack tip field in which the requirements for failure are met Graphite strength has been shown to be strongly volume dependent with small test section volumes exhibiting significantly higher strengths due to a lesser flaw content [57] Within the critical crack tip volume, strength is likely controlled by

a flaw content having a size regime sigtllficantly smaller than the artificial flaws

A 25 x 25 (FAILED AT ARTlFlCAL FLAW)

0 50 x 50 (FAILED AT ARTlFlCAL FLAW)

is that the crack tip inelastic zone was nearly twice that predicted by Eq 3, such that

If we take the nominal fracture toughness of IG-11 graphite to be 1 M P a F and the maximum stress in the process zone to be oCrit = 60.2 MPa according to the above analysis, we find that rim = 88 pm This value is virtually identical to r,, = 90pm, the process zone dimension determined using Eq 3 To summarize, the above analysis strongly supports a hypothesis that the maximum critical stress

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which defines the onset of failure for IG-11 graphite containing artificial flaws occurs at the boundary of the inelastic process zone The magnitude of this critical fracture stress is twice the nominal fracture stress due to the small stressed volume there

The experiments and analysis discussed above quantifies the fracture behavior of small flaws in two nuclear graphites for flaw sizes at and above intrinsic flaw dimensions The fracture behavior of graphite in the regime where intrinsic flaws control fracture and, therefore, the nominal tensile strength, is the topic of the next section In closing this discussion on the fracture behavior of small flaws, it is worthwhile reiterating the value of the stress versus crack length plot, as illustrated

in Fig 16, in assessing the failurehon-failure boundary for nuclear graphites It

is obvious that a single fracture criterion may not be adequate for the design of critical graphite components where there exists a significant probability of flaws greater than the transition dimension In this regime, the maximum principal stress failure criterion based on tensile strengths overestimates graphite strengths Consequently, care should be exercised when applying fracture mechanics below the transition crack length where the fracture toughness overestimates strengths

Fig 16 Fracture strength versus crack length diagram illustrating the failure / non-failure

boundary for fracture of a nuclear graphite

6.1 Modelling the graphite microstructure

The microstructure of graphite is considered to be comprised of an array of cubic particles, each particle being equal to the mean filler particle size (a) Each of these

particles is assumed to contain a plane of weakness oriented at some angle 0 to the

applied stress (Fig 17) Moreover, pores are randomly scattered about the microstructure, their size (cross-section) being log-normally distributed Such pores have a stress intensification factor (5) associated with them, which is quantifiable in terms of the applied stress and the pore half length c according to the principles of linear elastic fracture mechanics (LEFM) Cracks are considered

to initiate from these pores, and propagate by fracturing along the planes of weakness in the cubic particles of the graphite

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1,

Fig 17 Graphite microstructure as conceived in the Burchell fracture model

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A fracture mechanics criteria is developed to determine if fracture will occur on

such planes in the vicinity of pores A critical stress intensity factor (KIc) is

ascribed to the particles, and when this is exceeded by the stress intensity factor (KJ on the plane of weakness (KI is a function of JC, and 0) the particle is deemed

to have failed The initiating pore will then have grown from its original length c

to c+a, where a is the particle size In three dimensions the probability P(n) that the crack will traverse all n of the particles in the row ahead of it is P(n)=P(i)”, The probability P(f) that the defect c will grow to length c+ia, fracturing i rows of particles may then be determined Failure of the graphite is deemed to occur when

the stress intensity factor (K,) associated with the propagating defect, c+ia, exceeds

a critical value which is related to the particle KIc Moreover, there is a finite probability that the initiating defect (pore) exists, which is calculable from the pore size distribution These two probabilities may be combined to determine the failure probability The total failure probability that an inherent defect unll initiate a crack which propagates to cause fracture, which is a fimction of the number of pores in the specimen and their size distribution, may then be defined for any applied stress

6.2 A Facture criterion for particles in the graphite microstructure

The stress o‘, perpendicular to a plane x’ at angle 0 to the x-axis at a point distance

r from a crack tip (Fig 18) may be written as [ 11:

where K, is the stress intensity factor of the crack length 2c under applies stress 0

The stress dYy may also be defined in terms of the stress intensity factor KI of the plane x’ at angle 0 from x:

where

e

KI=K,cos3(-)

2

Fig 18 The stress u;Y acting on a point distance r along a plane x’ rotated through angle

8 from the plane x

At the moment of particle failure

Y

Fig 19 Schematic illustration of the fracture criterion

For as crack in tension K, is defined as

substituting equation (36) in equation (35) we get

(34)

From equation (37) above we see that 0 is a function of the applied stress, ow, and the crack half length c For a given (T and c the angle 0 below which a particle may

be assumed to have failed is given by equation (37) For simplicity the orientation

of these planes about the x-axis (Fig 18) is considered to be uniform Therefore, the fraction ofthe particle that is potentially fractured is given by 20/180" or 20/n

This is also the probability, Pi, that the particle has failed Thus we may write:

Substituting for 0 from equation (35):

i e ~ particle cleavage is certain to occur This happens when: