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ReviewCite this article: Armstrong RW. 2015Material grain size and crack size influences oncleavage fracturing. Phil. Trans. R. Soc. A 373:20140124.http://dx.doi.org/10.1098/rsta.2014.0124

One contribution of 19 to a theme issue‘Fracturing across the multi-scalesof diverse materials’.

Subject Areas:mechanical engineering, materials science

Keywords:grain size, cleavage, dislocation pile-upmechanics, ductile–brittle transition, cracksize, fracture mechanics

Author for correspondence:Ronald W. Armstronge-mail: rona@umd.edu

Material grain size and cracksize influences on cleavagefracturingRonald W. Armstrong

Center for Energetic Concepts Development, Department ofMechanical Engineering, University of Maryland, College Park,MD 21842, USA

A review is given of the analogous dependenceon reciprocal square root of grain size or cracksize of fracture strength measurements reportedfor steel and other potentially brittle materials.The two dependencies have much in common. Foronset of cleavage in steel, attention is focused onrelationship of the essentially athermal fracture stresscompared with a quite different viscoplastic yieldstress behaviour. Both grain-size-dependent stressesare accounted for in terms of dislocation pile-upmechanics. Lowering of the cleavage stress occursin steel because of carbide cracking. For crack sizedependence, there is complication of localized cracktip plasticity in fracture mechanics measurements.Crack-size-dependent conventional and indentationfracture mechanics measurements are describedalso for results obtained on the diverse materials:polymethylmethacrylate, silicon crystals, aluminapolycrystals and WC-Co (cermet) composites.

1. IntroductionMuch of the analysis presented here is attributed tomentoring received from Clarence Zener, Norman Petchand George Irwin, as will be made clear in the referencedmaterial. To begin with, Zener, as Director of theWestinghouse Research Laboratories, appointed me to asummer research position in East Pittsburgh in 1955, justprior to attending graduate school at Carnegie Instituteof Technology. A post-doctoral year was spent during1958–1959 under the direction of Norman Petch, thenat Leeds University, and research association continuedfor many years. Research interaction with George Irwinbegan in 1982 when he became research professor at

2015 The Author(s) Published by the Royal Society. All rights reserved.

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the University of Maryland after distinguished careers both at the US Naval Research Laboratoryand at Lehigh University.

There is a much longer history of science and engineering concern with fracturing of materials,particularly of brittle fracturing that can occur without warning. A historical example is providedby the late-nineteenth-century editorial comment from Wahl, who stated: ‘from the experimentsthus made with perfect iron the strength of materials was formulated; but practical experienceshowed that large forgings and castings were subjected to blow-holes and honeycombs, whichthe most vigilant inspection would fail to discover—and these defects would lead to breakage atpoints of strain where theory said the metal should bear the given pressure. The matter becameof so much importance in view of the large number of iron girders used in modern buildings,that in New York a law was passed requiring an actual test of beams, girders, etc.’. [1]. Wahl’scommentary was published only just more than one decade before Sorby [2] introduced a seminalpaper on the microscopic examination of steel microstructures.

Just after turn of the twentieth century, Inglis [3] took up investigation of the stress stateassociated with a hole or crack and Griffith used the result to establish an energy balance forcritical crack growth that follows the reciprocal square root of crack size dependence mentionedabove and provides the basis for modern fracture mechanics [4]. Establishment of an explicitinfluence of grain size on the fracture strength of crack-free steel and related metals wasslower in development. Taylor [5], co-inventor of the crystal dislocation, had suggested thatthe yield stress was independent of grain size, at least for aluminium. Kelly [6] has provideda valuable commentary on W.L. Bragg, of X-ray diffraction fame, suggesting a reciprocal grainsize dependence of the plastic flow stress. Zener [7] was first to establish a connection between amodel of a slip band and a shear crack, leading to prediction of a reciprocal square root of graindiameter dependence for plastic yielding, and later pointed to crack initiation occurring at thetip of a dislocation pile-up [8]. Eshelby et al. [9] produced the key model description of pile-upparameters that Hall [10] and Petch [11] employed to establish a reciprocal square root of graindiameter, �−1/2, dependence for measurements of the tensile yield, σy, and cleavage, σC, stresses,respectively, of steel in the same-type relationship

σ = σ0 + k�−1/2. (1.1)

In the Hall–Petch (H–P) equation (1.1), the experimental constants are σ0y and ky for the loweryield point stress and are σ0C and kC for the cleavage fracture stress.

Thus begins the present story. The review starts with close examination of Petch’sexperimental measurements for the yield and cleavage fracture stress, followed by generalizationof equation (1.1) to describe the complete stress–strain behaviour of mild steel at ambienttemperature, with true flow stress, σε , at true strain, ε. Additional extension was to other body-centred (bcc), face-centred cubic (fcc) and hexagonal close-packed (hcp) metals and alloys. Animportant early application of the H–P relationship came from Cottrell [12] and Petch [13] intheir specification of the ductile-to-brittle transition temperature (DBTT) exhibited by steel andrelated metals. The temperature and strain rate dependence of σ0y is of paramount importance indetermining the DBTT. The period of development occurred nearly simultaneously with the timeof Irwin and co-workers extending Griffith’s work to establish the subject of fracture mechanics[14]. And further development by Irwin [15] and by Bilby et al. [16] dealt with modelling ofcrack tip plasticity that is shown to provide interpretation of modern fracture mechanics resultsobtained on additional steel, polymethylmethacrylate (PMMA), silicon, alumina and compositeWC-Co (cermet) alloy materials.

2. Hall–Petch relations for steelPetch measured at liquid nitrogen temperature the tensile yield and cleavage fracture stressesof mild steel, ingot iron and spectrographic iron materials having different polycrystal grainsizes [11]. Additional measurements were reported of the reduction in area exhibited by the

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20

250

500

750

1000

1250

4�–1/2 (mm–1/2)

s l.y.p

., s C

(M

Pa)

mild steelingol ironspectrographic iron

6 8 10

Figure 1. Combined tensile (lower) yield point and (higher) cleavage fracture stressmeasurements reportedbyPetch for severaliron and steel materials [11].

failed specimens. The yield and cleavage strength measurements are combined in figure 1. Themeasurements shown on the left side of the segmented vertical line are solely cleavage fracturestresses. The right-side short line segments connect the clearly separated measurements at smallergrain sizes.

Within experimental accuracy, Petch chose to extrapolate the yield and cleavage stresses infigure 1 to a same value of σ0C. However, the true fracture stresses were reduced at smaller grainsizes to correct for an increased plastic strain preceding fracture, thus producing a relatively lowvalue of kC {ε = 0} ≈ 80 MPa mm1/2. The value compares with kC ≈ 104 MPa mm1/2 (and σ0C ≈350 MPa) as given by Madhava [17] in a comprehensive compilation of grain-size-dependent σCmeasurements, including (lower) intercrystalline failure stresses and (higher) post-yield cleavagemeasurements (see [18]). The issue relates importantly to the question of whether yielding wouldalways precede fracturing, as will be discussed. Low [19] produced similar results to those ofPetch at larger grain size but, in contrast, reported a much steeper grain-size-dependent cleavagestress, uncorrected for strain dependence, and taken to extrapolate to σ0C = 0, thus mimicking aGriffith-type dependence.

In figure 1, the lower dashed line applies for the H–P dependence of the ambient temperaturelower yield point stress of mild steel, σl.y.p. [20]. The result is shown more fully in figure 2 toinclude also a subsequent flow stress, σε , dependence on grain size with constants, σ0ε andkε , such that kε< kl.y.p, and leading to the earlier reported ductile fracture stress dependenceon grain size [21]. For this material, the value of σ0.18 was very nearly equal to the true stressat maximum load, and so the increase in stress until reaching a strain and ‘necking’-correctedductile fracture stress, σd.f., was largely accounted for by strain hardening during the neckingstrain. Comparison of results in figures 1 and 2 shows that kd.f. < kC even for Petch’s lowered σCdependence. The strain hardening during uniform straining past the lower yield point is seen tobe mainly contained in σ0ε that is taken to measure plastic flow within the grain volumes.

The level of the lower yield point stress in figure 2 is linked through its temperature, T,and strain rate, (dε/dt), dependence to the generally higher value of the athermal cleavagestress (figure 1). Reference is often made to Zener & Hollomon [22] for describing the coupledtemperature–strain rate dependencies of σε on the basis of a chemical rate theory type ofthermal activation description. The summer appointment mentioned earlier was to investigatethe coupled relationship for molybdenum material tested in compression so as to avoid brittlefracture in tension [23,24]. Later development with Zerilli led to the so-called Z-A dislocation-mechanics-based constitutive relations for material dynamics calculations and are expressed forbcc metals in the form [25]

σε = σG + B0exp[−βT] + Aεn + kε�−1/2. (2.1)

In equation (1.1), σG is an athermal stress component determined by the presence of solute,precipitates and an initial dislocation density; B0 is a reference stress at temperature, T =0 K; β is an exponential temperature coefficient, and A and n are coefficients for power

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0 1 2 3 4 5

�–1/2, mm–1/2

s (M

Pa)

6 7 8 9 10 11 12

250

500

mild steel

ductile fracture

l.y.p.

5%10%

e = 18%

Petch [21]Armstrong et al. [20]

750

1000

Figure 2. Hall–Petch relations for mild steel stress–strain behaviour [20,21].

law strain hardening. The coupled T and strain rate dependence are in β that follows therelationship

β = β0 − β1ln(

dε

dt

). (2.2)

In equation (2.2), β0 and β1 provide linkage in equation (2.1) of the coupled T and (dε/dt)dependencies. The first three terms on the right-hand side of equation (2.1) are normallycombined in the single ‘friction stress’, σ0ε , in equation (1.1). Different steel materials were shownto give reasonably constant values of B0, β0 and β1 [26]. The dislocation mechanics model of kl.y.p.,though containing a local concentrated shear stress, τC, has been found to be essentially athermalfor steel. The situation is quite different for an important thermal dependence in τC for fcc and hcpmetals [27]. The higher kl.y.p. shown in figure 1 for the yield stress at liquid nitrogen temperatureis possibly attributed to deformation twinning that follows its own (athermal) H–P dependencewith constants σOT and kT, such that kT < kC [28]. The relative k-values establish that deformationtwinning is not responsible for cleavage.

Equations (1.1) and (2.1) were employed in figure 3 to illustrate the occurrence of a tensileDBTT at σy = σC for mild steel [29], similar to the reported compression and tensile stressmeasurements made for molybdenum [24]. The yield stress parameters for steels A and B inequation (2.1) were taken as B0 = 1780 MPa, β = 0.0143 K−1, σG = 30 MPa and ky = 23 MPa mm1/2;the cleavage stress parameters were σ0C = 330 MPa and kC = 104 MPa mm1/2. Steel A has a grainsize of 100 µms and steel B, of 5 µms. The yield stress of A was raised to that of A* by addition ofan athermal �σ ∗

0y component of 260 MPa as might occur from neutron irradiation damage.Figure 3 shows that the smaller grain size steel B has a lower DBTT because of the H–P

inequality: kC > ky, whereas steel A∗, which has an appreciably raised σy because of the athermalcomponent �σ ∗

0y that leaves σC unchanged, has a much higher DBTT owing to the raised frictionstress. The situation relates to important measurements made on neutron irradiation damage insteel by Hull & Mogford [30]. The danger indicated in figure 3 for this case, which was deemedimportant at the time, was that steels B and A∗ could have the same ambient temperature yieldstrengths and temperature dependencies of them while exhibiting appreciably different values

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0

5

10kcD(�–1/2)

Ds*Oy

kyD(�–1/2)

ORNL-DWG 66-7452

compressive yield stress

tensile yield stressmild steel, e =10–4 s–1

brittle fracture stress15

20

50

TB TA TA*

100temperature (°K)

stre

ss (

109

dyne

scm

–2)

150 200

A

A*

B

250

1000

psi

0

50

100

Figure 3. Calculated yield and cleavage fracture stresses formild steel over a range of temperatures at twograin sizes; 109 dynescm−2 = 100 MPa [29].

of the DBTT. Wechsler et al. [31] have reported quantitative measurements made on irradiationembrittlement of A212B reactor pressure vessel steel material.

3. Grain-size-dependent ductile–brittle transitionCottrell and Petch produced a theoretical model description of the DBTT for steel and related bccmetals involving employment of the H–P dependencies and introducing a surface energy, γ , forcleavage [12,13]. Cottrell’s version of the DBTT was expressed in terms of the H–P parameters �,σ0y and ky as [12]

ky(σ0y�1/2 + ky) = CGγ . (3.1)

On the right-hand side of equation (1.1), G is the shear modulus and C is a numerical constant.By increasing ky, σ0y or �, the condition for cleavage cracking would be met more quickly andfailure would occur. Equation (3.1) was shown to be in agreement with the Hull & Mogford [30]results giving an increase in the DBTT produced by an increase in σ0y. Al Mundheri et al. [32] havedemonstrated that neutron irradiation clearly raises the yield stress by a constant �σ ∗

oy withoutchanging the cleavage fracture stress.

Petch [13] obtained an explicit temperature dependence of the DBTT. The temperature-dependent component of σy was employed with the stress coefficient, B0, and exponentialtemperature coefficient, β, as in equation (2.1). Petch also employed in place of σy the true ductilefracture stress, σf, with constants, σ0f and kf [21]. On such basis, an equation was obtained forTC as

TC =(

1β

)[lnB0 − ln

{(CGγ

kf

)− kf

}− ln�−1/2

]. (3.2)

Equation (3.2) provides indication in the β factor of the importance of the exponential temperaturecoefficient that is reduced at higher strain rate in accordance with equation (2.2). Experimentalsupport was reported by Heslop & Petch [33]. It may be noted that the figure 3 DBTT descriptionremoves the need for the relatively uncertain parameter, γ [34].

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0

Leslie et al. [36]

2 × 10–4 s–1

103 s–1

100

100

1000

200T (K)

s y–

100

(MPa

)

300 400

Figure 4. The thermal component of stress dependence obtained from data in [36].

(a) Charpy v-notch transition resultsIn a later quantitative assessment of the DBTT measured in Charpy v-notch (CVN) tests, Petch [35]dealt with the influence on β and B0 of the substitutional alloying elements: manganese, nickeland silicon. The assessment connected with valuable experimental results that had been reportedmuch earlier by Leslie et al. [36] for titanium-gettered interstitial-free steel and with other resultsreported later by Sandström & Bergström [37], including in the latter case a description of DBTTmeasurements reported for CVN tests. For relation to results reported by Petch, figure 4 leadsfor a 1.5% manganese steel to values of B0 = 1570 MPa, β0 = 0.0092 K−1 and β1 = 0.00045 K−1, asdetermined from the measurements reported by Leslie et al. [36]. For the titanium-gettered low-carbon steel, ky was reduced to 5.5 MPa mm1/2, σG = 70 MPa and �−1/2 = 5.6 mm−1/2, thus leadingto an athermal stress of 100 MPa being subtracted from the measured σy values. Leslie et al. [36]had remarked that the lowered carbon concentration (of 0.004 wgt. %) had largely eliminated mostof the grain size dependence of the DBTT, presumably because of the reduction in ky, otherwise,Petch had noted as well the important influence of ky lowering [35].

A valuable historical note is that Orowan [38] had drawn attention to the condition of theyield stress being raised to the level of the cleavage fracture stress as a criterion for the onset ofbrittleness in steel. With a plastic constraint factor, α, taken for the notch effect in a CVN test, theDBTT is specified as

σC = ασy. (3.3)

With α = 1.0, the condition is the same as described for the tensile result in figure 3. Krafft et al. [39]had commented previously on the importance of the strain rate in accounting for CVN impacttest results. Chlup et al. [40] have described application of higher strain rate testing to morebrittle cast-basalt and soda-lime glass materials. Otherwise, equation (3.3) provides a quantitativedescription of the DBTT for a notched impact test [41,42] analogous to that given by Cottrell [12]and Petch [13].

Figure 5 illustrates the use of equation (3.3) in evaluating very complete σy, σC and CVNmeasurements reported by Sandström and Bergström [37,41,42]. The 10 and 65 µm grain-size-dependent CVN results shown in figure 5 are seen to provide a counterpart description ofthe tensile transition shown in figure 3. A value of σ0C = 330 MPa and kC = 107 MPa mm1/2

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100

–200 –100 0 100

200

Charpyresults

a sy2

a sy1 {400 s–1}

sc1

sc2

aky D�–1/2

kc D�–1/2

Tensileresults

T (K)

CV

N, Jl1 = 10 mm

l1 = 10 mm

l2 = 65 mm

l2 = 65 mm

T (°C)

s (G

Pa)

0

0.5

1.0

1.5

3000

50

100

Sandström and Bergström0.15C 1.32 Mn mild steel

at DBTT: sc = a sy {Tc, e.}

Figure 5. Hall–Petch-type assessment of CVN measurements for mild steel [37,41,42].

had been measured by Sandström and Bergström. The constants σG = 130 MPa, β0 = 0.0075 K−1,β1 = 0.00040 K−1 and B0 = 1000 MPa, and ky = 14.9 MPa mm1/2 were determined from tabulatedT, (dε/dt) and grain size measurements. The β0 and B0 parameters, though different from thosedetermined from figure 4, were shown to be in excellent agreement with corresponding valuesreported for other results by Curry [43] and by Zerilli & Armstrong [25]. The value of β1 is in goodagreement with the value determined from the Leslie et al. [36] measurements. An effective strainrate of 400 s−1 was determined to apply in the Charpy test and a value of α = 1.94 was reported forthe v-notch plastic constraint. The value of TC was taken to be in good agreement with a lineardecrease in ln(�−1/2) as prescribed in equation (3.2). A relatively low value of γ = 3 J m−2 wasestimated. The higher than predicted measurement of TC for the smaller grain size microalloyedsteel has been tracked to a role for carbide cracking. The importance of large grains in initiatingcleavage in bimodal grain size distributions was established by Chakrabarti et al. [44]. OtherCharpy impact energy results have been reported for (Brazilian) ASTM A508 class 3A dual phasesteel for which the true strain at fracture was shown to follow an H–P dependence on effectivegrain size [45], in line with previous prediction [46].

(b) Carbide crackingAn indication of progress past Wahl’s historical editorial comment on the need for testing toavoid the use of defective material [1] is indicated by the finer scale focus over the interveningyears on microstructural elements involved in determining the fracture strength of engineeringmaterials. McMahon & Cohen [47] were pioneers in investigating the initiation of cleavage causedby cracking of carbide plates at grain boundaries. In a later publication dedicated to Orowan,Cohen and Vukcevik reported a model analysis for such carbide-initiated cleavage of ferrite inwhich the H–P yield stress was equated to the Griffith stress for carbide cracking, expressed in

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the latter case as [48]

σ ∗C =

[2EγC

π (1 − ν2)tmax

]1/2, (3.4)

where E is Young’s modulus, γC is the surface energy of the cracked carbide, ν is Poisson’s ratioand tmax is the maximum carbide plate thickness. The model gave a similar grain size dependenceof TC to that given by Petch [13] except for TC increasing with increase in carbide thickness.More recently, Roberts, Noronha, Wilkinson and Hirsch have employed an analogous relation toequation (3.4), with addition of a plastic work term, wp, to γC and specified σ ∗

C to be determinedby emission of dislocations from the cracked carbide [49]. Consideration of any grain size effectwas excluded.

Petch [50] too had earlier considered an influence of carbide cracking on the predicted grainsize dependence of TC. Emphasis was given to the dynamics of slip-induced cracking of grainboundary carbide that then required critical growth of cleavage through the ferrite even forrelatively thick carbide plates. Substitution of 2γP for γC was made in equation (3.4). Calculationswere presented for carbide modification of σC and reversal of the decreasing TC dependence on�−1/2 for different carbide plate thicknesses in the range of 0.25 to 5 µms. Interesting associationof larger carbide thickness and reduced sharpness of the temperature range for the transitionwas pointed out by Petch, thus adding credence to the modified model of predicting TC. In fact,Sandström & Bergström [37] had reported a relatively thicker 0.8 µm carbide thickness for the10 µm grain size microalloyed steel and this has very probably contributed to the higher thanpredicted value of TC shown in figure 5.

The calculations reported by Petch for carbide modification of the grain size dependence of σ ∗C

were re-cast by Armstrong et al. [26] in terms of the reverse consideration of dependence on t. Themodel description compared favourably with notched test results reported by Bowen & Knott[51]. At the smallest t-values, a comparison was made with Petch’s DBTT predictions for the σ ∗

Cdependence on �−1/2. The indication was that carbide control for conventional grain size materialis normally restricted to carbide thicknesses of approximately 1 µm or larger. A statistical analysisof main crack linkage with larger carbide particles in a spheroidized steel material was presentedby Strnadel et al. [52]. Relatively larger γP values were required to describe the measurements,presumably because of neglect of any slip-induced stress concentration effect. Armstrong [53] hassuggested that σ ∗

C should enter into the concentrated stress term in the H–P dislocation pile-upmodel relation for the cleavage stress, σC’, that is expressed as

σ ′C = σ ′

C0 + m[

πGbα∗�

]1/2σC∗1/2, (3.5)

where m is an orientation factor, b is dislocation Burgers vector and α∗ ≈ 0.8 is for an averagedislocation character. Equation (3.5) provides for possibility of part of the scatter observed inσC* measurements to be accounted for by influence of local grain size and crystallographicgrain orientation.

Zhang, Armstrong and Irwin investigated carbide or inclusion cracking influence on thebrittleness transition for an opposite condition of the first vestiges of isolated cleavage regions(ICRs) occurring in A533B pressure vessel and related steels either at the upper shelf level ofCVN specimens or from spring-loading of side-grooved compact tensile specimens [54]. In theproject, large-scale structural inhomogeneity influences associated with dendritic solidificationeffects had been tracked on the cleavage surface of cast thick-plate material through mismatchedsurface elevation measurements revealed on mated fracture surfaces [55]. The ICRs were foundto be initiated often at localized regions of ductile hole-joining failures [56].

Figure 6 was developed in a model description of the observations [57]. Variations inmicrostructure were taken to produce irregularities in the local stress–strain behaviour typified,for example, by a dislocation pile-up stress concentration at a relatively stiffer particle clump thatwould undergo abrupt plastic instability thus transferring rapid load increase onto an adjacentgrain that was favourably oriented to produce a cleavage crack in the manner described by

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cleavageembryo

particleclump

pile-upgrain

ductilecrack front

Figure 6. Model of cleavage initiation in a sea of ductile hole-joining failures [57].

Cottrell [12]. A large initiating grain size and rapid load transfer of a plastically unstable particleclump were important features of the model. The several requirements are in line with expectationof an ICR being a relatively rare occurrence in a sea of ductile fracturing. Less favourabilityof the same conditions in the encompassing microstructure halts the spread of any ICR. Themodel relates to another analysis of the Cottrell mechanism using a dislocation/crack interactionsolution [58]. And the figure 6 model description may be compared also with a model descriptiongiven by Ritchie et al. [59] for cleavage initiation at the DBTT.

4. Crack size in fracture mechanicsGriffith [4] tested his theory against experimental results obtained on soda lime silicate glassmaterial. Early difficulty in directly applying the theory to metal fracturing is well-known.Recently, Wiederhorn et al. [60] reviewed the case for cracking in silicate glass materials. Emphasiswas given to new nanoscale measurement capabilities that are being applied to characterizingsuch cracks, especially with regard to determining the nonlinear size and structure of the distortedregion at the crack tip. The experimental evidence for crack tip plasticity is overwhelming anddetailed mechanisms have been reported, for example, proceeding from a simple calculation ofdislocation nucleation in the plane of the crack [61] to a more general description of crack-assisteddislocation multiplication linked with the pre-existing dislocation density [62]. Antolovich &Armstrong [63] have reviewed the issue.

(a) Modification by plastic zone sizeA continuum mechanics basis for critical crack growth with a leading plastic zone at the cracktip was reported by Dugdale [64], Irwin [15] and Bilby et al. [16]. Dugdale provided connectionwith results on mild steel. Modification of the inverse square root of crack size, c, dependence was

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10 1.0 0.1 0.04

0

1

2

s (k

gm

m–2

)

3

4

5

6

7

8

9

1 2 3c–1/2, mm–1/2

c, mm

polymethylmethacrylate

Berry [67]

Williams et al. [68]

4 5 6

sT

sY

Figure 7. Fracture stress dependence on crack size, c, with crack tip plasticity [66–68].

obtained from the Bilby et al. analysis in terms of a leading plastic zone size, s, in the form [65,66]:

σF =(

81/2

π

)σF0

[s

c + s

]1/2. (4.1)

Figure 7 shows application of equation (4.1) to results reported for PMMA material. Crack-freeyield and tensile fracture stresses are shown on the ordinate scale axis. Heald et al. [69] reportedsimilar application of the same-type analysis to measurements obtained on other polymericmaterials as well as on the mild steel sheet material investigated by Dugdale, and to tungstenfibre-reinforced copper composite materials.

For (s/c) � 1.0, equation (4.1) follows the Griffith inverse square root of crack size dependenceas shown in figure 7. By comparison with the fracture mechanics relation, σF = K/[πc]1/2, theresult is obtained that K = σF0(8 s/π )1/2. A value of K = 32.3 MPa mm1/2 applies in figure 7 for themeasurements of Berry [67]. A close value of K = 36.4 MPa mm1/2 is obtained for the Williams &Ewing [68] measurements. The equation (4.1) dependence has been employed to describe a DBTTfor material with different crack sizes in comparison with a constitutive equation description ofthe material strain rate dependence [70]. At crack velocities greater than approximately 330 m s−1

imposed on PMMA materials, the relatively constant fracturing energy increases because ofinitiation of multiple small crack branchings [71]. Recently, Carpenteri & Taplin [72] have reporteda larger range in K-values for polypropylene material relating to test specimen width effects indetermining the plane strain fracture mechanics stress intensity, K1c.

In equation (4.1), the value of σF0 was specified as the crack-free fracture stress thus providingfor incorporation of H–P grain size dependence. The same relationship was shown to be obtainedfor critical crack tip opening displacement [66]. For comparison with Irwin’s result [15], his plasticzone size radius, r = (s/2). Conversion of equation (4.1) to plane strain for a von Mises yield stresscondition leads to the relationship [53]

K1c =(

83π

)1/2 [σ0C + kC�−1/2

]s1/2. (4.2)

Experimental verification of the H–P-type prediction in equation (4.2) was demonstrated for acompilation of iron and steel measurements [65,73–77]. The large plastic zone sizes associatedwith the measurements were found to be relatively independent of the material grain size. Thecondition may be altered according to the acuity of the pre-crack size as will be discussed here,and later in connection with indentation fracture mechanics measurements. Yokobori & Konosu

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[78] had investigated the effect of notch size and acuity for steel material tested at 77 K. Forrelatively blunt notches, a positive H–P dependence was obtained for K, whereas for fatigue pre-cracked material with a very large ratio of crack size to grain size, a negative H–P dependencewas obtained. Armstrong explained the reversed effect as being caused by a reduction in s,then becoming proportional to the material grain size [66]. More recently, Zeng & Hartmaier[79] have reported a (dislocation dynamics/cohesive zone) model for evaluating the fracturetoughness of tungsten material in which the fracture toughness decreased with increase in grainsize. The result was attributed to obtainment of relatively small plastic zone sizes being limitedby grain boundaries.

Petch & Armstrong [80] investigated the influences both of σ0y and ky on KC measurementsreported for cleavage of a number of steel materials. A role for work-hardening was establishedin a model description of plastic yielding being initiated at the root of a notch in a fracturemechanics test and essentially athermal work hardening ensuing until the cleavage fracture stresswas reached and the test specimen failed. Such bcc-type athermal work hardening characteristichas been confirmed, for example, for tantalum material in line with application of equation (2.1)[81]. For grain size dependence, the yield stress is increased, but the fracture stress is increasedmore because of kC > ky, and a greater amount of work hardening is required to reach a higherstress intensity, KC. An H–P dependence for KC was shown by Petch and Armstrong to apply for0.08 carbon steel results reported by Lin et al. [82] in tests at −80 and −120◦ C.

An opposite effect on KC occurs for increase in σ0y at fixed grain size, because such increasedoes not change the cleavage fracture stress, as described for figure 3. Consideration of therelationship between work-hardening and crack tip opening displacement led to prediction ofa logarithmic decrease in KC with increase in σy according to the expression [79]

lnKC = lnK′C − σy

2n′A′ (4.3)

In equation (4.3), n′ and A′ are power law work-hardening coefficients analogous to n and Ain equation (2.1) except being taken to represent the total stress. Petch and Armstrong obtainedreasonable agreement with equation (4.3) over a wide range of σy measurements made on varioussteels. Based on a ‘weakest link’ model, Tyson [83] also accounted for a dependence of KC on σy

with involvement of an important role for n′. A log/log dependence of KC on σy was obtainedfor cleavage-initiated failure measurements reported for AISI 1020 plain carbon steel material byCouque et al. [84]. Figure 8 shows agreement of equation (4.3) with the same results. The T1, T7and T9 designations apply for σy values obtained in different thermal treatments. As noted infigure 8, Couque et al. had reported test results for both quasi-static and dynamic loading.

There is additional complication to be expected for both grain size and crack size effectsin ductile fracture toughness measurements. Srinivas et al. [85] reported an H–P dependenceof ductile fracture toughness measurements made for Armco iron material, beginning from alowest K determined for cleavage initiation at a largest millimetre-scale grain size and extendingdownwards to fully ductile hole-joining failure occurring at a grain size, � = 38 µms. A value ofkf = 276 MPa mm1/2 was determined for a necking-corrected σf that corresponded to reductionsin area increasing from approximately 80% to approximately 90% with decreasing grain size, fargreater than Petch’s approximately 65% for mild steel. An H–P dependence was determined forKJC on the basis of J-integral methods involving critical stretch width zone measurements andload line displacement plots. Estimation of plastic zone sizes were compared with measurementsobtained in microhardness tests. Chen et al. [86] have reviewed the J-integral procedure fordetermination of crack growth resistance (J–R) curves.

(b) Indentation fracture mechanicsBeyond the multiscale dimensional aspects of �, c and s, there is diversity both in the availablemethods of fracture mechanics testing and in the different materials tested. Frank & Lawn[87] produced a Griffith-based breakthrough description of long-standing observations made

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100

15

30

50

100

150

Couque et al. [84]static dynamic

T1T7T9

200

KC

(M

Pam

1/2 )

300sy (MPa)

400 500

Figure 8. Yield stress dependence of cleavage-initiated fracture toughness [80,83,84].

of ‘cone-cracking’ configurations produced at hardness indentations in otherwise brittle glassmaterials. A dependence of load on crack length to the (3/2) power was predicted. The theorywas soon extended to radial and median crack geometries associated with indentations madein essentially brittle glass and crystalline ceramic-type materials. Atkinson et al. [88] havereviewed the theory. Figure 9 shows the (3/2) dependence in relation to elastic and plastic loadmeasurements determined for indentation tests made on silicon single crystals [89].

The indentation test method allows determination of KIC measurements from the (3/2)dependence. Related careful measurements were reported for polycrystalline alumina materialsover similar ranges of grain size by Franco et al. [90] and by Muchtar & Lim [91]. Figure 10 showsan H–P dependence of KIC for the combined measurements [92]. A range of KIC measurementsfor single crystals is shown on the ordinate axis [93]. Thermal strain influences occur for largergrain sizes. The three larger filled-circle points in figure 10 were determined for Hertzian-typering cracks formed under indenter loads of 30–50 kgf applied to sapphire spheres of 5 mmdiameter [90]. Ring crack depths of 2–13 µms were measured for surface crack diameters ofapproximately 300–400 µms. The same trend of measurements was obtained for smaller radialcracks of approximately 10 µm length initiated at a smaller load of 0.5 kgf applied in diamondpyramid indentation tests. The more extensive smaller-filled-circle measurements shown in thesame figure were obtained for radial cracking initiated by diamond pyramid indentions made atincreasing loads of 0.3, 0.5, 1.0 and 2.0 kgf [91].

Very interestingly, Franco et al. [90] found at larger loads of 1.0–25 kgf in the diamond pyramidhardness tests that the grain size dependence of KIC was gradually reduced and then reversed atthe highest applied load. Armstrong and Cazacu accounted for the reversal in terms of the plasticzone size factor, s, in equation (4.2) changing over to become proportional to the grain size [92].Computed values of s were determined from the limiting Griffith form of equation (4.1) usingknown H–P values for σ0F and the KIC measurements. Despite the opposite trend for s, a positiveH–P relation was obtained for σF. In other work, Belenky et al. [94] have reported a relatively low

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10–210–5

10–4

10–3

10–2

10–2

10–1

102

103

sharp indentations on silicon crystalTsai and Mecholsky, (001)

Lankford and Davidson

Shin, Singer and Armstrong, (111)Shin, Singer and Armstrong, (001)

Chiang, Marshall and EvansPethica, Hutchings and Oliver

104

105

1

1010–1

1

10

102

103

10–1

3.0

Hertzian prediction

2.0

1.5

D = 0.1 mm

0.6 mm

1.8 mm

5.7 mm

11.9 mm

24.6 mm

244.4 mm

4.7 mm

elastic

elastic/plastic

cracking

1

d, di, or dC (mm)

P (

N)

P (

gf)

10 102 103

Figure 9. Elastic, plastic, cracking measurements: d is Hertzian contact diameter; D is ball diameter; di is hardness diagonallength and dC is tip-to-tip crack length [89].

10

thermalstrain

influences

single crystals

fine-grained alumina

0 10 20 30�–1/2 (mm–1/2)

�, mm

KIc

(M

Pam

1/2 )

40 50 60

5

4

3

2

1

1 0.3

Figure 10. K1C measurements for single crystal and polycrystalline alumina [90–93].

value of KIc = 3.12 MPa m1/2 for static and dynamic nanograined alumina material tested in theform of pre-indentation-cracked bend bars.

Other application of equation (4.2) has involved both indentation fracture mechanics andconventional bend-stress-type fracture mechanics measurements reported for the industriallyimportant WC-Co (cermet) composite system [95]. WC particle size, d, and Co binder mean-freepath length, λ, influences had been accounted for by individual H–P relations being establishedfrom microhardness measurements made on each component [96]. The strength and ductility of

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00

5

10

15

20

0.2

K (MPa m1/2)

l1/2 mm1/2

Sigl & Fischmeister

R&vR

d > 0.8 mm (Jia et al.)d ~ 0.070 mm (Jia et al.)

0.4 0.6 0.8 1.0

Figure 11. K dependence on Co mean-free path in WC-Co composite material [95].

the composite material is known to be controlled by the Co phase. Armstrong and Cazacu showedthat λ could be substituted both for � and s in equation (4.2) and so give reasonable fit in figure 11of combined experimental K measurements reported in bend tests by Sigl & Fischmeister [97] andin indentation fracture mechanics tests by Jia et al. [98]. The single crystal WC measurement wastaken from Richter & Ruthendorf [99]. Kotoul [100] had developed a theoretical multi-ligamentplastic zone size description for unstable crack control of K that was shown to follow the sameλ1/2 dependence [101].

5. SummaryGrain size and crack size aspects of cleavage fracturing, the ductile–brittle transition and fracturemechanics stress intensity properties of steel and several other potentially brittle materials havebeen reviewed. In the preceding description:

— a quantitative assessment is given of reported measurements for the H–P kC valuedetermined for the brittle cleavage strength of a number of steel materials;

— extension of the H–P description to the plastic stress–strain behaviour of mild steel isdescribed, with kl.y.p and kC being athermal and thermal dependence in σ0ε ;

— analysis has been presented for the influence of higher kl.y.p, or ky, and σ0ε on increase inthe DBTT in tension and in CVN test results;

— a deleterious effect of cracking produced at grain boundary carbide plates is incorporatedwithin the DBTT analysis;

— a model description is given of transition to first vestiges of cleavage in steel uponlowering of temperature from the higher regime of ductile fracturing;

— a plastic-zone-modified Griffith-type inverse square root of crack size dependence isdescribed for the fracture mechanics stress intensity, K;

— grain size and σy influences are incorporated within the fracture mechanics descriptionof K, including an important role for the plastic zone size, s; and

— indentation fracture mechanics measurements of K are presented for diverse siliconcrystal, alumina polycrystal and WC-Co composite materials.

Acknowledgements. Appreciation is expressed to the Royal Society for sponsorship of the current theme issueon fracturing, and including the author’s participation in it. X. Jie Zhang is thanked for producing an earlyversion of figure 10. Stephen Walley is thanked for providing a number of important reference articles.

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