Dislocation viscoplasticity aspects of material fracturing

$Page 1: Dislocation viscoplasticity aspects of material fracturing$
Engineering Fracture Mechanics 77 (2010) 1348–1359

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Engineering Fracture Mechanics

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Dislocation viscoplasticity aspects of material fracturing q

R.W. ArmstrongCenter for Energetic Concepts Development, Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, United States

a r t i c l e i n f o

Article history:Received 20 November 2009Received in revised form 12 February 2010Accepted 19 February 2010Available online 23 February 2010

Keywords:Fracture mechanicsMicrostructureDislocationsViscoplasticityNuclear pressure vessel steelsGrain sizeCleavageDuctile-brittle transitionBrittlenessWC-Co compositesHardness

0013-7944/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.engfracmech.2010.02.019

q An article to honor Professor G.R. Irwin after thE-mail address: [email protected]

a b s t r a c t

Dislocation viscoplasticity, whether occurring before, during, or after the appearance ofcracking, and with energy requirement additional to needed crack surface energy, hasproved to be the bane of simplistic strength/energy evaluations applied to the fracturingproperties of solid materials. George Irwin, in later years, turned his attention to suchcrack-effected viscoplasticity consideration in relation to the complex microstructuralaspects of fracturing behaviors obtained for nuclear pressure vessel steels. The work isreviewed here with a broader research focus on dislocation plasticity aspects of fracturingbehaviors, at ever smaller crack sizes, for a wider range of materials, especially, of relativelygreater hardness.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

A relatively early 1951 photograph in the career of George Rankin Irwin, when Head of the Ballistics Branch, US NavalResearch Laboratory (NRL), and taken of him standing among leading post-World War II naval-related research scientists,is shown in Fig. 1, as reported by Frederick Seitz [1]. Not very long before, in 1941, Seitz and Read [2] had published a seriesof articles on the coming advent of dislocation mechanics in plasticity. And not much later, in 1957 when Irwin was Super-intendent of the Mechanics Division at NRL, Krafft et al. [3] were to make a plasticity connection between the static and dy-namic properties of a low carbon steel material, when in the latter case, subjected to fracturing induced by Charpy v-notchimpact testing. The authors suggested that their results would be of interest ‘‘to establish association of fracture strength ofmetals with dislocations, imperfections and metallurgical structure”. As known now, the background research activities ledto Irwin’s important establishment shortly afterwards of the fundamentals of fracture mechanics [4,5] and, in particular, ledto the relationship worked-out in one way with colleague, J.A. Kies, of the cleavage fracture stress, rC, dependence on theinverse square root of pre-crack half-length, c, for an internal, through thickness, crack in the form

rC ¼ K1C=ffiffið

ppcÞ ð1Þ

in which K1C is the fracture mechanics (FM) stress intensity for a test in plane strain.

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e centennial celebration of his birth.

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Nomenclature

rc polycrystal cleavage fracture stressK1C fracture mechanics stress intensity for plane strainc crack half-lengthl average grain diameterr0C cleavage stress at infinite grain diameterkC microstructural stress intensity for cleavagery yield stressr0Y stress intercept ‘‘friction stress”kY microstructural stress intensity for plastic yieldings�C concentrated shear stressn number of pile-up dislocationssE = s-s0 effective shear stress acting on pile-up dislocationss the applied shear stresss0 the friction shear stress resistance individual dislocation movementm Taylor orientation factorG shear modulusb dislocation Burgers vectora 2(1–m)/(2–m)m Poisson’s ratior�C concentrated tensile stress at pile-up tipmS Sachs orientation factorE Young’s modulusc effective surface energydp particle diameterr true stressdr/de strain hardening coefficiente true strainra athermal stress componentB thermal stress coefficient at T = 0 K.b temperature coefficient of thermal stressB0 coefficient for strain dependenceer reference strain for dynamic recoveryrG grain size independent athermal stresske microstructural stress intensity at strain equal to ebo strain-rate-independent temperature coefficientb1 strain-rate-dependent temperature coefficientde/dt strain rateA coefficient of power law strain dependencen0 power law exponenteu true strain at maximum tensile loada0 plastic notch constraint factorr{T, (de/dt), l} temperature-, strain rate-, and grain size-dependent tensile stresscS true surface energycUS unstable stacking energys strip-type plastic zone sizer crack tip plastic zone radiusrF plastic zone affected fracture stressHv Vickers diamond pyramid hardnessHwc Vickers hardness of tungsten carbideVWC volume fraction of tungsten carbideC contiguity of tungsten carbide particlesHm Vickers hardness of cobalt matrixk mean free path of the cobalt binder phaseH Hertzian elastic contact stressms; mB the Poisson’s ratios for specimen and ball, respectivelyEs, EB the Young’s modulus for ball and specimen, respectivelyd actual ball contact diameter or effective diameterD actual, or effective, ball diameterHC critical hardness for crackingðj2

1 þ j22Þ ¼ 2� 105 is a numerical constant

R.W. Armstrong / Engineering Fracture Mechanics 77 (2010) 1348–1359 1349

Fig. 1. A historical photograph of George R. Irwin among a distinguished group of naval research panel scientists.

1350 R.W. Armstrong / Engineering Fracture Mechanics 77 (2010) 1348–1359

The referred-to researches by Irwin and colleagues took place not long after Petch [6] had demonstrated an inverse squareroot of polycrystal grain diameter, ‘, dependence for the tensile cleavage fracture stress of several initially crack-free iron andsteel materials tested at liquid nitrogen temperature, as expressed in the relationship

rC ¼ r0C þ kCl�1=2 ð2Þ

in which r0C is the friction stress for pre-cleavage slip and kC is now known as the microstructural stress intensity for cleav-age fracturing. Petch found the same form of grain size dependence to apply for the low temperature yield stress, ry, withconstants r0y and ky, as had been reported also by Hall [7] with experimental constants determined at ambient temperature.The so-called Hall–Petch (H–P) relationship in Eq. (2) is now known to apply quite widely for body-centered cubic (bcc),face-centered cubic (fcc) and hexagonal close-packed (hcp) metals and alloys tested at different temperatures and strainrates, at constant strain values [8]. The purpose here, in honoring the pioneering researches of Irwin, is to add to the storyof plasticity influence on the fracturing properties of steel and various other solid materials, as will be described, in follow-upto the related pre-crack and crack-free material behaviors initially signaled, respectively, by Eqs. (1) and (2).

2. Isolated cleavage regions

In later years, as research professor at the University of Maryland, George Irwin turned his attention, in part, to investi-gating the microstructural mechanisms responsible for the loss of ductility occurring when the first vestiges of isolatedcleavage regions (ICRs) occurred, for example, either in side-grooved compact tension type specimens of A533B nuclear pres-sure vessel steel tested with a spring in series or in Charpy v-notch specimens tested at the earliest stage of reduction fromthe upper shelf energy [9]. In one case, ferrite and prior austenite grain size, as well as larger scale inhomogeneity influences


caused by dendritic segregation occurring in the original thick plate casting, were tracked through mismatched elevationsmeasured on the mated fracture surfaces of a broken specimen, including the determination of sequential stages of graincleavages, separations, and joined openings induced ahead of the main crack propagation. The work was followed by thePh.D. Thesis researches of Zhang [10] and several M.Sc. Theses [11–13] all addressed to specification of various microstruc-tural aspects of the fibrous to cleavage fracturing transition encountered in both nuclear pressure vessel steels and their weldmetals.

Two important outcomes of the foregoing researches were: first, a somewhat surprising observation of ICRs often beinginitiated by a localized region of ductile hole-joining failures as compared with individual particle fractures [14]; and, sec-ond, development of a model characterization of three successive stages occurring at the microstructural level in the tran-sition from fibrous fracturing to cleavage initiations [15]. Fig. 2 shows an ICR within a surrounding region of fibrousfracturing at a 20 lm scale marking for a Charpy v-notch specimen tested at 295 K, also, with an inset relatively higher2 lm scale magnification of the initiation site of associated particle fracturing and local hole-joining separations. A tabula-tion of ICRs was provided of the more often case of their being initiated at particle clumps (PCs) as compared with singlebroken particles (BPs). In later researches, Zhang developed the technique of stereo (scanning electron microscopy) sectionfractography for quantitatively associating ICR initiation sites with their underlying microstructural elements [16]. Suchexperimental results, and analysis of the factors most probably contributing to the observations, led to the model descriptionshown in Fig. 3, as reported by Gudas et al. [17]. John Gudas had completed his Ph.D. Thesis researches at Johns HopkinsUniversity with Irwin on his advisory committee. In the model development of Fig. 3, a main presumption is that a particleclump, when loaded rapidly, can separate sufficiently abruptly to generate a tensile stress high enough for cleavage initiationin a favorably oriented, adjacent grain. The combined conditional requirements are in line with the proposition that an ICR isa relatively rare occurrence within a sea of ductile fracturing elements. The variations in microstructure produce under stresscorresponding irregularities of local strains and grain separations during ductile crack growth that lead to rapid and rela-tively brittle fracturing of the plastically-stiffer, carbide particle clump, and are enhanced by H–P type dislocation pile-ups with greater effect in a larger grain. The higher strength particle clumps undergo relatively smaller uniform strain beforebecoming plastically unstable. Rapid load transfer is shifted to an adjacent grain of suitable crystallographic orientation andcontaining a potential nucleus for cleavage initiation and growth. Lesser favorability of the same conditions in the surround-ing microstructural environment halts the spread of the ICR. Lower test temperatures and/or enhancement of the strain rateenable more ICRs to form and others to spread further. In development of the model description by Gudas et al., referencewas made to important research results obtained in this subject area by C.J. McMahon, M. Cohen, E. Smith, J.F. Knott, R.O.Ritchie, J.R. Rice, D.A. Curry, K. Wallin, T. Saario, K. Torronen, A.G. Evans, T. Lin, S. Aihara, T. Haze, A.R. Rosenfield, G.T. Hahn,W.F. Flanagan, D.C. Drucker, L.B. Freund, and J.W. Hutchinson, among others [17].

3. Model evaluations

Dislocation pile-up representations of slip band stress concentrations are shown in the ‘‘pile-up grain” of Fig. 3. Theenhancement of the local shear stress, s�C , at the tip of a circular pile-up driven by the effective shear stress component,sE, is obtained as

Fig. 2.ductile

s�C ¼ ðn=2ÞsE ð3Þ

in which n is the number of dislocations in the pile-up and is linearly dependent on sE and, very importantly, on the graindiameter, ‘. Also, the effective shear stress, sE is obtained as the applied shear stress, s, minus a friction stress, s0, so that the

Isolated cleavage region (ICR) initiated at a fractured inclusion particle linked to a group of smaller particle hole-joining failures, all enclosed within afracture matrix of a pressure vessel steel.

Fig. 3. Model for initiation of an ICR within a matrix of ductile fracturing.


model description of an inverse square root of cleavage stress dependence on ‘, as given experimentally in Eq. (2), is ex-pressed in the relationship

rC ¼ r0C þm½pGb=al�1=2r�1=2C ð4Þ

in which r0 = ms0, r�C ¼ mSs�C , m and mS are Taylor and Sachs tensor orientation factors, respectively, G is the shear modulus,b is the dislocation Burgers vector, and a = 2(1–m)/(2–m) = �0.8, with m being Poisson’s ratio involved in determining the aver-age dislocation character [8]. In the case that r�C is determined by Griffith-type cracking of a brittle inclusion or carbide par-ticle [18],

r�C ¼ ½4Ec=fð1� m2Þpdpg�1=2 ð5Þ

in which E is the Young’s modulus, c is the effective crack surface energy, and dp is the particle diameter. Wu and Knott [19]have reported on the statistical nature of r�C determinations for as-received and intentionally degraded weld metal speci-mens. An exponential (�1/2) or (�1/4) dependence of rC on ‘ or dp, respectively, is obtained from the combination ofEqs. (4) and (5), as appears to be in reasonable agreement with both grain size and particle size results obtained on C–Mn and other steels, as reported by Armstrong et al. [20]. A more detailed description of the combined influences of grainboundary carbide and grain size on both cleavage strengths and Charpy v-notch temperatures was given by Petch [21]. Morerecently, Wang et al. [22] have reported on cleavage initiation in a C–Mn steel, as affected by single or ”stringed” inclusions,while making the observation that fracture was controlled by propagation into the steel matrix of a ferrite grain-sized crack;see Ref. [17] and Fig. 3. Most recently, dislocation and precipitation hardening influences, among other microstructuralparameters, have been described in connection with fissuring at elongated grain surfaces for HSLA (high strength low alloy),control rolled, ferrite/pearlite steels [23].

The rapid load transfer condition for the ductile failure of a locally strengthened particle clump, as schematically depictedalso in Fig. 3, is assessed on the basis of the Considere condition between the locally applied true stress, r, and the true strainhardening, as

r ¼ dr=de ð6Þ

In relation to viscoplasticity influences implicitly contained in Eq. (6), Zerilli and Armstrong [24] have produced a set ofso-called Z–A constitutive equation descriptions for the deformation properties of bcc, fcc and hcp metals and alloys, partic-ularly, for use in material dynamics computations, as have been recently reviewed [25]. The following relationships apply forthe bcc case; see also Zerilli [26]

Fig. 4.tests; o(CVN) s


r� ¼ ra þ B expð�bTÞ þ B0fer ½1� expð�e=erÞ�g1=2 ð7aÞ

in which

ra ¼ rG þ kel�1=2 ð7bÞ

and

b ¼ b0 � b1 lnðde=dtÞ ð7cÞ

At normal tensile straining, the third term in Eq. (7a) may be approximated by Taylor type, parabolic strain hardening or,as often employed in the bcc case, of an exponential strain hardening term, Aen0. With the latter dependence, Armstrong andZerilli [27] have shown that the uniform strain to the maximum load point, eu, may be expressed as a decreasing function ofdecreasing n0 and with increasing ratio of the combined first two terms in Eq. (7a) and the strain hardening coefficient, A.Increasing the strain rate (or lowering the temperature) was shown to reduce eu, hence, being in agreement with the con-sideration depicted in Fig. 3 of relatively greater plastic instability being associated with the ductile hole-joining failure ofa particle clump. Bandstra et al. [28] have provided a model analysis of void coalescence at MnS inclusion-initiated voidsdeveloped during the ductile fracturing of a high yield HY- 100, for 100 ksi, (about 690 MPa), steel material. A high densityof clusters of small voids, as modeled here in Fig. 3, was associated with intense strain localization.

For bcc and related metals, Eqs. (2) and (7a,b,c) are usefully equated to specify the onset of a brittle to ductile transition inbehavior, for example, as observed beginning from the low temperature end of Charpy v-notch tests; also, as suggested byOrowan in an early article [29], as

a0rfT; ðde=dtÞ; lg ¼ rCflg ð8Þ

in which a0 is a notch constraint factor to take account of the increase of yield stress in the triaxial state of stress. In this case,as mentioned, the focus is on the transition in behavior occurring at the (opposite) foot of the change in Charpy v-notch mea-surements beginning from their lowest levels. With e = 0 for the yield condition in Eq. (7a), and a0 = 1.94 and (de/dt) = 400 s�1, very complete Charpy v-notch ductile to brittle transition measurements plus yield and cleavage stress mea-surements made by Sandstrom and Bergstrom [30] were shown to be in agreement with the predictions of Eq. (8), allowingalso for an influence of larger carbide particles in the finer grade steel [31]. Fig. 4 shows a second application of the Eq. (8)relationship to tensile yield and cleavage stresses plus Charpy v-notch measurements obtained on a high yield stress, HY-130, (897 MPa), steel weld metal. In this case, the combined effects of accounting for the notch constraint and greater strainrate in the Charpy v-notch test transform a rather innocuous temperature dependence of the largely athermally-basedstrength of the weld metal, when tested at a conventional strain rate, into the stronger temperature-dependent viscoplasticform shown for the predicted stresses operative in the Charpy v-notch measurements. The thus-described brittle to ductiletransition analysis was shown to be of similar form to that developed directly on a dislocation mechanics model basis byCottrell [32] and by Petch [33] and were of interest at the time in structural safety aspects pertaining to neutron irradiationembrittlement of nuclear pressure vessel steels. Wechsler has reviewed such pressure vessel steel considerations involved innuclear power developments [34,35].

Ductile–brittle transition description for HY – 130 steel weld metal: filled diamond points are cleavage fracture stress measurements made in bendpen diamond points are yield stress measurements employed in the computation of the raised yield stress dependence at higher Charpy v-notchtrain rate and with notch constraint factor; and, half-filled diamond points are CVN measurements.


4. Intrinsic brittleness

At a more fundamental level, there is interest in assessing the role of dislocation-based plasticity in affecting the intrinsicbrittleness of a material starting from the basic description of the crack tip environment and at the earliest stage of crackgrowth. Barenblatt is credited in 1959, near to the time of Irwin’s 1958 report [4],with focusing on the role of cohesion forcesat a sharp crack tip in determining a finite limit load condition for crack growth, as relatively recently reviewed [36]. Possibleplasticity is proposed to enter the analysis of crack growth then for quasi-brittle fracturing through concentration ‘‘in a small‘head’ of the crack near its contour”. In a relatively simple model assessment of a penny-shaped crack growing either throughthe Griffith mechanism or by a simple mechanism of dislocation nucleation at the crack tip, Armstrong [37] arrived at a mea-sure of the intrinsic susceptibility to brittleness of the various elements in solid form as [cS/Gb]1/2; in which cS is the truesurface energy and the product Gb is defined in Eq. (4). In more detailed analyses, Rice and colleagues have re-examinedthe model consideration in a quantitative manner, as recently reviewed [38], and a ratio of cS and a newly defined, disloca-tion shear-based, ‘‘unstable stacking energy”, cUS, was obtained to evaluate the brittle vs ductile response of bcc and fcc met-als. Fig. 5 shows a comparison of the two criteria for a number of materials [39]. A recent evaluation of [cS/Gb]1/2 = 0.24 hasbeen given for the potential intrinsic brittleness of hcp rhenium metal that also should suffer in polycrystal form from sig-nificant thermally-induced residual strains generated by the thermal expansion anisotropy [40]. Most recently, Xu has pro-vided a comprehensive analysis of a number of model considerations involved in dislocation nucleation at a crack tip and, onsuch limiting scale, the consequences for a brittle to ductile transition [41]. Zhu, Li and Yip have reported on the detailedatomic mechanism by which dislocation loops may be emitted at a crack tip in a model fcc lattice [42].

5. Crack tip induced plastic zone

The issue of plastic flow at the crack tip enters also into Irwin’s extended development of Eq. (1) as investigated at NRLand at the University of Maryland by Stonesifer [43,44]. It was proposed for quasi-brittle fracturing under a controlled pre-crack condition, assumed to be described by an equation of the same form as Eq. (1), that a plastic zone size of radius, 2r,should be added to the physical crack radius in order to produce a larger effective crack diameter and, then too, the FM stressintensity K value is implicitly increased, also. An analogous model consideration of anti-plane strain loading of a crack with astrip type plastic zone at its tip, of width s = 2r, was independently investigated on an infinitesimal distributions of disloca-tions model basis by Bilby et al. [45]. The condition for unstable crack growth was specified in the transcendental equationfor the ratio of s to c, in this case, of the form

ðs=cÞ ¼ ½secðprF=2ryÞ� � l ð9Þ

in which rF is the corresponding crack size dependent fracture stress. A series expansion of the secant function and rear-rangement of terms led to the well-approximated relationship [46]

rF ¼ ð81=2=pÞryðs=½c þ s�Þl=2 ð10Þ

For s� c, the value of s can be neglected in [c + s] and Eq. (10) has the same crack size dependence as given in Eq. (1) while,for s approaching c, the crack size dependence is equally well-approximated by

rF ¼ ryðs=½c þ s�Þ�l=2 ð11Þ

Fig. 5. Assessment of the intrinsic susceptibility to brittleness among different solid elements.


It was proposed further for Eqs. (10) and (11) that ry should be replaced by a crack-free fracture stress that for cleavagewould be rC. Fig. 6 shows early application of Eq. (11) to a compilation of results obtained in two reports of tensile fracturestress measurements made as a function of crack size for polymethylmethacrylate (PMMA) material.

On such basis of including a plastic zone correction for FM measurements, and making allowance for the conversion ofanti-plane strain to plane strain on a von Mises yield stress comparison, then, equating Eqs. (1) and (10) provides a connec-tion between K1C and the microstructural stress intensity kC value as

Fig. 6.(PMMAand, W

K1C ¼ ð8=3pÞ1=2½r0C þ kCl�l=2�sl=2 ð12Þ

In metals, and for relatively constant plastic zone size, that is generally much larger than the material grain size, Eq. (12)provides an explanation of K1C being much larger than kC. At constant plastic zone size, K1C is predicted to be linearly depen-dent on the inverse square root of the material grain size, ‘, in agreement with a compilation of measurements made on var-ious steel materials [47,48]. Fig. 7 shows a relatively more recent application of Eq. (12) to the description of indentationfracture mechanics (IFM) K1C measurements reported at small crack sizes for alumina material [49]. At larger crack sizesin alumina, corresponding to larger applied indentation loads, a reversal in the grain size dependence of K1C was measuredand was attributed by Armstrong and Cazacu [50] to s increasing with increase in the material grain size, in line with tab-ulated grain size and crack size measurements coupled with plastic zone size computations; however, the combined effect ofthe changed values of the material parameters was to continue to produce the normal expectation of a positive kC in Eq. (2).Naturally, the plastic zone has its own viscoplastic character that must be taken into account in a fuller description of mate-rial behaviors. Barenblatt [51] has provided a numerical assessment of estimated material parameters involved in a non-localmodel of ‘‘damage accumulation” associated with a previously damaged (crack or notch) site, including an Arrhenius typekinetic flow law involved in the accumulation of further damage leading to fracture; see Eqs. (7a,b,c). Qiao and Argon[52] have reported on a microstructural model of the polycrystal fracture toughness properties of several steel materialsbased on the crystallographic propagation of grain-to-grain cleavages. The model gave only a weak increase of toughness

Application of approximated Bilby/Cottrell/Swinden fracture stress dependence on crack size to results reported for polymethylmethacrylate). Experimental results are from: Berry JP. In: Rosen B, editor. Fracture processes in polymeric solids. New York: Interscience Wiley; 1964. p. 195ff;illiams JD, Ewing RD. In: Fracture 1969. London (UK): Chapman and Hall; 1969. p. 11/1ff.

Fig. 7. Dependence on grain size of alumina K1C measurements obtained by the method of indentation fracture mechanics (IFM).


with grain size reduction and was concluded to lead, at small grain sizes, to a constant value close to three times the singlecrystal value.

6. WC-Co composites

Armstrong and Cazacu [50] also investigated the application of Eq. (12) to research results reported for industriallyimportant composite WC-Co materials whose exceptionally high hardness properties were known to be significantly depen-dent on the size of WC particles embedded in a thinly encompassing Co binder phase. Lee and Gurland [53] had provided arelationship for the diamond pyramid hardness, HV, properties of different composite compositions and sizes of the constit-uents as

Fig. 8.ExperimGalloisvon Ru

HV ¼ HWCVWCC þ HMð1� VWCCÞ ð13Þ

in which HWC and HM for the hardnesses of the WC and matrix Co phases, respectively, were both shown to follow their ownH–P dependences that were measured separately, VWC is the volume fraction of WC particles, and C is the contiguity of theWC particles, specified as the relative number of WC to WC particle contacts. In [50], a simple graphical procedure was devel-oped to demonstrate the combined influences of the H–P size dependencies for the WC particles and Co matrices along withthe contiguity-modified volume fraction of WC particles in determining the composite hardness value. Of greater interesthere, however, was the application of Eq. (12) to describing a combination of IFM and FM measurements reported for theK dependence on mean free path of the Co binder phase, k, taken both as the appropriate size parameter for the H–P depen-dence and for the plastic zone size, s, for several reported composite formulations; see Eq. (17) of reference [47]. Fig. 8 showsthe reasonable agreement obtained for the dependence on k1/2 of previously reported K measurements in which fracturingwas attributed to the properties of the Co binder phase. Sigl and Fischmeister [54], in particular, had shown previouslythrough other microstructural analyses and model assessments that the FM toughness properties were controlled by theCo binder phase. Recently, Shanmugam et al. [55], in distinguishing between morphological aspects of pearlite influencesaccompanying thermal processing of niobium- and vanadium-microalloyed steels, gave emphasis to the contiguity of theferrite phase in affecting the material impact toughness properties.

7. Cracking on a hardness stress–strain basis

The IFM method of determining material toughness properties, according to measurements of a variety of indentation-produced cracking behaviors, has been usefully applied to a number of ceramic-like material systems, basically of relativelygreater hardness but generally of relatively lower material toughnesses, following on from the pioneering researches of Lawn[56], in extension of earlier Griffith-type indentation cracking analysis done with F.C. Frank. Lawn’s experiments on singlecrystal silicon were re-visited by Armstrong et al. [57] who presented evidence for plastic flow being involved in the crackingproduced in ambient temperature hardness measurements. Even for high hardness plus high toughness plasma-transferredarc-welding (PTAW) steel material, however, Branagan et al. [58] have employed K1C measurements obtained via Palmquist-type indentation testing to assess toughness measurements relating to the material use in industrial hardfacing applicationsrequiring resistance to abrasion and impact. Armstrong and Elban [59] have also employed the IFM method to provide anindentation hardness stress–strain method of analysis to assess the elastic, plastic and cracking behavior of materials; seeFig. 9 for the hardness stress, H, expressed as the mean pressure on a ball indenter, vs a hardness strain defined as the ratioof the contact diameter, d, and indenter ball diameter, D. At initial loading, and corresponding small elastic strains, H is spec-ified on a Hertzian indentation basis as

H ¼ ð4=3pÞf½ð1� m2S Þ=ES� þ ½ð1� m2

BÞ=EB�g�1ðd=DÞ ð14Þ

FM and IFM stress intensity, K, measurements as a function of Co mean free path, k, for several WC-Co (cemented carbide) alloy materials.ental results are from: Sigl LS, Fischmeister HF. On the fracture toughness of cemented carbides. Acta Metall 1988;36:887–97; Jia K, Fischer TE,

B. Microstructure, hardness, and toughness of nanostructures and conventional WC-Co composites. Nanostruct Mater 1998;10:875–91; Richter V,thendorf M. On hardness and toughness of ultrafine and nanocrystalline hard materials. Int J Refract Met Hard Mater 1999;17:141–52.

Fig. 9. Indentation hardness stress – strain basis for assessing the elastic, plastic, cracking behaviors of materials; the Hertzian elastic loading lines arecomputed for a steel ball indenter.


in which the subscript indices on the previously defined parameters are for the specimen and ball, respectively. On the samebasis, the critical hardness, HC, for IFM-modeled Griffith-type cracking in the indentation stress field is specified along theelastic loading line [56] as

HC ¼ f4ESc=½pDð1� m2S Þðj2

1 þ j22Þ�g

1=2ðd=DÞ�1=2 ð15Þ

in which ðj21 þ j2

2Þ was specified as a dimensionless numerical factor equal to 2.5 � 10�5.In Fig. 9 then, a number of dashed Hertzian elastic loading lines are shown for a variety of materials, including results for

the oxidizer crystal, AP (ammonium perchlorate), and other loading lines for a number of energetic crystal materials: PETN(pentaerythritol tetranitrate), RDX (cyclotrimethylenetrinitramine), and HMX (cyclotetramethylenetetranitramine). ForNaCl, the solid elastic loading line and subsequent curve for plastic straining were determined in a continuous indentationhardness test performed with a 1.59 mm steel ball indenter pressed into an (0 0 1) crystal surface. At the top end of a numberof the dashed elastic loading lines and the NaCl one are values of HC (=rC) computed for particular values of D. The top-mostpoint on the MgO line, of D = 0.124 mm, is the equivalent ball diameter value corresponding to the averaged diamond pyr-amid indentation hardness measurements that are shown at lower hardness pressures. At larger D = 6.35 mm, a lower com-puted MgO cracking stress is plotted, as predicted by Eq. (15). The actual hardness measurements for MgO and othermaterials are plotted in the figure at an effective value of (d/D) = 0.375. The abscissa displacement of the average MgO hard-ness point from the elastic loading line is a measure of the effective plastic strain produced in the hardness test. In this way,the indication for MgO, and other even harder materials, is that the elastic and plastic indentation strains are comparable invalue and, thus, have influence, as is known experimentally, on the residual indentation shapes [60]. In this regard, the rel-atively recent advent of instrumented nanoindentation hardness testing has provided for quantitative evaluation of Eq. (14)in a number of investigations, for example, as reported by Armstrong, Ferranti and Thadhani [61].

The results presented in Fig. 9 for energetic crystals, that are molecularly-bonded, have the interesting mechanical prop-erty characteristics of being elastically compliant, plastically hard, and relatively brittle. Such property characteristics,including assessment of potential viscoplastic aspects of deformation preceding cracking of the crystals, are usefully assessedby comparison of the results shown for RDX and NaCl. First, the elastic loading of RDX occurs at lower hardness stress thanfor NaCl, as shown, in agreement with the respective ES and mS values contained in Eq. (14). The RDX crystal lattice is molec-ularly-bonded and, therefore, is more elastically compliant than is ionically-bonded NaCl. On the other hand, the plastichardness of RDX is greater than that of NaCl because of molecular configurational restrictions on dislocation plasticity inthe RDX lattice structure, for example, as modeled by Armstrong and Elban [62]. Lastly, there is the lower cracking stressshown on the elastic loading line for RDX, compared to NaCl, in line with a low value of the RDX susceptibility to brittlenessfactor, [cS/Gb]1/2 = 0.066; see Fig. 5. And, for both materials, the range in hardness stress between the plastic flow and crack-ing values is available for possible viscoplastic influence. The much lower range in stress between plastic hardness and brittle


(cleavage) cracking for RDX provides an explanation, in part, for the positive correlation reported for energetic material hard-ness values and the sensitivity of the materials to explosive initiation in impact tests.

8. Summary

Pioneering concerns about viscoplasticity aspects of the microstructurally-influenced cracking and fracturing behavior ofmaterials, as raised by George Irwin in his own researches and in interaction with colleagues, have been traced along a num-ber of paths through a significant range of material-type cracking and fracturing considerations. For pressure vessel and re-lated steel materials, several Irwin-influenced model descriptions have been given for evaluation of the ductile to brittletransition in behavior, and its reverse, also bearing on importantly related, microstructurally-based, researches of NormanPetch and colleagues. Issues of dislocation plasticity enter into models of cracking beginning from the earliest stages ofgrowth. Additional fracture mechanics based dislocation viscoplasticity influences are highlighted in the description of re-sults obtained via the method of indentation fracture mechanics.

Acknowledgements

It’s a great pleasure to recall the very positive weekly discussions held on these topics over several years with ProfessorGeorge Irwin and Dr. X. Jie Zhang at the University of Maryland, as briefly mentioned previously [63]. Dr. John Griffiths,CSIRO Process Science and Engineering, Queensland, Australia, is thanked for providing a number of helpful editorialcomments.

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Dislocation viscoplasticity aspects of material fracturing