Strength vs. toughness optimization of microstructured composites

Transcription

1 Chaos, Solitons and Fratals 39 (29) Strength vs. toughness optimization of mirostrutured omposites Alberto Carpinteri *, Niola Pugno, Simone Puzzi Department of Strutural Engineering and Geotehnis, Politenio di Torino, Corso Dua degli Abruzzi 24, 1129 Torino, Italy Aepted 25 May 27 Abstrat Aim of this paper is to present a new fratal approah linking the marosopi mehanial properties of miro- and nano-strutured materials with the main parameters: omposition, grain size and strutural dimension, as well as ontiguity and mean free path. Assuming the key role played by the interfaes, the proposed fratal energy approah unifies the influenes of all the above parameters, through the introdution of a fratal strutural parameter (FSP), whih represents an extension of the Gurland s strutural parameter. This modeling approah is assessed through an extensive omparison with experimental data on poly rystalline diamond (PCD) and WC Co alloys. The results learly show that the theoretial fratal preditions are in a fairly good agreement with the experiments on both hardness and toughness. This new syntheti parameter is thus proposed to investigate, design and optimize new miro- and nano-grained materials. Eventually, FSP-based optimization maps are developed, that allow to design new materials with high hardness and toughness. Ó 27 Elsevier Ltd. All rights reserved. 1. Introdution Composites onsisting of a brittle hard material, whether erami or intermetalli, reinfored with a dutile phase, are very important in several industrial appliations, due to their mehanial properties, whih inlude high hardness, wear resistane and toughness. From an experimental point of view, a diret relationship between the fabriation proess, the mirostruture and the properties of these materials has been evidened [1 3]). In the Literature several approahes, experimental, numerial and theoretial, have been proposed to orrelate the marosopi properties with the material mirostruture, but a general theory for the dependene of hardness and toughness of omposites suh as WC Co alloy or poly-rystalline diamond (PCD) on mirostrutural parameters has not been presented yet. In partiular, tungsten arbide obalt (WC Co) alloys, due to their importane in industrial appliations, have been investigated in great detail by several authors, who analysed the mirostrutural details and tried to pinpoint the most important parameters affeting the marosopi properties [4 14]. Several * Corresponding author. Tel.: ; fax: address: alberto.arpinteri@polito.it (A. Carpinteri). URL: (A. Carpinteri) /$ - see front matter Ó 27 Elsevier Ltd. All rights reserved. doi:1.116/j.haos

2 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) Nomenlature C D d d r H h K l N p(æ) R r r min r max S g V V g v v r W W a k r k K k Kr r ontiguity fratal exponent mean grain size referene grain size hardness mean free path toughness harateristi rak length number of grains probability density funtion strutural size grain size minimum grain size maximum grain size total grain surfae volume total grain volume volumetri fration of grains referene volumetri fration of grains total dissipated energy dissipated energy per unit volume saling exponent fratal exponent model parameter model parameter model parameter strength different approahes have been proposed for modeling both hardness (or strength) and toughness of these materials, aiming at tailoring the mehanial properties of the omposite materials by ontrolling the mirostruture all along the fabriation proess [15 21]. Regarding WC Co alloys, the relation between frature strength (and frature toughness) and mirostruture is a very attrating question, sine experiments have learly shown that these properties depend not only on the mean grain size d of WC but also on its volume fration v, the mean free path h of the Co binder and the ontiguity C of WC partiles [15,16]. The hoie of the independent parameters is still not lear and matter of researh, partiularly regarding the role of ontiguity [2]. In their pioneering paper, Lee and Gurland [16] first pointed out the role of ontiguity C and onluded that this parameter ould not be left out in any modeling attempt, whilst more reently Makhele-Lekala et al. [2] expliitly aimed at the derivation of an empirial formula relating hardness to WC grain size d and mean free path h of Co, without expliit inlusion of ontiguity. A similar result was obtained by Liu et al. [22] on the basis of a miromehanial model for polyrystalline aggregates with annular flaws surrounding the grains. Golovhan and Litoshenko [23] strengthened the view whih onsiders ontiguity not as a fundamental parameter, evidening a dependeny of C on the oeffiient of variation of the WC grain size, whilst Armstrong and Cazau [21] again inlude it in their analysis. The role of Cobalt mean free path h in ontrolling the frature toughness of WC Co alloys, first pointed out by Sigl and Fishmeister [24], is also not ompletely understood. In spite of the large number of theoretial and experimental investigations, a unified approah to the strength and toughness of these materials is at its early stages [25]. Aim of this paper is the generalization and unifiation of the influenes of the effets of all the above parameters, for both strength and toughness. In addition, the proposed model will also inlude the effet of the strutural dimension R [26], that will be defined later. It is worthwhile noting that in the fratal model, that we are going to present, the mean free path h of the Co binder and the ontiguity C of WC partiles are not inluded diretly, sine their effets are impliitly taken into aount by the proposed fratal strutural parameter.

3 1212 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) Composition, ontiguity, grain size, mean free path and strutural size The effet of omposition an be taken into aount by means of mixture rules. Within these rules, the mehanial behaviour of the mixture is usually assumed to be independent of the grain and strutural sizes. On the other hand, this result is learly an oversimplifiation, as experiments demonstrate. Even if more sophistiated estimates of the overall tensor mehanial properties of heterogeneous solids and a omparison with the most ommon rules of mixture have been reently proposed [27,28], a lear understanding of the phenomenon is still absent in the literature. Nevertheless, indiating with r the strength of a miro- or nano-strutured material, whih is omposed by a matrix having strength r hmatrixi rule) would predit: and grains with volumetri fration v and strength r hgraini, the simplest rule of mixture (diret r ¼ð1 vþr hmatrixi þ vr hgraini : ð1þ A similar mixture rule may also be written for the frature toughness K : K ¼ð1 vþk hmatrixi þ vk hgraini : ð2þ The role of ontiguity C, whih is defined, aording to Gurland [15], as the average fration of surfae area shared by a grain with all neighbouring partiles of the same phase, may be taken into aount by simply substituting the real volume fration v of eah phase with its ontinuous fration v (C), that an be evaluated through the ontiguity C as v ðcþ ¼ vc: As a onsequene, the role of the ontiguity an be simply onsidered substituting in the rules of mixture, at the plae of the real volume fration v, the effetive one v (C) [16]. This approah is still popular and re-proposed also in the paper by Armstrong and Cazau [21], although it is quite lear that the ontiguity is not an independent parameter [23], sine it depends on d and v, see Appendix A. The effet of the grain size may also play a onsiderable role. As the grain size dereases, there is a signifiant inrease in the volume fration of grain boundaries or interfaes. This harateristi strongly influenes the hemial and physial properties of the material. In materials that are onventionally quite strong but very brittle, suh as intermetalli ompounds and eramis, grain size redution enhanes dutility through the inreased probability of grain boundary sliding, offering onsiderable proessing advantage and performane improvements. In the ase of metals, on the other hand, the derease in grain size may bring to signifiant inreases in yield strength and elasti modulus, through the introdution of additional grain boundaries that an at as effetive barriers to disloation motion. This effet is usually desribed by means of the well-known Hall Peth law [29,3]: ð3þ r k / d 1=2 ; ð4þ where d is the mean grain size (and k is a onstant). The effet of the Co mean free path h on strength is again of the Hall Peth type and similar to that reported in Eq. (4) if d is replaed by h [16]. Its effet on the frature toughness is less lear and various dependenies of K on h have been proposed: aording to Ravihandran [19], K k /h (k is a onstant), whilst Armstrong and Cazau [21] propose a dependene of the form: K k /h 1=2. This unertainty will be overome in the forthoming treatment, in whih the effet of the mean free path is not onsidered, sine this variable depends on the volume fration v of hard phase and on the grain size d. The effet of all these parameters are usually dealt with in a asade-like approah [16,21]. In the ase of hardness (or strength), for instane, first a mixture rule of the type of Eq. (1) is written, then the effets of grain size d and mean free path h are inluded by replaing r hmatrixi and r hgraini with Hall Peth empirial relationships (see Eq. (4)). The effet of ontiguity C may be taken into aount by replaing the volume fration with the ontinuous volume fration v (C) of Eq. (3), as already stated. Regarding the strutural size R, whih is a harateristi length of the struture (e.g. the beam height in a three-point bending speimen, or the diameter in a ylindrial speimen), there is a large literature on its effet, whih is usually alled the size-sale effet. Dealing speifially with grained materials, it has been found experimentally that strength dereases with the strutural size, whereas frature energy inreases [31,32]). All these effets will be inluded in the fratal treatment, that we are going to present.

4 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) Energy saling In order to predit the influene of grain size, strutural dimension and omposition (volumetri grain ontent) in a unified manner, a fratal approah is herein onsidered. Contiguity and mean free path are onsidered dependent parameters. Aordingly, we assume a fratal (or self-similar) distribution of reinforement grains, for whih the probability density funtion p(r), desribing their distribution in size r in the range ½r min ; r max Š has to be written as [33 35]: D pðrþ ¼ r D min D r ffi D rd min Dþ1 r Dþ1 1 rmax r min where 2 6 D ¼ is the fratal exponent, r min is the size of the smallest grain and r max that of the largest grain. Notie that we assume r min r max, so that the first denominator in the previous equation is very lose to unity and the above approximation is justified. On the other hand, different types of grain size distribution an be onsidered as limit ases of the fratal approah, as disussed for the Gaussian by Carpinteri and Pugno [25]. The main assumption of the theory is that energy dissipation due to the presene of the reinforement phase ours only at the interfaes. It onsists of different phenomena, suh as onstrained deformation of the matrix between grains, plastiity and frature propagation. In partiular we assume that the energy dissipation W, onneted to the presene of the grains, is statistially expeted to be proportional to their total surfae area S g [33], i.e.: W / S g / Z rmax r min r 2 dn / Z rmax r min N r 2 pðrþdr / N r 2 min ; ð6þ where r max is the size of the largest grain and N is the total number of grains. On the other hand, the total volume V g of the grains is V g / Z rmax r min r 3 dn / Z rmax r min N r 3 pðrþdr / N r D min r3 D max : Notie that both equations are obtained under the assumption that r min r max. Deriving the expression of N from the previous equation and putting it into Eq. (6), yields: W / V g r 2 D min rd 3 max : As a onsequene, the energy dissipated in the grains per unit volume V of the speimen beomes: ð5þ ð7þ ð8þ W C ¼ W V / vr2 D min rd 3 max ; ð9þ where v p ¼ V g /V represents the volumetri fration of the grains. Assuming the statistial self-similarity hypothesis, i.e., r max / ffiffiffiffiffi 3 V g (the larger the total volume of grains, the larger the largest grain), the effet of the strutural size (size effet) an be obtained as W C / vr 2 D min V D 3 3 g : ð1þ Usually r min is assumed to be a onstant, i.e., a material property. The mean value of the external surfae of grains, proportional to hr 2 i, as well as of their volume, proportional to hr 3 i, are substantially estimated in Eqs. (6) and (7). On the other hand, the evaluation for the mean value of the grain size r gives: Z rmax hri / 1 Z rmax rdn / rpðrþdr / r : min ð11þ N r min r min Introduing D ¼ 3 and Eq. (11) into Eq. (1), noting that V R 3 with R strutural size, and d hri being the grain size, we an write: W C / v d 2 3 R 3ð 1Þ ; that represents the saling law of the additional energy density dissipated as a onsequene of the presene of the grains and interfaes. It is worthwhile noting that this relation impliitly onsiders the grains embedded in the matrix; as a onsequene, this energy density is not intrinsi to the grains only, but to the grains in the matrix. Eq. (12) is of great importane to predit the inrement in strength due to the presene of grains, as disussed below. It is also notieable that this result is onsistent with findings by Carpinteri and Pugno [33 35] from fragmentation theories, see Appendix C. ð12þ

5 1214 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) Grain and strutural size effets Aording to the fratal approah, the relationship of Eq. (12) represents the unified law to evaluate the grain and strutural size-effets, as well as the influene of omposition, on the dissipated energy density. Let us fous our attention onto a struture of size R, onstituted by a two-phase material, with only one grained phase of mean grain size d and volumetri perentage v. Noting that the square root of the dissipated energy due to the presene of the grains per unit speimen volume (if we assume negligible size-effets on the elasti modulus) an be onsidered to be proportional to the strength of the grains (not intrinsi, but in the matrix), and oupling Eq. (12) with a lassial rule of mixture gives the following unified saling law: r ¼ð1 vþr hmatrixi þ vr hgraini ; ð13aþ r hgraini k r v 3 2 R ð 1Þ 2 d ; ð13bþ where k r is a onstant and r hmatrixi is the strength of the matrix. It is interesting to observe that, onsidering self-similar strutures with the same saling for the grain sizes, i.e., d / R, v = onst., the size effet on strength does not vanish see Eq. (13): r / R 1 2 ; if we assume that the seond term in Eq. (13a) is the dominating one. On the other hand, the exponent disappears. This seems to suggest that the size-effet of Eq. (14) is intrinsi, whereas the hange of the exponents in the saling laws ( geometrial multifratality) is a onsequene of the ompetition between the grain size and the strutural dimension, as demonstrated by means of a statistial approah by Carpinteri et al. [36,37]. The developed fratal theory unifies the grain and strutural size effets and is in agreement with well-known sizeeffet laws [25], as well as with the Hall Peth relationship ( = 1; note that this orresponds to different grain size distributions, suh as Gaussian [25], see Appendix B). We also note that Eq. (13) represent an extension of the Gurland [15] strutural parameter, defined as r / k þ v 1 3 d 1 2 and more generally an extension inluding the volumetri ontent of the Hall Peth relationship [25]. For the hardness (or for the wear resistane) the same saling of Eq. (13) is expeted sine: H / r : The relationship analogous to the Hall Peth law, has been experimentally onfirmed for the hardness H, as an empirial law [15,16] for arbon-based materials, in agreement with the saling of Eq. (13b). An approximated law also an be proposed for the frature toughness, with respet to the p ffiffisize and the volumetri ontent of grains. Aording to Griffith, if the harateristi rak length is l, K hgraini / r hgraini l. Two plausible hypotheses for l are l / R or l / d; thus, in general, we assume l / d 1 2a R 2a where a; < 2a < 1, is a onstant. Aordingly, a lassial rule of mixture yields: K ð1 vþk hmatrixi K hgraini þ vk hgraini ; ð17aþ k K r hgraini d 1=2 a R a ; ð17bþ where k K is a onstant and K hmatrixi is the frature toughness of the matrix. ð14þ ð15þ ð16þ 5. Fratal strutural parameter and deviations from the empirial Hall Peth law The preditions of the fratal approah for the strength or hardness and for the frature toughness are desribed by Eqs. (13) and (16) Aordingly, we an define a Fratal Strutural Parameter (FSP) as for whih: FSP ¼ v 2 R 3 2 ð 1Þ d r hgraini / FSP: ð19þ Previous investigations on fragment size distributions [34] seem to suggest = 2/3 at small sale, as well as = 1 at large sale. These limit ases, together with their intermediate referene ase, orrespond respetively to: ð18þ

6 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) ¼ 2=3 : FSP / v 1 3 R 1=2 d ; ð2aþ ¼ 5=6 : FSP / v 5 12 R 1=4 d 1=4 ; ð2bþ ¼ 1 : FSP / v 1 2 R d 1=2 : ð2þ Sine r C / k þ FSP, Eqs. (2) indiate a deviation of the exponent in the Hall Peth relationship of Eq. (4). It is interesting to observe that Masamura et al. [38] desribe a similar deviation for the exponent of the Hall Peth relationship. The experiments on fine grained miro- and nano-strutured materials seem in fat to suggest three different regions for the exponent of the Hall Peth relationship: a region for single rystal to a grain size of about 1 lm, where the exponent seems to be lose to 1/2. a region for grain sizes ranging from about 1 lm to 3 nm, showing exponents lower than 1/2. a region below a very small ritial grain size (of about 3 nm), with the exponent equal to zero. Note that, in this ase, the reinforement grain size effet is null, whereas the size-sale effet (i.e. the effet of the strutural size R) is maximum. Thus, it appears that the fratal approah is not only able to justify the empirial Hall Peth law, but also to predit the deviation of its exponent. The inverse Hall Peth behaviour (i.e. softening assoiated with grain refinement), experimentally observed by several Authors, may be interpreted in the present framework by using values of smaller than 2/ 3. In this ase the dependeny of strength on the grain size beomes positive, sine the exponent 1 ð3=2þ in Eq. (18) hanges its sign. On the other hand, it is lear that both saling laws (Eqs. (13) and (16)) are monotoni laws, therefore not apt to desribe a transition from the usual Hall Peth behaviour to the anomalous, inverse one. Therefore, in the following assessment on experimental data, only monotoni data sets will be onsidered. 6. Experimental assessment of the FSP on WC Co alloys and PCD It has been already shown that the speimen size effets predited by Eqs. (18) and (19) agree with the experiments satisfatorily (see [35]), so that we are herein more interested in the effets of grain size and volumetri ontent of grains. Eqs. (13) for strength (or hardness) may be rewritten as y ¼ ðr ð1 vþr hmatrixi ÞR 3 2 ¼ k vdr 3 r v R2d 3 2 ¼ kr x : ð21aþ 2 Therefore, the fratal approah implies a power law between y and x with an exponent omprised between 2/3 and 1. This is equivalent to verify the validity of the FSP-based approah. The advantage here is that the exponent, as well as the onstant of proportionality k r, an be easily obtained as best fit parameters. When the material strength is predited, then the frature toughness an be also estimated by Eqs. (16): y ¼ ðk ð1 vþk hmatrixi ÞR 3 2 þ ¼ k v 1þ 2d 3 3 K k r R 3 2 d a 2 ¼ kkr x þ2 3 a ; ð21bþ 2 R2 by fitting k Kr, a. We refer to the experimental investigation on fine grained PCD, reported by Lammer [39] and Huang et al. [4]. In addition, to assess the FSP on a larger data set, we also refer to the extensive experimental investigations on miro- and nano-grained WC Co alloys available in the Literature [16,9,24,7,4,8,5,11,6,1,41,12]. In Fig. 1, the experimental assessment of the fratal theory aording to Eq. (21) (assuming a unit speimen size; all the parameters are expressed in [SI] units), is presented for the strength and toughness of PCD. Data set [39] has been shifted vertially by unity for larity reasons. Suh experiments strongly onfirm the predition of the fratal approah. The exponents, whih are the slopes in the strength diagrams, are found belonging to the theoretial range for eah onsidered material. In Fig. 2 the same analysis is arried out for the strength and toughness of WC Co alloys, onfirming the same findings. The reported data sets have been taken from: [4 6,8 1,12,15 17,2,24,41]. Also in this ase the data sets have been shifted vertially by unity for larity reasons. In eah ase, the agreement between FSP-based mixture rule and experiments is very good, as shown also by values of the regression parameter R 2. Three of these data sets are not omplete and report hardness data only [15,16,2].

7 1216 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) a PCD Strength: omparison theory vs. experiments b PCD Toughness: omparison theory vs. experiments log(y) log(x) ref. [39] y=.8964 x R 2 =.985 ref. [4] y=.7812 x+8.39 R 2 =.996 log(y) log(x) ref. [39] y=.9361 x R 2 =.942 ref. [4] y=.8695 x R 2 =.994 Fig. 1. PCD Comparison between theory (straight line) and experiments (points): strength (a) and toughness (b). Data set [39] has been shifted vertially by unity for larity reasons. a log(y) WC Co alloy Strength: omparison theory vs. experiments 28 ref. [4] y=.7973 x+8.27; R 2 =.955 ref. [5] 26 y=.754 x ; R 2 =.93 ref. [6] y=.733 x+9.31; R 2 =.7178 ref. [8] 24 y=.77 x ; R 2 =.9952 ref. [9] y=.7197 x+8.852; R 2 =.9194 ref. [1] 22 y=.6696 x ; R 2 =.9864 ref. [12] y=.7643 x ; R 2 =.998 ref. [15] 2 y=.9369 x ; R 2 =.9863 ref. [16] y=.9459 x ; R 2 =.9541 ref. [17] 18 y=.7492 x ; R 2 =.975 ref. [2] y=.832 x ; R 2 = ref. [24] y=.775 x ; R 2 =.993 ref. [41] y=.7845 x ; R 2 = log(x) b WC Co alloy Toughness: omparison theory vs. experiments log(y) ref. [4] 28 y=.6114x ; R 2 =.788 ref. [5] y=.5939 x+1.48; R 2 =.937 ref. [6] 26 y=.8318 x ; R 2 =.935 ref. [8] y=.6264 x+1.316; R 2 = ref. [9] y=.5478 x ; R 2 =.528 ref. [1] y=.6998 x ; R 2 = ref. [12] y=.8657 x ; R 2 = ref. [17] y=.7137 x ; R 2 = ref. [24] 16 y=.7661 x ; R 2 =.99 ref. [41] y=.6876 x ; R 2 = log(x) Fig. 2. WC Co alloys Comparison between theory (straight line) and experiments (points): strength (a) and toughness (b). Data relative to the inversion of the Hall Peth law have been disarded and reported in Fig. 3. All data sets (exept [41]) have been shifted vertially for larity reasons. It is worthwhile noting that the FSP-based mixture rules fit the experiments on hardness better than those on frature toughness; this trend is partiularly evident with the data sets by Nakamura and Gurland [9] and Sadahiro and Takatsu [1]. As already stated, both saling laws (Eqs. (13) and (16)) are monotoni laws, therefore not apt to fit data desribing a transition from the usual Hall Peth behaviour to the anomalous, inverse one. As a onsequene, only data sets

8 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) a WC Co alloy Strength: omparison theory vs. experiments log(y) ref. [7] y=.6317 x ; R 2 =.94 ref. [11] y=.4819 x+1.863; R 2 =.864 ref. [17] y=.6341 x+9.528; R 2 =.913 ref. [24] y=.6284 x ; R 2 = ref. [15, v=.69] y=.5297 x+1.64; R 2 =.995 ref. [15, v=.75] y=.6223 x ; R 2 = ref. [15, v=.81] y=.5284 x+1.483; R 2 =.989 ref. [15, v=.9] 17 y=.525 x+1.544; R 2 = log(x) b log(y) WC Co alloy Toughness: omparison theory vs. experiments log(x) ref. [7] y=.6327 x+1.274; R 2 =.934 ref. [11] y=.9292 x+7.771; R 2 =.975 ref. [17] y=.735 x+9.729; R 2 =.7617 ref. [24] y=.7851 x+9.49; R 2 =.989 Fig. 3. WC Co alloys Comparison between theory (straight line) and experiments (points): strength (a) and toughness (b). Data relative to the inversion of the Hall Peth law only. All data sets (exept [15]) have been shifted vertially for larity reasons. displaying a monotoni trend in hardness (or strength) vs. grain size d have been fitted. In the ase of data sets displaying both Hall Peth ranges [15,17,24], data relative to the inverse Hall Peth behaviour have been disarded and then fitted separately in Fig. 3, together with two more data sets [7,11], whih display the inverse Hall Peth behaviour only. One again, different data sets have been shifted vertially by unity for larity reasons. Also in this ase the agreement between the proposed approah and the experiments is remarkably good. It is worthwhile noting that the inverse Hall Peth behaviour is desribed onsistently by a exponent falling outside the anonial range, i.e., lower than 2/3 (see Fig. 3a). In the ase of the data set by Gurland [16], however, it results that the effet of omposition v is not aptured by the FSP (see Fig. 3a). In other words, data do not ollapse onto the same line and different values of v arrange themselves on different lines. We also notie that, with these data sets, the FSP-based mixture rules fit the data of frature toughness better than those of hardness, in ontrast to what happened with the previous data sets. Regarding toughness, if we onsider the data sets by Sigl and Fishmeister [24] and Pikens and Gurland [17], whih display both Hall Peth ranges, it may be noted that the slope in the toughness diagram does not hange signifiantly (ompare Fig. 2b with Fig. 3b). Summarizing, the FSP of Eq. (18) represents an important tool to predit grain size effets, as well as the influene of the volumetri grain ontent, for miro- and nano-grained materials. In addition, the FSP may also desribe the influene of the strutural size on material properties. 7. Material design and optimization From the FSP-based laws (Eqs. (13) and (16)), strength and toughness an be derived as funtions of volumetri grain ontent, grain size and strutural dimension. Aording to the FSP-based laws in whih the onstant of proportionality and the exponents are fitted from the experiments by Huang et al. [4], the following laws for PCD (in [SI]) are found: r ðd; vþ ð1 vþr hmatrixi þ v 1:391 d :172 ; ð22aþ K ðd; vþ ð1 vþk hmatrixi þ v 1:391 d :196 ; ð22bþ with r hmatrixi 5 MPa, K hmatrixi p 2 MPa ffiffiffiffi m. The strutural dimension R is here negleted (and fititiously assumed to be equal to the unity in [SI]) as a onsequene of the idential sizes for the investigated speimens. In addition, onsidering a referene PCD with grain size d r and volumetri ontent v r, we an dedue the inrements in strength (or hardness H / r Þ and frature toughness expeted by using a new PCD material, designed with different grain size d and volumetri ontent v. For example, for the hardness we would have to evaluate the ratio

9 1218 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) Fig. 4. PCD optimization map. Iso-hardness lines are drawn in blue and iso-toughness lines in red. Numbers along the urves indiate hardness and frature toughness inrements % [4]. ½Hðd; vþ Hðd r ; v r ÞŠ=Hðd r ; v r Þ1%. InFig. 4, the inrements of the hardness and of the frature toughness for a standard PCD (v r :9, d r 3 lm, r hmatrixi 5 MPa, K hmatrixi p 2 MPa ffiffiffiffi m Þ are reported. The experimental range for the grain size is from 5 to 9 lm, whilst that for the volumetri ontent is from.89 to.95; thus, large part of the graph is, stritly speaking, an extrapolation. This diagram an be onsidered as a PCD optimization map. Note the small (orange) region where both the inrements are positive: a new PCD belonging to this region will be harder and tougher at the same time (with respet to the referene one). Basially, the inrement in hardness is given by the inrease in the volume fration of hard phase, whih ounterbalanes the inrease in d. The inrement in frature toughness, on the other hand, is given by the rising of the grain size d, whih ounterbalanes the derease of the dutile phase volume fration. The same operation has been performed also in the ase of the WC Co alloy. Aording to the FSP-based laws in whih the onstant of proportionality and the exponents are fitted from the experiments on nano-strutured alloys reated by Jia et al. [12], the following laws (in [SI]) are found: r ðd; vþ ð1 vþr hmatrixi þ v 1:382 d :146 ; ð23aþ K ðd; vþ ð1 vþk hmatrixi þ v 1:382 d :21 ; ð23bþ with r hmatrixi 5 MPa, K hmatrixi p 2 MPa ffiffiffiffi m. The strutural dimension R is again negleted (and fititiously assumed to be equal to the unity in [SI]), sine all speimens ome from the same data set [12]. Considering a referene alloy with grain size d r and volumetri ontent v r, we an dedue the inrements in strength (or hardness H / r Þ and frature toughness expeted by using a new WC Co material, designed with different grain size d and volumetri ontent v. Fig. 5. WC Co alloy optimization map. Iso-hardness lines are drawn in blue and iso-toughness lines in red. Numbers along the urves indiate hardness and frature toughness inrements % [12].

10 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) In Fig. 5, the inrements of the hardness and of the frature toughness for a referene WC Co Alloy (v r :8, d r 2 lm, r hmatrixi 5 MPa, K hmatrixi p 2 MPa ffiffiffiffi m Þ are reported. This diagram may be onsidered as a WC Co alloy optimization map. In this ase, the experimental range for the grain size is from 7 nm to 2.5 lm, whilst that for the volumetri ontent is from.69 to.9. In omparison with the previous ase, where the range for the grain size d did not extend below 1 lm, the differene is given by the presene of two regions (marked in orange) where both the inrements are positive. One of them is similar to that in Fig. 4: in fat, it is haraterized by a simultaneous inrease in both d and v. The seond region, whih extends itself towards the left, i.e. towards nano-sized grains, is haraterized by a simultaneous derease in both d and v. Thus, the mehanisms whih produe better material properties are opposite to those operating in the other region. The hardness inreases thanks to the derease in the grain size, whih ounterbalanes the derease in the volume fration v of hard phase. On the other hand, frature toughness inreases, despite the grain size derease, thanks to the inrease in the dutile phase volume fration. This remark suggests that better nano-grained omposites should be produed by oupling nano-sized grains with volume perentages smaller than those usually employed in onventional miro-grained alloys. 8. Conlusions A generalization of the Hall Peth law for grain size effet on material strength, r C / k þ d 1=2, was introdued by Gurland [15], asr / k þ v 1 3d 1 2, to take into aount the volumetri grain ontent. Our approah represents a further generalization of the Gurland s parameter, introduing a fratal strutural parameter (FSP), in whih the saling related to the strutural dimension R is also inluded and in whih the fratal exponents are generalized, i.e., r C / k þ v2r 3 2 ð 1Þ d The fratal approah seems to present at the same time the fasinating advantages of inorporating the lassial approah with different types of grain size distributions (e.g., the Gaussian and the Log-normal), the influene of the omposition, as well as reasonable grain and strutural size-effets. In addition, it provides hints for the design of onventional, as well as nano-strutured brittle materials reinfored with a dutile phase. For these reasons, and in virtue of a very broad suessful assessment with experimental data from the Literature, the Fratal Strutural Parameter (FSP) ould help in designing optimized innovative miro- and nano-strutured materials. Aknowledgements The finanial support of the European Union to the Leonardo da Vini Projet I/6/B/F/PP Innovative Learning and Training on Frature (ILTOF) is gratefully aknowledged. Appendix A. The role of ontiguity and mean free path Let ontiguity C i be defined for a given phase i. We an extend the onept defining the ontiguity C ij of the phase i with respet to the phase j as the average fration of surfae area shared by the grains of the phase i with all neigh-boring partiles of the phase j. Aording to suh definition, an estimation of the ontiguity an be obtained assuming that the shared surfae area between the two phases i and j, S ij, is statistially proportional to the produt of their total surfae areas S i, S j : C ij ¼ S ij S i / S is j S i ¼ S j : ð24þ From Eq. (6), noting that S g W W C V, the relation between surfae and volume is S V ( D/3), or for a given phase j: S j / V j j ; so that Eq. (24) beomes: C ij / V j j : Considering that C i C ii and assuming i ¼ : C i ¼ v i : ð25þ ð26þ ð27þ

11 122 A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) In the fratal approah, the ontiguity effet is simply equivalent to onsider a different power of the volumetri fration of the phase. In other words, in the fratal approah the parameter plays the key role, sine also the ontiguity is not an independent parameter. It is interesting to observe that, aording to Eqs. (3) and (27), the lassial diret rule of mixture of Eqs. (1) and (2) for the generi property P, may be rewritten as P ¼ XN i¼1 P i v 1þ i and is lose to the empirial rule of mixture in whih the volumetri fration is raised to the exponent 2, obtained as a limit ase for = 1. Aordingly, the fratal approah provides the reason for the exponent 2 in the empirial mixture rule, whih is often used in order to have a better desription of the experimental data [42]. Regarding the mean free path h, it ould be shown that it is a funtion of the mean grain size d and of the volumetri fration of grains v, if a regular arrangement of grains is assumed. For instane, by assuming a tetrakaideahedra shape for the grains, [43]. provided a simple formula relating these three quantities: ðd hþ3 v ¼ ; ð29þ d 3 In the ase of more realisti distribution of grains, with dispersion of grain size d, it has been shown by means of stereology [44]. that h depends not only on d and v, but also on the ontiguity C: ð1 vþ h ¼ d vð1 CÞ : ð3þ Sine the ontiguity C itself depends on v and d (see Eq. (27) or the relations proposed by Golovhan and Litoshenko [23], it is lear that h also depends on these two variables. ð28þ Appendix B. The Gaussian and the Log-normal distributions From the fratal approah, the saling of the energy density has been obtained as reported in Eq. (12). In this appendix we show how suh a result is more general than expeted. In partiular, we show that, assuming a Gaussian distribution for the grain sizes, the energy density sales as predited by the fratal approah in the limit ase of = 1. Let us assume a Gaussian (or normal) distribution of grains, for whih the probability density funtion p(r) is " # pðrþ ¼pffiffiffiffiffi 1 exp ðr r mþ 2 2p r 2r 2 ; ð31þ where r is the standard deviation and r m is the mean grain size, i.e.: hri / 1 rdn / N 1 rpðrþdr ¼ r m : Note that the integrals evaluated between minus infinite and zero must give negligible ontributions to the final results, the variable r being defined as positive. The energy dissipation W onneted to the presene of the grains is statistially expeted to be proportional to their total surfae area, i.e.: W / S g / r 2 dn ¼ N r 2 pðrþdr ¼ N ðr 2 m þ r2 Þ: On the other hand, the total volume of the grains is V g / r 3 dn / N r 3 pðrþdr ¼ N ðr 3 m þ 3r mr 2 Þ: Deriving the expression of N from the previous equation and introduing it into Eq. (33), being r m d, yields the asymptoti behaviour: W / V g d 1 / vd 1 ; idential to Eq. (12) if = 1. The same result is provided if we onsider a Log-normal distribution, for whih the probability density funtion p(r) is " # pðrþ ¼ p 1 ffiffiffiffiffi exp ðlnðrþ MÞ2 ; ð36þ r 2p S 2S 2 ð32þ ð33þ ð34þ ð35þ

12 where M is the sale parameter and S P the shape parameter. The mean grain size r m and the standard deviation r are given by r m ¼ rpðrþdr ¼ exp M þ S2 ; ð37þ 2 r 2 ¼ 1 1 ðr r m Þ 2 pðrþdr ¼ expð2m þ S 2 Þ½expðS 2 Þ 1Š: As in the previous ase, the energy dissipation W onneted to the presene of the grains is statistially expeted to be proportional to their total surfae area, i.e.: W / S g / r 2 dn ¼ N r 2 pðrþdr ¼ N exp½2ðm þ S 2 ÞŠ: On the other hand, the total volume of the grains is V g / r 3 dn / N r 3 pðrþdr ¼ N exp 3M þ 9 2 S2 : ð4þ A. Carpinteri et al. / Chaos, Solitons and Fratals 39 (29) Deriving the expression of N from the previous equation and introduing it into Eq. (39), being r m d, yields the asymptoti behaviour: W / V g exp M þ 5 2 S2 / v 1 d 2 2 / vd 1 ; ð41þ d d 2 þ r 2 whih is idential to Eq. (12) if, again, =1. ð38þ ð39þ Appendix C. Agreement with fragmentation theory The saling law of the additional energy density dissipated as a onsequene of the presene of the grains and interfaes, Eq. (12), may be interpreted with findings by [33 35] from fragmentation theories. In a fragmentation proess of a mixture of total volume V, with phases of volumes V i, the total dissipated energy W an be evaluated as W ¼ XN i¼1 W i ¼ XN i¼1 C i V i i ¼ CV ; ð42þ where i ¼ 2=3 to 1 are the fratal exponents and C i are material onstants of the phases (the so-alled fratal fragmentation strengths). The parameters and C represent the fratal exponent and fragmentation strength of the mixture. The onstants C i are by definition the size-independent mehanial properties of the materials. For the partiular ase of i ¼ : C ¼ XN i¼1 C i v i ; where v i ¼ V i =V is the volumetri fration of the phase i. The exponent ¼ 2=3 desribes a rule of mixture governed by the surfaes of the omponents, as well as the exponent = 1 represents, as a limit ase, the lassial approah desribing volume dominated proesses. Eqs. (42) and (43) are onsequenes of the universal laws derived in the evaluation of the multisale energy dissipation during fragmentation and omminution [33]. Eq. (43) shows that fratal mehanial properties, rather than the lassial ones, should be onsidered as the real properties of the phases and of the mixture. Comparing Eqs. (12) and (42) yields the dependene of the material property on the grain size: C / d 2 3 : This equation, for multi-phase materials, would have to be applied to eah phase i, yielding: C i / d 2 3 i i : ð45þ Assuming i ¼ and introduing Eq. (45) into Eq. (43) yields: C ¼ XN i¼1 k i d 2 3 i v i ; ð46þ ð43þ ð44þ

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