Strengthening Micromechanisms in Cold-ChamberHigh-Pressure Die-Cast Mg-Al Alloys

KUN V. YANG, CARLOS H. CACERES, and MARK A. EASTON

The contributions from grain boundary, solid solution, and dispersion strengthening to the yieldstrength of cast-to-shape specimens were calculated for seven binary alloys with compositionsranging from very dilute (0.5 mass pct Al) to concentrated (12 mass pct Al). Experimentally andtheoretically determined parameters were used to explicitly account for the different micro-structures at the skin and core regions of specimens’ cross sections. Microhardness maps wereused to identify the specimens’ skin. The specimens’ strength was calculated as the weightedaddition of the respective strengths of skin and core. The calculated strengths reproduced wellthe experimental values for the dilute alloys but underestimated the strength of the most con-centrated alloys by as much as ~35 MPa. It is argued that the presence of the percolatingnetwork of Mg17Al12 eutectic intermetallic, particularly in the skin region, in conjunction withhighly efficient dispersion hardening due to the convoluted shape of the intermetallics, accountsfor the shortfall in the calculated strength.

DOI: 10.1007/s11661-014-2326-x� The Minerals, Metals & Materials Society and ASM International 2014

I. INTRODUCTION

HIGH-PRESSURE die-cast (HPDC) Mg alloys arenormally not heat treated, and, for a given alloy, theirmicrostructure, hence their mechanical properties aredetermined by the die geometry and the casting pro-cessing parameters.[1,2] The good thermal contactbetween the die wall and the liquid metal ensured bythe applied pressure, combined with the relatively lowheat capacity of molten magnesium result in very highsolidification rates, not only with the beneficial effect ofa generally finer microstructure in comparison withother forms of casting,[3] but also with strong gradientsof composition and distribution of intermetallics at boththe grain size and casting cross section scales.[4–14] Theheterogeneity of the cast microstructure is particularlymarked when a cold-chamber press is used, in whichcase the grain microstructure is strongly bimodal, withpredominantly fine grains at or near the surface of thecross section (or skin) and large dendritic grains, termedexternally solidified grains (ESGs), in the center (orcore).[7,13,15,16] The ESGs, formed by partial solidifica-tion of the melt in the unheated shot sleeve andrunner,[7,13,17,18] are introduced as minute crystals dis-

persed in the liquid during the filling of the die.[3,13,19,20]

The dynamics of die filling and that of solidificationitself[21] concentrates the ESGs at the core, particularlyfor the concentrated alloys.[22] The ESGs have a lowerthan average solute content, enriching the liquid thatfills the die cavity,[23,24] and the pronounced coring inthe grains solidified inside the die leads to the formationof eutectic in proportions well in excess of what thephase diagram predicts.[2,23,25,26] Thus, even for Alcontents as low as 2 mass pct, a microstructure of b-Mg17Al12 eutectic intermetallics delineates the dendriteand grain boundaries alike.[2,23,25,26] The net result,especially for the more concentrated alloys, is theformation of two well-differentiated regions across thethickness of the casting: the skin layer, which may be upto 500 microns thick, and the core encompassing therest. Although visually the two regions can easily bedifferentiated from each other on a polished crosssection, defining the boundary between them is still amoot point. Details of the different definitions of theskin can be found in recent publications by the presentauthors, where two complementary micromechanisticdefinitions of the skin have been also presented.[21,27,28]

The skin has higher integrity and strength than thecore, whereas the latter concentrates most of the ESGsand the microporosity.[7,9,21] The strength and theoverall deformation behavior of any casting sectioncan be thus expected to be a weighted average of thoseof its skin and core regions, and, by necessity, involvethe local non-uniformity in composition and grain sizeas well as the amount, scale, and morphology of theeutectic intermetallic compounds.[5,6,9,10,29] Being stron-ger, the skin imposes an elastic constraint upon the core,delaying the onset of full plasticity at the core, until theskin itself reaches the micro-yielding stage.[22,27]

Five strengthening mechanisms were identified in aprior work on sand-cast Mg-Al alloys,[30] namely: grain

KUN V. YANG, formerly with the ARC Centre of ExcellenceDesign in Light Metals, Materials Engineering, School of Engineering,The University of Queensland, St Lucia, QLD 4072, Australia, is nowPostdoctoral Research Fellow with Department of Materials Engi-neering, Monash University, Clayton, VIC 3168, Australia. CARLOSH. CACERES, Reader in Casting Technology, is with MaterialsEngineering, School of Engineering, The University of Queensland,QLD 4072, Australia. Contact e-mail: c.caceres@uq.edu.au MARKA. EASTON, formerly with Department of Materials Engineering,Monash University, is now Professor with School of Aerospace,Mechanical and Manufacturing Engineering, RMIT University,Melbourne, VIC 3083, Australia.

Manuscript submitted October 14, 2013.Article published online May 20, 2014

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boundary, solid solution, dispersion, Orowan looping,and dislocation strengthening. Out of them, only grainboundary, solid solution, and dispersion hardening arerelevant to HPDC Mg-Al alloys.[5,6,26] In turn, due tothe small grain size and Mg’s large strength coefficient ofthe Hall–Petch mechanism,[31,32] grain boundarystrengthening is usually considered dominant,[2,5–7,9,17,23,33,34] but, since the grain size does not changemuch with the solute content,[15] the solute-dependentmechanisms are the ones that set the alloys apart fromeach other.[5,6] In the more concentrated alloys, a closelyinterconnected spatial network of eutectic intermetallicforms,[29,35] and it has been argued in the literature thatthis network constitutes an additional strengtheningmechanism,[5,6,29,35–37] in Mg-Al as well as in otherHPDC Mg-based alloys. Recent FEM modeling byZhang et al.[35] based on FIB 3D data of the Mg17Al12intermetallic of alloy AZ91 showed that between 4 and7 MPa are added to the strength of the alloy’s skinregion.

In spite of these developments, the understanding ofthe origin of the alloys’ overall strength remainssomewhat sketchy and qualitative,[5,6] largely becausethe presence of the skin and core regions turns the yieldstrength into a structural property of the particularcasting rather than a material property. It follows that aworking microstructure-yield strength relationship forHPDC Mg alloys is feasible as long as the respectivecontributions of the skin and core regions are accountedfor explicitly through suitably determined area frac-tions.[21,22,27]

The aim of this work is twofold: to calculate thecontribution of the Hall–Petch, solid solution anddispersion-strengthening mechanisms to the strength ofthe skin and core regions of cast-to-shape tensilespecimens of (cold-chamber) HPDC Mg-Al alloys, andthe quantification, for the more concentrated alloys, ofany additional strengthening arising from the percolat-ing eutectic microstructure in excess of the expecteddispersion hardening. Calculations were done for theskin and core regions using available microstructuraldata and current theoretical models. Subsequently, thestrength of the whole specimen was calculated byweighted integration over the entire cross section ofthe specimens using the area fractions of the skin andcore as determined in Reference 22.

Seven Mg-Al binary alloys, with Al contents from~0.5 to ~12 mass pct, were cast for this study. The widerange of solute content was meant to produce micro-structures with no dispersed intermetallics and a uni-form grain size across the casting cross section at thelower concentrations, and a highly inhomogeneousgrain microstructure and a closely interconnected net-work of eutectic intermetallics, hence with well-differ-entiated skin and core regions, at the higherconcentrations. The solid solution and Hall–Petchhardening components of the strength appeared rela-tively straightforward to quantify[5,6] for the more dilutealloys because of their simpler microstructure; thesealloys were thus used as a stepping stone from where toincorporate the effects of the interconnected intermetal-lic in the higher content alloys.

Preliminary versions of the present analysis havealready been published with reference to AZ91Dalloy.[30,35] The current work extends these prior studiesby combining the theoretical analysis with experimentaldata collected following an incremental approach: aseries of binary Mg-Al alloys with increasing soluteconcentration were cast, ensuring that the solute-basedstrengthening mechanisms, namely, solid solution, pre-cipitation, and dispersion hardening and the 3D inter-connection, become operational one by one. Tensile andgrain size data plus the relative proportions of the skinand core for the same batch of alloys were determined inprior work.[15,16,22] It is made explicit at all times that thestrength being modeled is a structural rather than amaterial property.

II. MATERIALS AND DATA FOR ANALYSIS

A. Alloys and Specimens

Table I shows the Al content of the alloys studied.The experiments were done using dog-bone-shapedtensile specimens of rectangular and round cross sec-tions, cast-to-shape on a 250T cold-chamber HPDCpress. The rectangular specimens’ gage length was 3-mmthick, 5.75-mm wide, and 30-mm long. The cylindricalspecimens had a gage length of 35 mm and diameter5.75 mm. More details of the casting procedure can befound elsewhere.[38]

Samples for microstructural analysis were cut fromthe gage length of undeformed specimens. Metallo-graphic polishing was carried out down to 0.05 lmcolloidal silica by standard methods.

B. Microstructural Data

The grain size data for the analysis, shown inFigure 1, were obtained by EBSD* in a parallel study[15]

of the rectangular specimens of the alloys of Table I.The Scheil–Gulliver Equation[40,41]:

Cs ¼ kC0 1� fsð Þk�1; ½1�

where Cs is the concentration of solute in solid solution,k is the partition coefficient, C0 is the composition of thealloy, and fs is the fraction solid (or fraction of phase),

Table I. The Concentration of Al in the Alloys Studied,Measured Using Atomic Emission Spectroscopy (ICP-AES)[39]

Alloy 1 2 3 4 5 6 7

Al (mass pct) 0.47 0.93 1.82 4.37 5.51 8.77 11.6

*The data of Figure 1 are given without error bars as they werecollected on one set of grains, on a single specimen of each alloy. Theaccuracy of the data was reassured by the qualitative match with themicrographs of Figures 3 and 4, and the close quantitative agreementof the calculated values for the dilute alloys, whose strengths are lar-gely determined by the grain size (Figure 9).

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was used to produce the solute concentration profilesacross the grains and the volume fractions of eutectica-Mg and b-Mg17Al12 intermetallic, depicted inFigures 2(a) and (b), respectively, for the alloys ofTable I. It assumes instantaneous diffusion in the liquidphase and no diffusion in the solid phase, which is areasonable assumption for most substitutional diffusionprocesses. The Scheil–Gulliver equation was used inreferences[42] to predict the partition of solute indirectionally solidified Mg-Al alloys, and in the authors’prior work[29] to calculate the volume fraction of eutecticb-Mg17Al12 intermetallic in HPDC AZ91 alloy. In allcases, the calculated and measured values agreed closelyas can be seen in Figure 2(b) (cf. column fi in Table II).Chia et al.[43] also found good agreement betweenpredicted and calculated values applying the sameapproach to binary Mg-RE alloys. More details aboutthe calculations involving the Scheil–Gulliver equationas applied to this alloy system can be found in theauthors’ earlier work.[15,16]

The solute profiles (Figure 2(a)) were obtained for acylindrical grain symmetry and scaled on the radial-direction to match each alloy’s average grain size at thecasting surface, as per the data of Figure 1. The graingrowth restriction factors (1/Q, top x-axis), also sourcedfrom Reference 15, are included to show the extent towhich the solute concentration in the liquid determinesthe alloys’ grain sizes inside the die cavity. The calcu-lated mass and volume fraction of eutectic and eutecticintermetallic for all of the alloys are listed in Table IItogether with other relevant microstructural informa-tion.

C. Grain Microstructure

The SEM images of Figure 3 provide an overview ofthe microstructure over the entire cross section of thealloys studied.

For the most dilute alloys (Figure 3(a)) the grain sizewas relatively uniform over the entire cross section,unlike for the more concentrated ones (Figures 3(b) and

Fig. 1—The mean grain size at the core and near the surface of rect-angular samples as a function of the Al content, for the alloys ofTable I. Data reproduced from Ref. [15] (see Fig. 3(d) for the defini-tion of top and bottom surfaces).

Fig. 2—(a) The solute concentration as a function of the radial dis-tance from the grain center calculated using the Scheil–Gulliverequation[40] for the alloys of Table I. The grain radius for each alloymatches the corresponding grain size at the surface regions in Fig. 1;grain growth restriction factors (1/Q) are from Ref. [15]. The cross-ing of the lines for the most concentrated alloys reflects the respec-tive grain sizes in Fig. 1 (see text). (b) The solid symbols are thevolume fractions of eutectic and eutectic intermetallic for the alloysof Table I, according to the Scheil–Gulliver equation[40] (see valuesin Table II). The open symbols are volume fractions of b-Mg17Al12at the skin and core, respectively, of a 5-mm thick HPDC AZ91 al-loy specimen, determined by FIB in Ref. [29] or by image analysis inthis work.

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(c)) which showed an increasingly finer grain micro-structure at the surfaces and coarser at the core withincreasing solute content, following the trends depictedin Figure 1. The rapid decrease in grain size at both thesurfaces and the core with increasing solute at the lowerconcentration was ascribed to the growth restrictionfactor (1/Q) imposed by the solute (see top x-axis inFigure 1(a)). The formation of ESGs in the shot sleeveseems to require a solidification range of >293 K(20 �C), i.e., Al concentrations of >5 mass pct asindicated by the pronounced stepwise increase in thecore grain size at that concentration. It follows[15] thatonly the concentrated alloys (i.e., c> 5 mass pct Al)should be expected to exhibit a distinct (softer) core, asindeed observed.[21,27]

Figure 4 shows the microstructures at the skin and coreregions of the 5.51 and 11.6 mass pct Al alloys, bothconsisting of primary a-Mg (light gray), eutectic a-Mg(dark), and eutectic intermetallic b-Mg17Al12 (bright). Thevolume fractions of eutectic intermetallic quoted inTable II (column fi) were obtained for the two mostconcentrated alloys through quantitative image analysis ofsecondary electron images similar to those of Figure 4.**

D. Yield Strength Data

Figure 5 shows the (0.2 pct) proof stress of the alloys ofTable I (solid symbols) sourced from the authors’preliminary work.[15,22] Each data point is an average ofat least 3 tensile specimens (The stress–strain curves forall of the alloys of this study can be seen in Figure 3 ofReference 22). The alloys’ strength increased monoton-ically with the solute content, with a stepwise increase at

~4 mass pct Al, ascribed to the formation of a measur-able amount of eutectic b-Mg17Al12 intermetallic particleson the dendrite/grain boundaries (see Figure 2(b)), and asecond stepped increase at ~9 mass pct Al, ascribed to theformation of an increasingly interconnected eutectic b-Mg17Al12 intermetallic across the section of the cast-ing.[5,6,22] Data from the literature for commercial alloys(open symbols) of similar Al content are included forcomparison; both the absolute value and the overallsolute dependence of the commercial alloys’ strength areconsistent with those of the present binary alloys.

E. The Area Fractions of Skin and Core

The microhardness map of Figure 6 illustrates thedistribution of hardness over the cross section for the11.6 mass pct Al alloy. Similar maps for all of the alloysstudied were published elsewhere.[21,27] The dashed linein Figure 6(b) delineates an outer area of higherhardness, or skin, covering ~30 pct of the total crosssectional area, leaving ~70 pct for the softer parts, orcore. The apportioning of core and skin for all of thealloys is given in the relevant column (area fraction ofskin, f) of Table II. These values were determinedanalytically in earlier� work,[22] and they will be used

to calculate the overall strength by linearly adding theskin and core strength contributions in the next section.

F. Structural vs Local Yield Strength

Relating the 0.2 pct proof stress of the tensilespecimens, a global structural value involving the entirecross section, to the (local) yield strength of either the

Table II. Values Calculated with the Scheil–Gulliver Equation for the Alloys of Table I (See Also Figs. 2 and 8)

c (Mass Pct)

Mg-Al Eutectic fe

Eutectic b-PhaseIntermetallic fi

Average SoluteContent inSolution css

Average SoluteContent in

a-Mg Grains csa

Solute Contentin ESGs cE

Area Fractionof Skin f[22]

Mass Fraction[Vol. Fraction]

Mass Fraction[Vol. Fraction]

Mass Fraction[At. Fraction]

Mass Fraction[At. Fraction]

Mass Fraction[At. Fraction] —

0.47 0.001 [0.0009] 0.00069 [0.0060] 0.0044 [0.0040] — 0.0044 [0.0040] 00.93 0.004 [0.0036] 0.0028 [0.0024] 0.0083 [0.0075] — 0.0083 [0.0075] 0.101.82 0.01 [0.009] 0.0069 [0.0060] 0.0152 [0.0137] — 0.0152 [0.0137] 0.124.37 0.04 [0.037] 0.028 [0.024] 0.0340 [0.031] — 0.0171 [0.0154] 0.085.51 0.06 [0.111] 0.042 [0.036] 0.0402 [0.036] — 0.0209 [0.0189] 0.118.77 0.12 [0.111] 0.083 [0.073] s[0.11]*

c[0.06]*0.0574 [0.052] 0.078 [0.070] 0.033 [0.030] 0.17

11.6 0.19 [0.176] 0.13 [0.116] s[0.17]*c[0.09]*

0.0710 [0.064] 0.090 [0.082] 0.044 [0.040] 0.30

The mass and volume fraction of the Mg-Al eutectic, fe; the mass and volume fraction of eutectic b-Mg17Al12 intermetallic, fi; the average solute insolution, css; the average solute in solution in the a-Mg grains, csa (only for the two most concentrated alloys); the average solute in solution in theESGs, cE. The area fraction of skin, f, as determined in Ref. [22].

*Experimental (area fraction) fi-values for the skin (s) and core (c) regions obtained by image analysis on secondary electron images like those ofFig. 4.

**The specimens were slightly etched to increase the contrast, butwith care not to etch out the Mg17Al12 (the bright phase). The volumefractions measured by image analysis (Table II; Figure 2(b)) on theseimages matched the values obtained by other techniques, providingconfidence that the etching did not affect the integrity of the inter-metallic.

�The fraction of the cross section that remained elastic (thus definingthe skin) at the point of the stress–strain curve where the core becamefully plastic was determined analytically by comparing the strainhardening rate of the different alloys at yielding with that of the mostdilute alloy, which does not exhibit a differentiated skin region.[22]

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core or the skin regions requires separate testing of theskin and core, for instance by machining away one orthe other prior to testing. Doing so is not an easy taskdue to the small dimensions involved, although it hasbeen reported in the literature.[7] In the present work,this was attempted in two ways. The microhardnessmaps described in the preceding section were integratedto obtain the weighted average hardness (wHV) for eachalloy. These values are plotted in Figure 7 as a functionof the 0.2 pct proof stress data of Figure 5, with lines ofbest fit given by (in units of MPa):

wHV ¼ 0:43 � YSþ 10:6 rectangularð Þ; ½2a�

wHV ¼ 0:39 � YSþ 8:2 cylindricalð Þ: ½2b�

These relationships were used to assign yield strengthvalues to the alloys’ skin and core regions and comparedto the respective analytical values in the next section(Figure 9).Alternatively, the outer layer of rectangular specimens

of the two more concentrated alloys was machinedaway, and the specimens tested in tension. The data soacquired were reported in earlier work[22] and are usedhere as a reference for the alloys’ core strength (thecrosses in Figures 7 and 9(b) and (c)).

Fig. 3—(a) through (c): SEM secondary electron images of the grain microstructures of specimens of the alloys with (a) 0.47, (b) 5.51, (c)11.6 mass pct Al (rectangular cross section); and (d) schematic diagram of the cross section indicating the location of the images of (a) through(c). The numerals indicate 1, corner; 2, surface; 3, core. The wider side of the cross section in (d) corresponds to the top surface or lid side of thedie.

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III. RESULTS

The (local) strength of the skin and core of each alloyis assumed to stem from the three elements of themicrostructure schematized in Figure 8: ESGs at thecore, small primary a-Mg grains at the skin, and a layerof eutectic (a-Mg+ b-Mg17Al12) in between grains. Inturn, each of them was assumed to contribute to therespective strengths through solid solution, dispersion,and/or Hall–Petch hardening, as detailed below. Fig-ures 9(a) through (c) compare the calculated values withthe experimental data for the skin, the core, and thewhole specimen, respectively.

A. Solid Solution Hardening

Since the formation of ESGs involves only incipientsolidification in the shot sleeve,[13] to a first order, theconcentration of solute can be approximated by the y-intersect of the relevant concentration profile inFigure 2(a),

labeled cE in Figure 8 and Table II. These values applyfor Al concentrations above 5.51 mass pct, i.e., alloyswith a distinct core and skin. In turn, the richer liquidentering the die creates primary a-Mg grains with aslightly above average concentration, csa.

� The average

concentration, css, across the casting can be obtained byintegration of the alloy’s solute profile in Figure 2(a).§

See Table II for the actual values for csa, css, and cE.

Fig. 3—continued.

�The Al dissolved in the eutectic a-Mg was included in the calcu-lation of csa.

§Both the css- and the cE-values are immediate from the phase dia-gram, unlike the csa, which depends on the volume fraction of ESGs,or, in geometrical terms, on the area fraction covered by the core.

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In References 5 and 6, css was used to estimate thecontribution from solid solution to the total strength,i.e., the differences between skin and core were ignored.

In the present case, the css- and cE-values were used toobtain lower bounds§§ to the respective solid solution

strengthening at the skin and core as detailed next.Experiments show that the solid solution strengthen-

ing of Mg-Al solid solutions is well described[47] by

rss ffi B c2=3; ½3�

where B = 197.5 MPa, and c was taken as either css orcE (both values are listed in Table II) for the skin andcore regions to obtain the respective lower bounds to thesolid solution strength.–

The solid solution lines in Figures 9(a) and (b)show that hardening due to the solute in solu-tion accounts at the most for ~37 MPa in the skin

Fig. 4—Secondary electron images of the primary a-Mg (light gray), eutectic a-Mg (dark), and b-Mg17Al12 (bright) for (a) skin and (b) core re-gion of the Mg-5.51 mass pct Al alloy; (c) skin and (d) core region of Mg-11.6 mass pct Al alloy.

Fig. 5—The yield strength of HPDC Mg-Al alloys as a function ofthe Al content, for cylindrical and rectangular specimens. The solidsymbols are data for the present alloys; the open symbols weresourced from the literature[2,4,7,11,23,26,44–46].

§§Any departure from the predicted solute profiles of Figure 2 im-plies a stronger solid solution hardening effect, hence the reference tothis calculations as ‘‘lower bounds.’’

–The strength of the solid solution calculated using the grain’saverage solute concentration (csa values in Table II) is nearly identicalto the solid solution strength averaged over the grain’s diameter.

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region of the most concentrated alloy. Figure 9(c)shows that solute in solution accounts for astrength difference between skin and core of about12 MPa.

B. Dispersion Strengthening

The Mg-Al eutectic involves a mixture of eutectic a-Mg and b-Mg17Al12 intermetallic phase particles, in

Fig. 6—(a) The indentation pattern; (b) the microhardness post map of the cross section; and (c) microhardness mapping and color scale of thecross section of the 11.6 mass pct Al alloy. The dashed line in (b) separates the core from the skin region, as per the proportions (area fractionof skin, f = 0.30 for this alloy), given in Table II.

Fig. 7—Weighted average microhardness values (wHV) of the entirecross section as a function of the tensile 0.2 pct proof stress, forspecimens with rectangular and circular cross sections of the alloysstudied; see Fig. 2 for the proof stress data. The crosses correspondto two rectangular specimens with the skin machined away[22].

Fig. 8—Schematic representation of the average content of soluteacross two representative grains in the alloy with 8.77 mass pct Al. Thegrain on the left is a primary, solidified in-die, a-Mg grain of diameter8 lm, whereas the one on the right is an ESG of diameter 13 lm (Grainsizes as per Fig. 1; values for csa, css, and cE as per Table II).

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proportions (by volume), fe and fi, given in (Table II).The dispersion strengthening caused by a volumefraction of eutectic intermetallic particles was shown inReferences 6, 22, and 30, (at an (off-set) strain of 0.2 pct)to be given by:

rdisp ðMPaÞ ¼ 135 � fi; ½4�

where fi is the volume fraction of the dispersed b-Mg17Al12 intermetallic particles (Table II). As for thesolute in solution, a single average volume fraction ofthe intermetallic was assumed for the less concentratedalloys (5.51 mass pct and below), whereas separatevalues (as determined by image analysis, Table II) forthe skin and core were used for the two more concen-trated ones. Figures 9(a) and (b) show that dispersionhardening accounts for 15 to 20 MPa in the mostconcentrated alloy, whereas Figure 9(c) shows that itaccounts for ~8 MPa of the difference between skin andcore.

C. Grain Boundary Strengthening

Grain size effects were accounted for through theHall–Petch relationship[31,32,48]:

rhp ¼ ro þ kd�1=2; ½5�

where ro = 12 MPa[55,56] was assumed independent ofthe solute concentration. For pure Mg, experimentsshow that k � 0.22 MPa m1/2,[32,48] whereas it increasesslightly, to ~0.28 MPa m1/2[31] for alloy AM60. Equation[5] was thus applied assuming a monotonic increase in kvalue from 0.22 MPa m1/2 for the most dilute alloy to0.3 MPa m1/2 for the most concentrated one, with therelevant grain size for skin and core data from Figure 1.It is noted that in comparison with Mg-Al alloys, theHall–Petch strength coefficient of Mg-Zn alloys increasesmuch more rapidly, whereas the friction stress decreases,with the concentration of solute,[32] a reminder that thepresent calculations are alloy system-specific.The values obtained with Eq. [5] are labeled Hall–

Petch in Figure 9. The dip in the grain boundarystrengthening at the core region at ~5 mass pct Alreflects the profile of the average grain size at the core ofFigure 1. The Hall–Petch components account for thebulk of the strengths in Figures 9(a) and (b), as antic-ipated, but also their relative magnitudes are stronglyaffected by the appearance of ESGs at the core atintermediate concentrations and up. Figure 9(c) showsthat at intermediate concentrations the differentiationbetween skin and core due to grain size effects is largerthan those arising from the solute-based mechanisms.

D. Summation of Values

The dispersed eutectic b-Mg17Al12 particles and thegrain boundaries can be considered as hard discrete

Fig. 9—Experimental and calculated values of the strength of (a) theskin, (b) the core and (c) the whole specimens, as functions of the Alcontent. The diamonds (labeled tensile) are experimental yieldstrength values for the rectangular specimens of Fig. 5; the skin andcore values were obtained with Eq. [6b]; HV-skin and HV-core (tri-angles) were calculated using Eq. [2a]; total is the weighted sum ofthe skin and core strengths (Eq. [7]); the lines labeled solid solution,dispersion, and Hall–Petch were calculated with Eqs. [3] through [5],respectively; the crosses are the yield strength of specimens with theskin removed, as in Fig. 7.

b

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obstacles of comparable strength. A fraction of theeutectic intermetallics are located on the dendriteboundaries, i.e., are dispersed inside the grains whereasthe rest are located on the grain boundaries.–– Kocks

et al.[50] pointed out that strengthening effects due toimpenetrable obstacles of comparable strength, such asthe grain boundaries and particles of the present alloys,should be added using a root mean square law. Incontrast, solute in solid solution imparts a ‘‘smooth’’and uniform friction over the slip planes and can beadded linearly[51] to the combination of the other twocomponents. The total strength of the alloys thusbecomes

r ¼ rss þ ½ðrdispÞ2 þ ðrhpÞ2�1=2; ½6a�

where rss, rdisp, and rhp are given by Eqs. [3] through,[5]respectively.

Alternatively, it may be argued[51] that the interme-tallics and grain boundaries represent numerically fewobstacles far apart from each other when looked fromthe scale of dislocations and solute atoms, in which casethe solid solution hardening should be considered thebase component to which the others are added linearly,i.e.,

r ¼ rss þ rdisp þ rhp; ½6b�

The values obtained with Eq. [6b], labeled skin inFigure 9(a) and core in Figure 9(b), were larger andreproduced the experimental values much better thanthose obtained with Eq. [6a], hence they are used for therest of the analysis.*

The highest hardness value at the skin and the lowestones at the core of the rectangular specimens wereconverted to yield strength (YS) through Eq. [2a], andplotted, labeled HV-skin and HV-core, in Figures 9(a)and (b), respectively, in order to relate the calculatedvalues to experimental local values of the relevantstrengths. The calculated strength of the skin regionmatches the HV-skin values for the leaner alloys, but itis lower for the two concentrated alloys. Likewise, thecalculated strength of the core is increasingly insufficientabove ~5 mass pct Al. The crosses in Figures 9(b) and(c) represent the yield strength of specimens with theskin removed and also indicate a shortfall in thecalculated strength.

E. The Strength of the Cast Bars

The yield strength, YStotal, of the entire tensilespecimens, i.e., as a structural property for each casting,was calculated through weighted linear addition,**

YStotal ¼ YSskin � fþ YScore � ð1� fÞ; ½7�

with f given in Table II. The values so obtained areplotted in Figure 9(c), labeled total; again exhibited anincreasing shortfall for the more concentrated alloys,from ~6 MPa for 5.51 mass pct Al alloy to ~35 MPa forthe 11.6 mass pct Al alloy.

IV. DISCUSSION

The main issue to discuss is the shortfall of calculatedvalues of the strength. It will be argued that the shortfallcan be accounted for through intermetallics’ morphol-ogy and spatial interconnection effects.In keeping with the original formulation of the

dispersion-hardening models,[54] Eq. [4] assumes a dis-persion of isolated, disk-shaped particles. Figure 4 showsthat such an assumption is highly unrealistic already forthe 5.51 mass pct Al alloy and more so for the 11.6 masspct Al alloy due to the increasingly convoluted shape andthe spatial interconnection of the intermetallic. Brechetet al.[55] accounted for the effect of shape of thereinforcing particles in metal matrix composites througha multiplicative factor equal to the particles’ aspect ratioin front of Eq. [4]. Such an approach also proved correctin accounting for the dispersion-hardening effects ofeutectic Si particles in Al–Si–Mg casting alloys[56] andshould account for a significant part of the shortfall ofthe calculated strength in the present case.Likewise, due to the increasing degree of spatial

interconnection, the b-Mg17Al12 intermetallic can beexpected to behave as a ceramic skeleton, scaffold, ormicro-truss structure.[29] Evidence of the existence of aclosely interconnected intermetallic structure in alloyAZ91D, especially in the skin region, has been obtainedrecently using dual beam FIB 3D tomography,[29]

whereas numerical modeling by Zhang et al. indicatesthat the elastically deforming 3D scaffold in alloyAZ91D may account for about 7 MPa in the skinregion.[35] Due to the larger volume fraction of inter-metallic, this reinforcement should scale up� to about

––Hansen showed that whether the reinforcing particles are locatedalong the grain boundaries or uniformly dispersed inside the grainsmake little difference to the total strengthening.[49]

*Hansen and Ralph[52] also found a better agreement with theexperimental data in dispersion-hardened Cu alloys using a linearaddition law.

**The use of a linear addition of the strengths of skin and core isconsistent with Kurzydlowski and Bucki’s analysis[53] of the strength ofpolycrystals made of subpolycrystals of different grain sizes.

�The stiffness of a bending dominated cellular structure scales withthe square of the relative volume fraction of solid material.[35,57] FromTable 2, the intermetallic at the skin increases from 11 pct for the 8.77Al alloy to about 17 pct for the 11.6 Al alloy, increasing the local 3Dnetwork’s contribution to (7*(17/11)2) ~ 16.7 MPa.

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17 MPa for the 11.6 mass pct Al. Quantitative evidenceof the reinforcement introduced by the spatial intercon-nection was obtained for an HPDC Mg-Ce alloy usingshort anneal experiments by Zhang et al.[57] aimed atbreaking down the 3D eutectic branching. The sameexperiments showed that dispersion hardening andspatial interconnection contributed similar amounts tothe alloy’s total strength.

Zhang et al.[35,57] suggested yet another way in whichthe percolating scaffold may reinforce the alloy at yield:the small a-Mgcells ofFigure 4(c), being fully surroundedby the intermetallic, appear highly constrained and can beexpected to be much harder to deform than any ofsimilarly enclosed but larger dendritic cells and grains. Inpractice, those regions of compact cellularmicrostructureshould be equivalent to an increased volume fraction ofintermetallic, adding to the overall reinforcement effect ofthe intermetallic scaffold at very low strains.

By and large, it can be asserted that the percolatingeutectic intermetallic accounts for a measurable fractionof the structural strength of the castings, and therefore,it should be explicitly recognized in the microstructuraldesign of the most concentrated varieties of the presentalloys.

V. CONCLUSIONS

The strength of (3-mm thick) HPDC Mg-Al castingswith Al concentration up to 11.6 mass pct stemmingfrom grain boundary (Hall–Petch), solid solution, anddispersion strengthening was calculated using availabledata and current theoretical models, explicitly account-ing for the presence of the skin and core regions acrossthe casting thickness. The calculations show that

1. The strength of the more dilute alloys can be pre-dicted quite accurately through grain boundary andsolid solution hardening.

2. For intermediate concentrations (4 to 6 mass pctAl), solid solution and dispersion hardening ac-count for most of the differences in strength be-tween the alloys, whereas for given alloy the largergrain size at the core, resulting from the presence ofESGs, account for most of the difference betweenthe skin and the core.

3. For the most concentrated alloys, (~9 mass pct Aland above) the eutectic intermetallic appears to ac-count for as much as ~35 MPa of the casting’sstrength through profuse spatial interconnectionand more efficient dispersion hardening, especiallyat the skin.

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