Prediction models for the yield strength of particle-reinforcedunimodal pure magnesium (Mg) metal matrix nanocomposites(MMNCs)

Chang-Soo Kim Il Sohn Marjan Nezafati J. B. Ferguson Benjamin F. Schultz

Zahra Bajestani-Gohari Pradeep K. Rohatgi Kyu Cho

Received: 13 December 2012 / Accepted: 11 February 2013 / Published online: 23 February 2013

Springer Science+Business Media New York 2013

Abstract Particle-reinforced metal matrix nanocompos-

ites (MMNCs) have been lauded for their potentially

superior mechanical properties such as modulus, yield

strength, and ultimate tensile strength. Though these

materials have been synthesized using several modern

solid- or liquid-phase processes, the relationships between

material types, contents, processing conditions, and the

resultant mechanical properties are not well understood. In

this paper, we examine the yield strength of particle-rein-

forced MMNCs by considering individual strengthening

mechanism candidates and yield strength prediction mod-

els. We first introduce several strengthening mechanisms

that can account for increase in the yield strength in

MMNC materials, and address the features of currently

available yield strength superposition methods. We then

apply these prediction models to the existing dataset of

magnesium MMNCs. Through a series of quantitative

analyses, it is demonstrated that grain refinement plays a

significant role in determining the overall yield strength of

most of the MMNCs developed to date. Also, it is found

that the incorporation of the coefficient of thermal expan-

sion mismatch and modulus mismatch strengthening

mechanisms will considerably overestimate the experi-

mental yield strength. Finally, it is shown that work-hard-

ening during post-processing of MMNCs employed by

many researchers is in part responsible for improvement to

the yield strength of these materials.

Introduction

Recent advances in nanotechnology have enabled the

incorporation of nano-sized reinforcements in metal matrix

composite materials in an effort to achieve mechanical

performances (i.e., strength and modulus) that were unat-

tainable through the conventional micron-sized reinforce-

ments. These MMNCs containing nanoparticles (NPs) have

been claimed to exhibit several potentially advantageous

properties that might enable their use in applications for

automotive, aerospace, construction, and military sectors.

Aluminum (Al) and magnesium (Mg) are most commonly

used as base matrix materials in light-weight MMNCs as

they offer high specific mechanical properties due to their

low densities (2.7 and 1.7 g/cm3 for Al and Mg, respec-

tively) when compared to iron. The development and

application of high strength and light-weight MMNC

materials could have significant impact as a replacement

for heavier traditional metals or composites with resultant

savings in fuel economy. For these reasons, various

materials, synthesis, and processing methodologies and

conditions have been proposed to develop high strength,

particle-reinforced MMNCs [115].

To design and synthesize MMNCs with desirable prop-

erties, it is first necessary to develop a theoretical under-

standing of how the various potential strengthening

mechanisms combine to determine the strength of these

materials and under which conditions the desired

C.-S. Kim (&) M. Nezafati J. B. Ferguson B. F. Schultz Z. Bajestani-Gohari P. K. RohatgiMaterials Science and Engineering Department, University

of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA

e-mail: kimcs@uwm.edu

I. Sohn

Materials Science and Engineering Department,

Yonsei University, Seoul 120-749, South Korea

K. Cho

U.S. Army Research Laboratory, Weapons and Materials

Research Directorate, Aberdeen Proving Ground,

MD 21005, USA

123

J Mater Sci (2013) 48:41914204

DOI 10.1007/s10853-013-7232-x

microstructures can be produced. However, despite the

extensive efforts that have been made to synthesize MMNCs

with enhanced strength, a generally agreed upon theory to

describe the strengthening mechanism of these MMNCs has

not yet emerged. In the present work, we revisit the individual

strengthening mechanisms that may be present in particle-

reinforced MMNCs, and we test the previously proposed

theoretical models to predict the strength of MMNCs. Then,

we provide a thorough analysis based on the results from

prediction models and experimental observations. In

Strengthening mechanisms section, we review the candi-

date strengthening mechanisms that account for the various

aspects of strengthening of particle-reinforced MMNCs.

Next, various summation methods of contributions from

individual strengthening mechanisms to predict the strength

of synthesized MMNCs are introduced, and predicted

strengths using analytical models are compared with experi-

mental measurements. In the current work, we have focused

on unimodal MMNCs in examining the yield strength

improvement. Multimodal MMNCs (e.g., bimodal or trimo-

dal MMNCs) that are composed of fine and coarse matrix

crystals were not considered here because the strength of such

multimodal materials is routinely and relatively accurately

estimated by the rule of mixtures applied to the properties of

the fine and coarse phases. In our analysis, we primarily

employed the experimental data available for pure Mg

MMNCs. We analyzed the experimental data for pure Mg

MMNCs because, although there is a set of data for alloyed

Mg MMNCs [8, 10], we have only a few data points for each

of the alloy type. We, therefore, note that some of the material

parameters addressed in strengthening mechanisms and

strength prediction models are specifically relevant to predict

the properties of pure Mg-based MMNCs.

Strengthening mechanisms

Strength enhancement in MMNCs is commonly attributed

to the following three mechanisms: grain refinement,

Orowan strengthening, and coefficient of thermal expan-

sion (CTE) mismatch (a.k.a., Forest) strengthening.

Grain refinement strengthening (HallPetch

strengthening)

There is evidence that the introduction of NPs into a metal

matrix either by liquid- or solid-state processing results in a

grain refinement effect [16, 17]. The HallPetch empirical

relation has traditionally been used to describe the effect of

average grain size (D) on the yield strength (ry) and/orhardness (H) of metallic polycrystalline materials, e.g.,

ry r0 kyDp ; where r0 and ky are experimentally

determined parameters. The parameter r0 represents theyield strength of a single crystal in the absence of any

strengthening mechanisms except the solid solution effect,

and ky is the magnitude by which crystal boundaries in a

polycrystalline material resist slip. In cases where the

addition of particles reduces the size of the grains in the

MMNCs compared to the un-reinforced metal processed

under the same conditions, there will be an improvement to

yield strength due to grain refinement, DrGR:

DrGR ky 1DMMNC

p 1D0

p

1

where DMMNC and D0 are the average grain diameters of

polycrystalline matrix materials in the MMNC composite

and unreinforced material, respectively. This equation

assumes that the HallPetch parameters ky and r0 remainunchanged in the composites during processing. Unfortu-

nately, the HallPetch parameters in a given alloy in a

given state cannot be predicted, and must determined

empirically for a given microstructure taking into consid-

eration of such factors as precipitation condition and

increased dislocation density or texturing resulting from

mechanical post-processing. Thus, for the empirically

determined parameters to be valid for the purpose of cal-

culating the grain refinement strength contribution, only the

grain size can vary and all other influences on the strength

of the material must, as much as possible, remain

unchanged [1820].

Table 1 shows the list of available HallPetch parame-

ters for pure and alloyed Mg materials at room temperature

with available grain sizes that were synthesized by a

variety of processing routes, e.g., casting, rolling, extru-

sion, equal-channel angular processing (ECAP), powder

metallurgy (PM), etc. [19, 2130]. As demonstrated in

Table 1, it is generally accepted that r0 and ky values forpure Mg materials fall in the ranges of 17.735 MPa and

250300 MPa

lmp

; respectively, with few exceptions. In

the case of Mg alloys, the values of r0 and ky varied morewidely in the range of 3130 MPa and 150350 MPa

lmp

;

respectively.

While it can be shown that the addition of NPs to a

metal matrix acts to refine the grain size [16, 17], it should

be noted that the NPs may only restrict the size of the

grains in the MMNC rather than nucleate new grains. For

such indirect impacts of NP addition, the following Zener

formula [31] is sometimes used to describe the grain

boundary pinning of added particles during processing and

post-processing (e.g., hot extrusion).

Dm 4adp3Vp

2

where a is a proportionality constant, and Dm, dp, and Vpare the smallest grain size of matrix, reinforcement particle

4192 J Mater Sci (2013) 48:41914204

123

diameter, and volume fraction of particles, respectively.

However, it is highly questionable if Eq. (2) can be applied

directly to the processing of MMNCs. Rather, it is

empirically proposed that Dm will follow Eq. (3) that

does not explicitly include the effect of particle diameter,

dp, but is instead based on the grain size, D0, that results

from processing conditions when particles are absent. This

has been shown to provide a reasonably good prediction of

resultant grain size for a specific cooling rate in the liquid-

phase processing [17].

Dm D01 pVp 1

3

3

where p is the proportionality constant that describes the

refining power of the reinforcement, which can be empir-

ically determined.

Orowan strengthening

Orowan strengthening, sometimes called second-phase

particle strengthening, is another primary mechanism that

could result in considerable strength increases in particle-

reinforced MMNCs. The Orowan strengthening model

describes improvements in strength in the matrix material

based on the resistance of small hard particles against the

motion of dislocations. The dislocation movement pro-

ceeds with passing by these NP obstacles by first bowing,

then reconnecting, and finally forming a dislocation loop

around the particles. Formation of these loops in principle

leads to high work-hardening rates with improved strength

[3235]. In composites reinforced by spherical particles,

the maximum tensile and shear stresses occur at the surface

of the particles and they decrease with distance from the

surface of particles. Since the Orowan-type of strengthen-

ing is observed when the reinforcement particles are suf-

ficiently small, it is proposed that the strength increases are

only effective for the particles with sizes under 1 lm [36].The original Orowan formulation predicts an inverse rela-

tionship between the strength increase (DrOrowan) and theinterparticle mean free path (k) given by:

DrOrowan / Gmbk 4

where Gm and b are the shear modulus of matrix and the

magnitude of Burgers vector, respectively. The Ashby

Orowan formulation is a more advanced revision to Eq. (4)

that is widely used in predicting the strength improvements

[37] and takes the following form:

DrOrowan AGmb2pk

lndpr

5

Table 1 HallPetch parameters for pure and alloyed Mg materials

Alloy type Processing route Grain size (lm) r0 (MPa) ky (MPa

lmp

) Ref

Pure Mg Rolled 43172 35 291 [21]

Rolled 16 278 [22]

Extruded 1389 158 [22]

Cast 361500 17.7 250 [23]

AZ31 Rolled 1.4140 10131 160348 [24]

Extruded 2.511 80130 182303 [24]

ECAP 232 3085.2 170205 [24, 25]

EBW 1140 62 202 [19, 24]

FSP 2.66.1 10 161 [25]

FSW 16.4 119.5 [26]

AZ91 Extruded 0.33.7 133 [27]

Cast 0.35 166 [27]

Mg2Zn Cast 55340 3.1 470 [28]

Extruded [29]

MgAlMnCa Cast 23.3 192 [29]

Powder metallurgy 0.40.5 [29]

ZM21 725 154 80 [23]

Mg97Zn1Y2 Extruded 49 188 [30]

ZK30, ZE10, ZEK100 Hydrostatic procedure 320 200 210 [23]

AZ61 Extruded 8150 348 [22]

1100 247 [22]

ECAP equal-channel angular processing, EBW electron beam welding, FSP friction stir processing, FSW friction stir welding

J Mater Sci (2013) 48:41914204 4193

123

where A is a constant depending on the materials and types

of dislocations (e.g., edge or screw), dp is the reinforcement

particle diameter, and r is the dislocation core radius (i.e.,

the size of the dilated or diminished lattice with a dislo-

cation), respectively.

Several variations of the general form of the Ashby

Orowan equation (Eq. (5)) have been proposed to account

for the proportionality constant, particle distributions, and

lattice dilation effects in various materials [3234, 3843].

Table 2 summarizes the list of AshbyOrowan-type equa-

tions (Eqs. (6) through (10)) that have been previously

proposed for predicting strengthening effects of composites

based on pure or alloyed Mg materials. For most of these

relations, it was assumed that the reinforcement particles

are randomly distributed in the matrix, and one particular

slip system is uniquely dominant and more highly stressed

than others. In addition, to simplify the mathematical cal-

culations, the particles were assumed to be spherical;

otherwise, the derived formula must be corrected incor-

porating the particle geometry factor [37, 42]. Among the

expressions listed in Table 2, Eq. (6) along with the cor-

responding interparticle spacing (Eq. (6a)) is the most

widely used formula to estimate the Orowan-type contri-

bution to the yield strength improvement of particle-rein-

forced MMNCs [33, 38, 39]. The activation of the Orowan

strengthening mechanism is particularly important because

it does not necessarily involve the reduction of ductility of

MMNCs as other dislocation density-based strengthening

mechanisms, such as CTE and/or modulus mismatch

mechanisms do.

Coefficient of thermal expansion mismatch

strengthening

When a MMNC is quenched from the processing temperature

to room temperature, volumetric strain mismatch between the

monolithic matrix and reinforcement particles may occur due

to differences in CTE, which will subsequently produce

geometrically necessary dislocations (GND) around rein-

forcement particles to accommodate the CTE difference.

This mechanism is based on Taylor strengthening that

describes the increase in the flow stress in the matrix due to

the presence of the GND. When the length of a generated

dislocation loop is assumed as pdp, the CTE mismatchstrengthening (DrCTE) can be estimated by [44, 45]:

DrCTE

3p

bGmb

qCTEp

; 11

qCTE 12VpDaDT1 Vpbdp 11a

where b is the dislocation strengthening coefficient, qCTE is thedensity of dislocations generated from the CTE difference, Dais the CTE difference between the matrix and the reinforce-

ment particles, and DS is the temperature change, respectively.

Table 2 Orowan equations and material parameters used for particle-reinforced Mg-based alloys or MMNCs in the previous papers [3234,3842]

Formula Composition b (nm) m dp(nm)

Vp(vol%)

Gm(GPa)

Ref

DrOrowan 0:13Gmbk ln dp2b (6)In references [38, 39],

k dp1=2Vp1=3 1 6a

Pure Mg (4.9 wt% C-coated Ni) 0.35 20 0.4 11.60 [38]

99.9 % Mg [39]

Pure Mg (99.8 %)

AZ31 Mg (8 wt% Al2O3)

0.32 0.30 100 0.88 16.5 [33]

DrOrowan 6AGmb4pk ln Dr0 Bh i

7(A = 1/(1 - m), B = 0.6 for screw, A = 1, B = 0.7

for edge dislocations)

Mg, 5.25 wt% Y, 3.5 wt% RE,

0.5 wt% Zr

0.321 0.27 [32]

DrOrowan 1:33GmbV1=2p

dp(8)

AZ91 0.321 3.10 45 [42]

DrOrowan Gmb2pk 1mp

lndpb 9

In Ref. [40], k 0:953Vp

p 1

dp 9a

Mg5 wt% Zn 0.069 1011 0.29 15 [40]Mg3.0 at.% Zn 0.32 0.28 16.6 [41]

DrOrowan M 0:4Gmbpkln

db

1mp 10

d

23

q

dp 10a;k d

p4Vp

q

1

10b

ZK60 [34]

b, m, dp, k, Vp, and Gm denote the Burgers vector, Poissons ratio, average particle diameter, interparticle spacing, particle volume fraction, andshear modulus, respectively

4194 J Mater Sci (2013) 48:41914204

123

The activation of CTE mismatch strengthening mecha-

nism in MMNCs is, however, unclear because many

observations report the absence of CTE effects on the

strength improvement of MMNCs [4548]. For example,

Vogt et al. [45] tested the true stressstrain curves of

MMNC specimens with various heat treatments and

quenching conditions and they confirmed that the CTE

mismatch strengthening does not occur in MMNCs pro-

duced by PM techniques. Also, Redsten et al. [46] have

proposed that CTE mismatch strengthening can be ignored

below a critical reinforcement particle size limit, which, in

Mg/Y2O3 and Al/Al2O3 composites, corresponds to rein-

forcement particle sizes of approximately *70 and*80 nm, respectively, when the processing temperature isassumed as 300 C. Therefore, based on their work, if thereinforcement particle size is small enough (i.e., below the

range of 7080 nm), the contribution from CTE mismatch

strengthening may be considered negligible or minor

compared with those from grain refinement or Orowan

strengthening mechanisms.

Other strengthening mechanisms

In addition to grain refinement, Orowan, and CTE mis-

match mechanisms, the modulus mismatch and load-bear-

ing strengthening mechanisms have been offered as

mechanisms capable of improving the strength of MMNCs.

The modulus mismatch strengthening mechanism also

describes the generation of GND when a MMNC is sub-

jected to a compressive loading such as in hot extrusion.

Because of the presence of reinforcement NPs, many GND

must be created to accommodate the moduli difference

between the matrix and particles, and distortion deforma-

tion subsequently results during the post-processing.

Therefore, this modulus strengthening must be applied only

to the MMNCs where any suitable post-processing is

involved including compressive loading. If we again

assume that the length of a generated dislocation loop is

pdp, then the strength improvement by the modulus mis-match (DrModulus) is approximated by [44]:

DrModulus

3p

aGmb

qModulusp

; 12

qModulus 6Vpbdp

e 12a

where a is the material-specific coefficient, qModulus is thedensity of GND generated by the modulus mismatch

mechanism, and e is the bulk strain of composites,respectively. Activation of the modulus mismatch

strengthening mechanism is, however, also unclear for

reasons similar to those described in the CTE mismatch

strengthening section.

The load-bearing strengthening mechanism explains the

direct strengthening contribution from the presence of

reinforcement particles (i.e., Vprp, where Vp and rp are thevolume fraction and the yield strength of particles,

respectively). If we assume a well-bonded spherical rein-

forcement particle to the matrix, then the yield strength of

particles, rp, can be represented by 1/2rm, where rm is theyield strength of matrix. Therefore, the strength improve-

ment by the load-bearing mechanism (DrLoad) is expressedas [7, 38, 39, 49]:

DrLoad 12

Vprm 13

The influence of this load-bearing mechanism on the

MMNCs is, however, in general small enough to be ignored.

For example, when Vp = 0.010.05, the strength improvements

from the load-bearing mechanism are estimated as only 0.5

2.5 % of the original yield strength of matrix.

Strength prediction models for unimodal MMNCs

To predict the strength improvement of particle-reinforced

unimodal MMNCs, the following three superposition

approaches have been used: arithmetic summation, qua-

dratic summation, and compounding methods. The physi-

cal basis for arithmetic and quadratic summation methods

have been put forward based on dislocation theory applied

to single crystals where quadratic summation should be

applied for obstacles to dislocation motion on the same

structural scale, and arithmetic summation should be used

for obstacles at significantly different scales [50]. Though

this scheme seemed to be confirmed experimentally by

Eberling and Ashby [51], the latest experimental evidence

from Lagerpusch et al. [52] does not support this scheme.

The compounding method is based on the assumption that

the stress applied to the material is transferred from the

relatively weak matrix material to the significantly stronger

reinforcement, enabling the material to withstand higher

stresses before the matrix yields. This is based at root on

the modified shear lag mechanism originally proposed by

Nardone and Prewo [48] for metal matrix microcomposites

(MMCs). It is characteristic of this proposed mechanism

that the yield strength of the composite depends on the

yield strength of the matrix, but not the strength of the

reinforcement. Thus the increase in strength due to load

transfer to the reinforcement can be treated mathematically

by multiplying the matrix yield stress by an improvement

factor. The compounding method, therefore, treats all

strengthening mechanisms as load-transferring mecha-

nisms, which can be represented mathematically using a

series of improvement factors. Currently, there is no con-

sensus as to which model or physical basis best represents

J Mater Sci (2013) 48:41914204 4195

123

reality, with the result that the prediction of strength relies

on intuition and empiricism and often strength predictions

take a form similar to that used to describe conventional

MMCs [5356]:

r rm Dr 14where r, rm, and Dr are the yield strength of MMNCs,grain size adjusted yield strength of matrix (i.e.,

rm r0 ky=

DAlloyp

), and the strength improvement by

incorporation of NPs, respectively.

Arithmetic summation method

The arithmetic summation method simply adds the con-

tribution of individual strengthening mechanisms in a lin-

ear fashion. It assumes that different mechanisms do not

influence each other and they can independently contribute

to the final yield strength of composites.

Dr DrGR DrOrowan DrCTE DrModulus DrLoad15

The individual contributions of strengthening mechanisms

are indicated by subscripts GR (grain refinement),

Orowan, CTE, Modulus, and Load. This method

has not been applied in many cases simply because such

linear arithmetic summation has been shown to predict higher

yield strengths compared to experimental data [39, 44].

Quadratic summation method

The quadratic summation method was originated by Clyne

and co-workers [57, 58] for the composites comproed of

micron-sized reinforcements. They assumed that the indi-

vidual strengthening mechanisms interact with each other

and the sum of the squares of individual strengthening

contribution is proportional to the square of the total yield

strength improvement.

Dr

DrGR 2 DrOrowan 2 DrCTE 2 DrModulus 2 DrLoad 2q

16There are several prior efforts to estimate the yield

strength of particle-reinforced MMNCs based on this

quadratic summation approach [39, 44, 5759].

Depending on the processing routes and other conditions,

some of the strengthening factors are omitted in applying

Eq. (16). Several studies have reported that the quadratic

method shows better agreement between predictions and

experimental observations than other methods [39, 44, 60].

For example, Sanaty-Zadeh [60] has claimed that the

predicted strength using Eq. (16) gives the most accurate

values in reproducing the results of experimental testing.

However, the conclusion in [60] was based on several

inappropriate material properties and expressions including

a much too large single-crystal yield strength (r0) of pureMg matrix in the HallPetch expression.

Compounding method

The compounding approach differs from the previous two

summation methods in that the strengthening contributions

from the NPs are not added as Eq. (14) but are multiplied to

the original matrix yield strength. In other words, the

influences from each strengthening mechanism are

repeatedly counted as multiplication factors as follows:

r rm Drf ; 17Dr rm Drf 1 ; 17awith

Drf 1 DrGRrm

1 DrOrowanrm

1 DrCTErm

1 DrModulusrm

1 DrLoadrm

17b

As one of the representative models employing the

concept of compounding method given in Eqs. (17) and

(17b), Zhang and Chen [61] have demonstrated that their

approach produces the yield strength of composites very

close to the experimental findings of Mg/Y2O3 MMNCs

[62]; they proposed that the yield strength of MMNCs is

predicted by the following equation:

r 1 12

Vp

rm A B ABrm

; 18

A 1:25Gmb

12DaDTVpbdp1 Vp

s

; 18a

B 0:13Gmbdp

12Vp

131

lndp2b

18b

In this Zhang and Chens model [61], the effects of

Orowan, CTE mismatch, and load-bearing mechanisms are

considered, but grain refinement effects are ignored. In

other words, they used the monolithic yield strength of the

pure Mg material without NPs for rm for all materials, eventhough the monolithic Mg material contained grains with

much larger size (49 lm) than the MMNCs with Al2O3reinforcements (11 and 14 lm).

Experimental datasets and material properties for pure

Mg MMNCs

Table 3 summarizes the currently available microstructural

and mechanical property data for pure Mg MMNCs with

4196 J Mater Sci (2013) 48:41914204

123

corresponding references [5, 6370]. It includes the types,

sizes, and volume fractions of reinforcement NPs, processing

routes, Mg matrix grain sizes (dp), yield strengths (ry), ulti-mate tensile strengths (rUTS), and the stain to failure (ef). Forour analysis, Mg-based MMNCs are selected over Al-based

MMNCs to investigate the strengthening, because there is

little data available that include the necessary microstructural

and mechanical properties for Al MMNCs though they also

have been widely studied in recent years. Using the data in

Table 3, a thorough analysis has been carried out to examine

the individual strengthening mechanisms and the strength

summation methods previously addressed in Strengthening

mechanisms and Strength prediction models for unimodal

MMNCs sections. Throughout the analysis, we have par-

ticularly taken into account the strengthening improvement

from grain refinement and Orowan mechanisms, because they

are considered as primary strengthening factors to enhance the

overall strength of particle-reinforced MMNCs. In testing the

Table 3 Reinforcement types, sizes, and volume fractions, processing routes, matrix grain sizes, and observed mechanical properties of pure MgMMNCs from [5, 6370]

Reinforcement type (size) Process Reinforcement Grain Size Mechanical properties Ref.

VpA (vol%) VpB (vol%) D (lm) ry (MPa) rUTS (MPa) ef (%)

Al2O3 (50 nm) DMD ? hot extrusion,

(20.25:1) 150 ton

press T = 250 C

0.00 49 8 97 2 173 1 7.4 0.2 [5]

0.22 11 3 146 5 207 11 8.0 2.3

0.66 14 4 170 4 229 2 12.4 2.1

1.11 14 2 175 3 246 3 14.0 2.4

Al2O3 (300 nm) DMD 0.00 49 8 97 2 173 1 7.4 0.2 [63]

0.70 6 2 214 4 261 5 12.5 1.8

1.10 6 1 200 1 256 1 8.6 1.1

2.50 4 1 222 2 281 5 4.5 0.5

Al2O3 (A: 50 nm) ?Al2O3 (B: 300 nm)

DMD 0.00 0.00 36 4 116 11 168 10 9.0 0.3 [64]

0.50 4.50 24 8 139 26 187 28 1.9 0.2

0.75 4.25 27 9 138 13 189 15 2.4 0.6

1.00 4.00 31 7 157 20 211 21 3.0 0.3

Al2O3 (50 nm) PM ? hot extrusion,

(20.25:1) 150 ton

press T = 250 C

0.00 60 10 132 7 193 2 4.2 0.1 [65]

0.22 61 18 169 4 232 4 6.5 2.0

0.66 63 16 191 2 247 2 8.8 1.6

1.11 31 13 194 5 250 3 6.9 1.0

Y2O3 (29 nm) DMD ? hot extrusion,

(20.25:1) 150 ton

press T = 250 C

0.00 49 8 97 2 173 1 7.4 0.2 [66]

0.22 10 1 218 2 277 5 12.7 1.3

0.66 6 1 312 4 318 2 6.9 1.6

Y2O3 (29 nm) PM 0.00 60 10 132 7 193 2 4.2 0.1 [67]

0.20 25 3 156 1 211 1 15.8 0.7

0.70 13 2 151 2 202 2 12.0 1.0

ZrO2 (2968 nm) DMD ? hot extrusion,

(20.25:1) 150 ton

press T = 250 C

0.00 49 8 97 2 173 1 7.4 0.2 [68]

0.22 8 2 186 2 248 4 4.7 0.2

0.66 5 2 221 5 271 6 4.8 0.7

1.11 2 1 216 4 250 6 3.0 0.2

Al2O3 (300 nm) PM ? hot extrusion,

(20.25:1) 150 ton

press T = 250 C

1.10 11 4 182 3 237 1 12.1 1.4 [69]

Al2O3 (1.0 lm) 1.10 11 3 172 1 227 2 16.8 0.4

Y2O3 (3050 nm) PM ? microwave

sintering ? hot extrusion,

(25:1)

T = 250 C

0.00 20 3 134 7 193 1 7.5 2.5 [70]

0.17 19 3 214 4 214 4 8.0 2.8

0.70 18 3 244 1 244 1 8.6 1.2

In processing methods, PM and DMD indicate powder metallurgy and disintegrated melt deposition, respectively. VpA and VpB are the volumefractions of A (primary) and B (secondary, if any) reinforcement particle types, dp is the particle diameter, ry is the yield strength, rUTS is theultimate tensile strength, and ef is the stain to failure, respectively

J Mater Sci (2013) 48:41914204 4197

123

contributions of individual strengthening mechanisms and the

adequacy of the strength prediction methods, we must first

identify the proper material properties such as HallPetch and

Orowan parameters along with relevant strengthening mech-

anism expressions. For the grain refinement strengthening, the

HallPetch proportionality constant ky must be determined for

Eq. (1). In the current work, we have adopted ky =

291 MPa

lmp

based on Ono et al.s measurement [21] con-

sidering that the general consensus on ky for pure Mg is 250

300 MPa

lmp

. For Orowan, CTE, and modulus mismatch

strengthening mechanisms, we used the following data for

pure Mg based on the physical property values found in prior

literatures [33, 44, 61]: Gm = 16.5 GPa, Burgers vector

b = 0.32 nm, Poissons ratio m = 0.3, the dislocationstrengthening coefficient b = 1.25 in Eq. (11), and the CTEvalues of pure Mg aMg = 28.4 9 10

-6 K-1, Al2O3 aAl2O3 =8.1 9 10-6 K-1, Y2O3 aY2O3 = 9.3 9 10

-6 K-1, and ZrO2aZrO2 = 10.3 9 10

-6 K-1, respectively. We assumed that

CTEs are all constant over the entire processing and service

temperature ranges. This assumption will generate an error in

the range of at most *3.2 % depending on the processingtemperatures and materials types considering the CTE value

changes of *10 % with difference temperatures. Finally,Eq. (6) along with (6a) is used to estimate the Orowan con-

tribution, DrOrowan, because it is the most widely used formulaand is claimed to generate reasonable strength prediction

results consistent with experimental measurements [33, 38,

39, 61].

Strengthening prediction for pure Mg MMNCs

Figure 1 shows stacked bar graphs that quantify the pre-

dicted individual contribution of grain refinement (DrHP),Orowan (DrOrowan), and CTE mismatch (DrCTE)strengthening mechanisms colored by pink, yellow, and

orange contrasts, respectively, to the overall yield strength

improvement of the pure Mg MMNCs listed in Table 3.

These bars were generated using the representative grain

refinement (Eq. (1)), Orowan (Eqs. (6), (6a)), and CTE

mismatch (Eqs. (11), (11a)) theories, and the material/

processing parameters given in Experimental datasets and

material properties for pure Mg MMNCs section. One

experiment with a fixed NP volume fraction is represented

by single bar graph, and a set of experiments is clustered

together as indicated by the reinforcement types and ref-

erences along the x-axis. The potential strength improve-

ments from modulus mismatch and load-bearing

mechanisms are purposefully not included in this figure.

Contribution from the modulus mismatch strengthening is

relatively difficult to estimate because the bulk strain in

Eq. (12a) cannot be easily identified, and the load-bearing

effect is likely to provide only a minor contribution. From

Fig. 1, it is clear that the grain refinement strengthening

(pink bars) and the CTE mismatch strengthening (orange

bars) are both predicted to be large in magnitude, while the

effects of Orowan strengthening (yellow bars) are rela-

tively minor. Here, it should be mentioned that these plots

are based on the presumption of the full activation of the

three strengthening mechanisms.

Figure 2 shows a comparison chart between the predicted

yield strength increase (Drtheory) and the measured yieldstrength increase (Drexperiments). Drexperiments was simplyextracted from the measurement data in Table 3, and

Drtheory were calculated by the superposition of the three(i.e., grain refinement, Orowan, and CTE mismatch)

strengthening components, and the modulus mismatch effect

as shown in Fig. 1. A bulk strain (e) value of 0.001 was usedfor the modulus mismatch mechanism assuming only a

minimal contribution of this mechanism. As seen in the

figure, we tested the predicted yield strengths from the three

strength summation methods (i.e., arithmetic, quadratic, and

compounding), and additionally, we also examined the

Zhang and Chen (ZC) summation method [61]. Their

approach was particularly examined since, in some previous

reports, this model has been demonstrated to reasonably

reproduce the experimental yield strengths of MMNCs [7,

61]. In Fig. 2, the dotted diagonal line denotes the one-to-

one correspondence between the theory and experimental

measurements. From the vertical positions of symbols in

Fig. 2, it is apparent that the majority of theoretical pre-

dictions, even though the minimal contribution from the

modulus mismatch strengthening was approximated, sig-

nificantly overestimate the yield strengths of MMNCs (i.e.,

data points are above the dotted line), except some estimates

by quadratic summation and the ZC models. In general, the

compounding and the arithmetic summation models predict

Fig. 1 Contribution of different strengthening mechanisms, grainrefinement (DrGR), Orowan (DrOrowan), and CTE mismatch (DrCTE),based on the experimental data in Table 3 (Color figure online)

4198 J Mater Sci (2013) 48:41914204

123

the highest and the next higher Drtheory, and the quadraticsummation and ZC methods estimate relatively lower

Drtheory; Dr (compounding) [Dr (arithmetic) [Dr (qua-dratic) and Dr (ZC). This trend can be easily anticipatedfrom the nature of mathematical expression for these sum-

mation methods and from the deficiency of the grain

refinement component in ZC approach. The significance of

the missing grain refinement effect in the ZC model is

clearly seen in the chart as it consistently produces negative

deviation from the dotted line when Drexperiments increases(the orange triangle symbols in the Drexperiments range of75125 MPa). Higher Drexperiments generally related to thehigher grain refinement effect as will be subsequently

demonstrated; thus, it can be inferred that ZC approach

tends to show underestimation in the yield strengths as the

grain refinement effect increases. From Fig. 2, it is plausible

that the full activation of CTE and modulus mismatch

strengthening mechanisms is unlikely to occur due to the

overestimation.

Since most predicted yield strengths based on grain

refinement, Orowan, CTE, and modulus mismatch mech-

anisms exhibit considerable positive deviation from the

experimental observations, and it is not clear that CTE and

modulus mismatch mechanisms are truly active, we tested

the Drtheory based only on HallPetch and Orowan mech-anisms excluding the effects of CTE and modulus mis-

match strengthening. In Fig. 3, the predicted Drtheoryvalues are plotted with reference to the experiments using

arithmetic, quadratic, and compounding superposition

methods. The dotted diagonal line again represents exact

agreement between the theory and experiment. As appar-

ently seen from Fig. 3, the theory now predicts the correct

trend, but underestimates the yield strength of Mg MMNCs

for the majority of experiments listed in Table 3. This trend

showing negative deviations likely results from underesti-

mating the contribution from grain refinement and/or the

Orowan mechanism. In determining the grain refinement

contribution, we used ky = 291 MPa

lmp

treating

MMNCs as pure Mg materials; however, it may be that the

presence of NPs results in a higher ky in Mg MMNCs

causing a greater contribution of grain refinement that

would raise the predicted theoretical data points in Figs. 2

and 3. Since all materials analyzed have been subjected to

mechanical post-processing treatments, increased strength

due to work-hardening is another factor that may explain

the discrepancy in the predicted strength when only grain

refinement and Orowan strengthening are considered. Post-

processing using extrusion or rolling involves large

amounts of deformation and generates dislocations that

likely lead to improved yield strength in MMNCs. Such

work-hardening effect has apparently never been consid-

ered when strengthening mechanisms for MMNCs are

discussed, likely because it is the result of post-processing

and not directly attributable to the incorporation of NPs.

In Fig. 3, there are two distinct datasets as enclosed by

ellipses A and B to show the opposite extreme deviations in

the yield strength predictions. It is thought that Drtheory isoverestimated for the dataset A (ZrO2, Vp = 1.11 vol%,

DMMNC = 2 lm [68]) because of the significantly smallermatrix grain size; as the grain size decreases to near (or

below) 1 lm range, the effect of grain refinement isanticipated to abruptly increase by the HallPetch expres-

sion (Eq. (1)), but it seems that Eq. (1) may overestimate

the contribution of grain refinement for the dataset A as

Fig. 2 Comparison between experimental (Drexperiments) and theo-retical (Drtheory) yield strength improvements using arithmeticsummation, quadratic summation, compounding, and ZC [61] meth-

ods with grain refinement, Orowan, CTE, and modulus mismatch

strengthening mechanisms (Color figure online)

Fig. 3 Comparison between experimental (Drexperiments) and theoretical(Drtheory) yield strength improvements using arithmetic summation,quadratic summation, and compounding methods with grain refinement

and Orowan strengthening mechanisms (Color figure online)

J Mater Sci (2013) 48:41914204 4199

123

shown in the far right bar in Fig. 1. Although there is a

report to address that the HallPetch relation is valid down

to *75 nm in some metallic composites [71], it is cur-rently unclear if the HallPetch relation in MMNCs is still

active in the experimental dataset A. For the dataset

B (Y2O3, Vp = 0.66 vol%, DMMNC = 6 lm [66]) in Fig. 3,Drtheory is much lower than the experimental measurement.This can be presumably explained by the extensive gen-

eration of dislocations from the work-hardening. It is cur-

rently unclear why the work-hardening effect would be so

prevalent in only this sample, but the significantly reduced

ductility (i.e., strain to failure) from 12.7 to 6.9 % for this

system, as shown in Table 3, supports the generation of

massive dislocations by work-hardening.

To examine the effects of only grain refinement, we

have plotted the Drexperiments in Fig. 4 using the grain sizeand yield strength data in Table 3 as a function of the grain

size reduction, i.e., 1DMMNC

p 1D0

p

, where DMMNC and D0

denote the matrix grain size of MMNCs and pure Mg

materials without NPs, respectively. In the figure, the solid

line is drawn to represent only the contribution from grain

refinement strengthening to Drtheory, meaning that theslope of the solid line is 291 MPa

lmp

. Therefore, if the

Drexperiments data fall on the solid line, it can be understoodthat grain refinement is the only active mechanism. If the

experimental data lie above the solid line, it is then con-

sidered that mechanisms other than grain refinement

strengthening may influence the overall yield strength

improvement for the corresponding MMNCs. From Fig. 4,

it is seen that, in general, Drexperiments increases with the

degree of grain size refinement, 1DMMNC

p 1D0

p

, which

confirms the positive contribution from the grain refine-

ment strengthening mechanism to the overall yield

strength. This further indicates that as the grain size of

MMNCs decreases, the relative contribution from grain

refinement to the total Dr monotonically increases. InFig. 4, about two thirds of Drexperiments data are reasonablyexplained by the sum of grain refinement effect and a

constant offset caused by other strengthening mecha-

nism(s). In other words, when the solid HallPetch line is

shifted up by *35 MPa as indicated by the dotted line, itreasonably predicts two thirds of Drexperiments data, whichsignifies that the yield strength improvement of current

MMNC synthesis is highly dependent on the grain size of

the synthesized matrix. Further, this indicates that the

specimens showing much improved yield strength gener-

ally contain smaller matrix grain sizes. Hence, much of the

improvement is not directly attributable to the NPs them-

selves, but rather results from the reduced grain size that

comes from the incorporation of the NPs. One desirable

outlier from the dotted trend line is given by the symbol

C (the purple square near the top), which corresponds to the

datasets of ellipse B in Fig. 3. In this experiment, as

described in Fig. 3, Drexperiments is much higher than thedotted HallPetch trend line, meaning that a significant

portion of Drexperiments seems to be accounted for bymechanisms other than grain refinement. However, the

significantly reduced strain to failure indicates that the

improvement seems not to be coming from the desirable

Orowan mechanism but rather from the massive generation

of dislocations resulting from mechanical post-processing.

If the grain refinement mechanism predominantly deter-

mines the yield strength of MMNCs especially for the fine-

grain materials, it is very difficult to clarify or determine

which summation method will most truly predict the

experimental measurement values because all of the three

superposition models are identical in the case of only a

single strengthening mechanism. From the fact that the

yield strength improvements can be approximated by the

simple sum of grain refinement effect and a constant (i.e.,

*35 MPa) irrespective of the NP volume fractions orparticle size, it is doubtful that the Orowan strengthening

mechanism is in fact active in these materials. Given that

the *35 MPa seems to be roughly independent of NPcomposition, size, and volume fraction, it is similarly

doubtful that CTE or modulus mismatch mechanisms are

active and it would seem that increased dislocation density

from post-process work-hardening is the best and only

remaining candidate to account for this relatively modest

strength increase.

In Fig. 5, the predicted Drtheory are plotted as a functionof NP volume fraction, Vp, with two different particle

diameters, dp, of (a) 30 and (b) 100 nm, respectively.

Drtheory were predicted for the Orowan mechanism alone,as well as the three basic superposition methods using both

Fig. 4 Contribution from the grain refinement strengthening mech-anism to the overall yield strength improvement (Color figure online)

4200 J Mater Sci (2013) 48:41914204

123

Orowan and CTE mismatch mechanisms. Contribution

from the CTE mismatch strengthening was considered to

test the activation of the GND generation-based strength-

ening mechanism. The effect of grain refinement strength-

ening is not included for now because it is not directly

related to the NP addition as mentioned before. To deter-

mine the contribution from CTE mismatch strengthening,

we used the temperature difference DT of 280 K, and theCTE difference Da of 19 9 10-6 K-1 for Eq. (11a). FromFig. 5a, b, it is clear that the Drtheory is strongly influencedby the particle diameter, dp, as well as the volume fraction,

Vp. As shown by the black curves (i.e., Orowan), Orowan

strengthening mechanism predicts the yield strength

improvement of 50 and 20 MPa with 2 vol% of NP addi-

tions when the dps are 30 and 100 nm, respectively. It is,

therefore, critical to add fine NPs into the matrix in uti-

lizing the Orowan-type mechanism. By the activation of

CTE mismatch mechanism, the improvement is expected

to be greatly increased; with the same condition (i.e.,

Vp = 2 vol%, and dp = 30 and 100 nm), Drtheory predicts130 and 70 MPa using the quadratic summation. These

predictions are already close to the observed yield strength

improvements for the majority of samples in Table 3,

which disproves the activation of GND generation-type

strengthening mechanisms.

Despite the grain refinement strength enhancement of

MMNCs being caused indirectly by the addition of NPs, it is

worth theoretically examining a possible effect of NP

additions on ky. As stated earlier, a possible change in ky due

directly to NPs may account for the disagreement between

predicted and experimental yield stress improvements. Fig-

ure 6 shows the predicted yield strength increase by the

grain refinement mechanism, DrGR, with fixed values of theHallPetch slope, ky, at 150, 291, and 350 MPa

lmp

. The

minimum 150 MPa lmp and maximum 350 MPa lmp values for the ky were selected from Table 1 treating them as

lower and upper bounds for the DrGR prediction in MgMMNCs. The red curve is based on ky 291 MPa lmp ;which was used for the analyses in Figs. 24 based on Ono

et al.s report [21]. DrGR in Fig. 6 were computed assumingthat the elemental metal matrix without NPs (i.e., D0 in Eq.

(1)) has the original grain size of 50 lm. When the matrixgrain size of MMNCs (DMMNC) is decreased from 50 to

5 lm, DrGR is estimated to be 45115 MPa, which isalready greater than the strength improvement factors other

than the grain refinement effect based on the properties of

pure Mg observed in Fig. 4 (i.e., the difference between the

solid and dotted lines, about 35 MPa). If the grain size is

reduced below 5 lm, DrGR is estimated to exhibit an abruptincrease due to the fine size effect, but it is not certain that

the HallPetch expression with constant ky in Eq. (1) can

still be applied to the MMNCs with such fine grain sizes

(i.e., under *1 lm) because it may overestimate the effectof grain refinement as illustrated in the data points sur-

rounded in the ellipse A in Fig. 3. Here, we want to mention

that the grain refinement effect in Mg MMNCs would be

much greater than that in Al-based MMNCs, because the

HallPetch slope, ky, for Al materials 68 MPa lmp

is

in general much smaller compared to the slope for Mg

because of the multiple slip systems for Al [72]. Therefore,

the contribution from the grain refinement in Al MMNCs is

expected to be smaller than that in Mg MMNCs, but it still

would occupy a considerable portion in the yield strength

increase.

Finally, 3D prediction maps for the yield strength

increase in particle-reinforced Mg MMNCs using (a) grain

refinement and Orowan strengthening, and (b) grain

Fig. 5 Expected yield strength improvement (Drtheory) using arith-metic summation, quadratic summation, and compounding methods

as a function of particle volume fractions (Vp) with particle diameter(dp) of a 30 nm and b 100 nm (Color figure online)

J Mater Sci (2013) 48:41914204 4201

123

refinement strengthening mechanisms are presented in

Fig. 7a, b, respectively. The map in Fig. 7a was con-

structed using the quadratic summation method, and the

prediction in Fig. 7b is based on the sum of grain refine-

ment effect and 35 MPa, i.e., the contribution from other

sources as indicated in Fig. 4. The color legend represents

the predicted Dr in the unit of MPa by the NP volumefraction, Vp, and the matrix grain size, D. We used same

processing/material conditions (DS = 280 K, Da = 19.09 10-6 K-1 in Eqs. (11) and (11a)) and assumed the

particle diameter, dp, as 30 nm for Fig. 7a. The map based

only on grain refinement and Orowan (Fig. 7a) will act as

the prediction for desired strength improvement that would

be attained without loss of ductility, and the other map

including the grain refinement effect and a constant sum

(Fig. 7b) can be used to reasonably approximate the

observed yield strength improvement of Mg MMNCs

resulting from the grain refinement effect and probably

from work-hardening. From Fig. 7a, in theory, the yield

strength improvement will reach *280 MPa with submi-cron grain size (*1 lm) and 7.5 vol% NP additions.Given that the baseline of yield strength of monolithic

polycrystalline Mg materials is around 100 MPa, and the

maximum strength improvement is found as *130 MPa inFigs. 2 and 3 with the exception of one outlier, Dr of*280 MPa would be regarded as a significant strengthimprovement in the MMNC community. It must be noted

that the predictions presented in Fig. 7a are based on the

assumption that the NPs are not agglomerated and evenly

distributed during the synthesis procedure, i.e., the Orowan

strengthening mechanism is fully active. It is, however, still

very challenging to obtain such ideal microstructures even

using the best currently available synthesis technologies for

solid- and liquid-phase processing routes. Therefore, it is

required that more advanced synthesis technologies must be

developed to prevent agglomeration of NPs with high vol-

ume contents and to effectively reduce grain size of matrix

without generation of massive deformation dislocations.

Summary

In this work, we have reviewed the theoretical mechanisms and

summation methods to predict the yield strength of particle-

reinforced MMNCs, and we have performed analyses using Mg

MMNCs to study these mechanisms and summation methods.

The following are the main points addressed in the current

document.

It was found that all of the arithmetic, quadratic, andcompounding methods using grain refinement, Orowan,

Fig. 6 Expected grain refinement contribution (DrGR) as a functionof matrix grain diameter of MMNC, DMMNC. Initial diameter, D0, wasassumed as 50 lm (Color figure online)

Fig. 7 3D prediction maps for the yield strength increase in particle-reinforced Mg MMNCs using a grain refinement and Orowanstrengthening, and b grain refinement strengthening mechanisms(Color figure online)

4202 J Mater Sci (2013) 48:41914204

123

CTE, and modulus mismatch strengthening mecha-

nisms in general overestimate the yield strength, and

ZC method underestimates the strength especially when

the effect of matrix grain size becomes dominant. On

the other hand, it is demonstrated that the three

conventional summation methods using only grain

refinement and Orowan mechanisms produce yield

strength values lower than the experimental ones.

Since the activation of CTE and modulus mismatchmechanisms are unlikely to occur, and the yield strength

prediction based only on grain refinement and Orowan

mechanism generally showed underestimation, it is

inferred that work-hardening from post-processing of

MMNCs may contribute to the overall strength improve-

ment. Such work-hardening effect is supported by the

reduction in failure strains of MMNCs.

Our analysis shows that, in most samples, the measuredyield strength increases linearly with the inverse of the

square root of matrix grain size, which confirms that the

grain refinement effect predominantly determines the

strength of MMNCs, in which case all summation

methods become identical to the HallPetch relation

with about a 35 MPa adjustment to r0. Therefore, giventhe current lack of a sufficient strengthening contribu-

tion by other mechanisms it is currently not feasible to

test which summation method produces the most

consistent results with experimental observations.

We present the prediction map for the yield strengthimprovement of Mg MMNCs. In theory, the yield

strength of Mg MMNCs would reach 380 MPa with a

submicron matrix grain size and 7.5 vol% particle

addition. In attaining MMNC products with such theo-

retical strength, it is, however, still considered as a huge

roadblock for MMNC synthesis technologies to minimize

the particle agglomeration while maintaining uniform

distributions, and also to prevent excessive generation of

dislocations in effectively reducing matrix grain size.

Acknowledgements This work is primarily supported by theResearch Growth Initiative (RGI) Award from University of Wis-

consin-Milwaukee (UWM). Partial support from the U.S. Army

Research Laboratory (US ARL) under Cooperative Agreement No.

W911NF-08-2-0014 is also acknowledged. The views, opinions, and

conclusions made in this document are those of the authors and

should not be interpreted as representing the official policies, either

expressed or implied, of Army Research Laboratory or the U.S.

Government. The U.S. Government is authorized to reproduce and

distribute reprints for Government purposes notwithstanding any

copyright notation herein.

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Prediction models for the yield strength of particle-reinforced unimodal pure magnesium (Mg) metal matrix nanocomposites (MMNCs)AbstractIntroductionStrengthening mechanismsGrain refinement strengthening (Hall--Petch strengthening)Orowan strengtheningCoefficient of thermal expansion mismatch strengtheningOther strengthening mechanisms

Strength prediction models for unimodal MMNCsArithmetic summation methodQuadratic summation methodCompounding methodExperimental datasets and material properties for pure Mg MMNCs

Strengthening prediction for pure Mg MMNCsSummaryAcknowledgementsReferences