Extreme Mechanics Letters 8 (2016) 213–219

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Extreme Mechanics Letters

journal homepage: www.elsevier.com/locate/eml

Gradient plasticity in gradient nano-grained metalsZhi Zeng a, Xiaoyan Li b, Dongsheng Xu c, Lei Lu c, Huajian Gao d, Ting Zhu a,∗

a Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USAb Center for Advanced Mechanics and Materials, Applied Mechanics Laboratory, Department of Engineering Mechanics,Tsinghua University, Beijing 100084, Chinac Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, Chinad School of Engineering, Brown University, Providence, RI 02912, USA

a r t i c l e i n f o

Article history:Received 13 November 2015Received in revised form 11 December 2015Accepted 13 December 2015Available online 14 December 2015

Keywords:Gradient plasticityCrystal plasticityFinite element

a b s t r a c t

Gradient nano-grained (GNG) metals are a unique class of materials with spatial gradientsin grain size, typically from the surface to the bulk. Here the gradient mechanical behaviorinGNG copper is studied by a crystal plasticity finite elementmodel that accounts for grain-size-dependent yield strengths. The associated finite element simulations reveal both thegradient stress and gradient plastic strain in the cross section of GNG copper subjected toaxial tension. These spatial gradients arise due to progressive yielding of gradient grainsunder an overall uniform deformation. They stand in stark contrast to the widely studiedstrain gradient plasticity induced by imposing a non-uniform deformation such as torsion,bending, and indentation. Our work suggests a new material strengthening mechanismthrough the introduction of plastic strain gradients via gradient microstructures.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Gradient nano-grained (GNG) metals are a new classof materials with unique polycrystalline microstructures[1,2]. Their grain sizes typically change with a gradientvariation between tens to hundreds of nanometers, fromthe surface to the bulk. GNG metals promise to achievean unprecedented combination of strength, ductility andtoughness. They have great potential in engineering ap-plications [1–6] and also motivate exploration on variousother kinds of gradient nanostructured materials [7–10].

Experiments have been performed to process the GNGmetals and further study their mechanical behavior. Fanget al. [1] utilized surface mechanical grinding treatment toprepare a GNG surface layer in a bulk coarse-grained (CG)rod of face-centered cubic (FCC) Cu. The topmost layer ofthe GNG structure, up to a depth of 60 µm, consisted of

∗ Corresponding author.E-mail address: ting.zhu@me.gatech.edu (T. Zhu).

http://dx.doi.org/10.1016/j.eml.2015.12.0052352-4316/© 2015 Elsevier Ltd. All rights reserved.

nano-grains with an average grain size of about 20 nm. Thegrain size gradually increased to about 300 nm in the depthof 60 µm–150 µm. Below the depth of 150µm, the grainsize continued to increase to that of coarse grains at themicrometer scale. The tensile yield strength of the GNG/CGCu was two times that of CG Cu, and the yield strength ofthe free-standing GNG foil was ten times that of CG Cu.Recently, Wu et al. [5] used surface mechanical attritiontreatment to prepare GNG samples of body-centered cubic(BCC) steel with a sandwich sheet structure, i.e., a CG corein between two GNG layers. The tensile tests showed thatthe gradient structure induced an extra strain hardening,which led to a high ductility. This extra strain hardeningwas attributed to the effects of the macroscopic straingradients associated with multi-axial stress states in GNGstructures.

The experiments described above have shown the en-hanced mechanical properties of GNG metals. However,the detailed mechanisms underlying the observed me-chanical behavior of GNG metals remain little understood[11–13]. In this letter, we employ a crystal plasticity

214 Z. Zeng et al. / Extreme Mechanics Letters 8 (2016) 213–219

finite element (CPFE) model to investigate the mechanicalresponses of GNG Cu. We build a GNG structure by adapt-ing the conventional Voronoi tessellationmethod [14]. Wealso extend the classical crystal plasticity theory to in-corporate the grain-size-dependent constitutive relations.The associated finite element simulations reveal a noveltype of gradient stress and gradient plastic strain in thecross section of GNG samples subjected to axial tension.These spatial gradients arise due to progressive yieldingof gradient grains under an overall uniform deformation.They stand in stark contrast to the widely studied straingradient plasticity induced by imposing non-uniform de-formations. Our work has important implications for ma-terial strengthening through the introduction of plasticstrain gradients via gradient microstructures.

2. Modeling methods

2.1. Grain-size-dependent crystal plasticity theory

To model the constitutive response of GNG Cu, we ex-tend the classical crystal plasticity theory by incorporat-ing the grain size dependence of yield strength. The rate-dependent finite strain crystal plasticity theory adoptedhere can be traced to the work by Rice [15], Asaro andRice [16], and Kalidindi et al. [17]. According to Kalidindiet al. [17], the plastic shearing rate on the slip system α is

γ α= γo

τ α

sα

1/m sign (τ α) (1)

where τ α is the resolved shear stress, sα is the slip resis-tance, γo is the reference shearing rate, and m is the strainrate sensitivity parameter. The initial value of sα is de-noted as s0. During plastic deformation, the slip resistancesα evolves according to

s =

β

hαβγ β

, hαβ= qαβh(β),

h(β)= h0

1 − sβ/ssat

a (2)

where qαβ are the components of amatrix which describesthe latent hardening behavior of the crystal, and h0, a andssat are the hardening parameters taken to be identical forall slip systems.

Our crystal plasticity model is dependent on grain size.For each grain, the slip resistance parameters, includ-ing s0, h0, a, ssat, m, are taken as functions of grainsize [18,19]. According to the classical Hall–Petch relation,the overall yield strength is inversely proportional to thesquare root of grain size [20,21]; such a relation is valid forgrain sizes greater than ∼20 nm [22,23]. In this work, weassume that all the slip resistance parameters at the singlecrystal level are inversely proportional to the square rootof the size of the local grain D,

s0(D), h0(D), a(D), ssat(D),m(D) ∼ D−1/2. (3)

The numerical values of grain-size-dependent s0 arereadily estimated from the available experimental data inthe literature, but the determination of the strain hard-ening related parameters h0, a, ssat requires certain

assumptions. Specifically, the values of s0 are determinedbased on the experimentally measured yield stresses ofnanocrystalline Cu with uniform grain size, i.e., from860 MPa to 400 MPa for D from 20 nm to 110 nm [24–26].Dividing these macroscopic yield stresses by Taylor’s fac-tor of 3, we estimate s0 in between 286 MPa and 133 MPa.Nanocrystalline Cu exhibits little hardening in experi-ments. However, we assign small values of h0, a, ssat,so as to produce a weak hardening for ensuring numeri-cal stability in CPFE simulations. For the 20 nm grain, wetake h0 = 102 MPa, a = 2.0, ssat = 600 MPa; and for the110 nm grain, h0 = 48 MPa, a = 1.8, ssat = 287 MPa.Furthermore, experiments show that the strain rate sen-sitivity exponent m of nanocrystalline Cu with uniformgrain size varies from 0.04 to 0.022 for D from 20 nm to110 nm [27,28]. To evaluate s0 for intermediate grain sizes,we use the above bounding values to fit the formula ofs0 = B + C · D−1/2 where B and C are the fitting constants.Along the same line, we also fit other slip resistance pa-rameters of h0, a, ssat, m for intermediate grain sizes.The following fitting formulas are obtained:

s0(MPa) = 19.3 + 1196 · D−1/2,

h0(MPa) = 8.68 + 415.8 · D−1/2,(4)

a = 1.65 + 1.56 · D−1/2,

ssat(MPa) = 53.7 + 2443 · D−1/2,

m = 0.0086 + 0.1403 · D−1/2.

Other material properties, including elastic constants(C11, C12, C44), twelve 111 ⟨110⟩ slip systems, andthe latent hardening matrix

qαβ

, are assumed to be

independent of grain size. For FCC Cu, we take C11 =

170 GPa, C12 = 124 GPa and C44 = 75 GPa; qαβ= 1.0

if the slip systems α and β are coplanar and qαβ= 1.4 if

they are non-coplanar [17].

2.2. Finite element model

We construct a two-dimensional GNG structure withcolumnar grains by adapting the Voronoi tessellationmethod inMatlab. The geometrical information of the GNGstructure is then used to develop the corresponding fi-nite element model in ABAQUS/CAE with a Python script.Fig. 1 shows an example of the GNG structure generatedin ABAQUS/CAE. In this case, the grain size D increaseslinearly from ∼20 nm in the top/bottom surface layer to∼110 nm in the central region. The overall sample geom-etry is 640 nm in length and 1120 nm in width. As such,the spatial gradient in grain size, |dD/dy|, is about 0.1.The orientation of grains is assigned randomly in termsof three Euler angles, θ, ϕ, Ω, representing rotationsfrom the crystal basis to the global basis [17]. The sampleis meshed with the non-structured plane strain elements.As a result, most elements have four nodes (CPE4R) and asmall fraction three nodes (CPE3). Displacements and trac-tions are continuous at grain boundaries, meaning no sep-aration or sliding between every pair of adjoining grains.A user material subroutine VUMAT [29] is developed inABAQUS/EXPLICIT to implement the grain-size-dependentcrystal plasticity model described above. All the slip

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Fig. 1. Crystal plastic finite element model of GNG Cu, which consistsof an assembly of quasi-2D columnar grains having a linear variation ofgrain size from ∼20 nm in the top/bottom surface layer to ∼110 nm inthe central region.

resistance parameters at each integration point dependon the local grain size D according to Eq. (4). Using theGNG structure in Fig. 1, CPFE simulations are performedto investigate the axial tensile responses of GNG Cu un-der the plane-strain condition. The boundary conditions ofthe GNG sample are prescribed as follows: the upper andlower surfaces are traction free; on the left side (x = 0),the displacement in the x direction is zero (ux = 0); on theright side (x = 640 nm), the velocity in the x direction isconstant (vx = 0.64 nm/s), corresponding to an appliedtensile strain rate of 0.001/s.

3. Results and discussion

Fig. 2 shows the simulated stress–strain response of theGNG Cu sample subjected to axial tension under the plane-strain condition. Deformation begins with a linear elasticresponse. When the elastic strain reaches about 0.27%,the GNG Cu starts to yield, i.e., deviating from the linearstress–strain response, at about 390MPa. Further strainingresults in a smooth and gradual increase of the axialtensile stress. Such a yielding behavior can be attributedto progressive attainment of yield strengths in grains withrandom orientations and, more importantly, with gradientsizes having gradient slip resistances. When the appliedtensile strain reaches about 0.5%, the entire GNGCu sampleyields and the stress–strain curve attains a plateau of about500 MPa.

Importantly, our CPFE simulation reveals the gradientstresses in the cross section of GNG Cu, which arise due

Fig. 2. CPFE simulated stress–strain curve for a GNG Cu sample subjectedto axial tension. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

to progressive yielding of grains with gradient sizes. InFig. 3(a), we plot the distributions of the axial stress inthe sample cross section at various applied strains. Togenerate these plots, we divide the GNG Cu sample into 60slabs in the y direction and then calculate the average ofnodal axial stresses in each slab. As such, the data pointsin Fig. 3(a) correspond to the average axial stresses atdifferent locations of the cross section, and the solid linesare the fitting curves. It is seen that at small applied strains,the distribution of axial stresses in the cross section isnearly uniform, e.g., the pink curve at a strain of 0.2%.This is because all the grains undergo elastic deformationand the linear elastic constitutive relation is independentof grain size. As the applied strain continues to increase,large grains in the central region of the cross sectionbegin to yield, causing a drastic slowdown of increaseof axial stresses in the central region. In contrast, smallgrains near the surface continue the elastic deformation,resulting in a more pronounced increase of axial stressesnear the surface. Due to the gradient grain sizes betweenthe central and surface regions, the cross-sectional stressexhibits a smooth gradient variation, e.g., the blue curve ata strain of 0.33%. Such gradient stress evolves as the appliedstrain increases, e.g., the green curve at a strain of 0.5%.Meanwhile, the plastically yielded domain expands fromthe central region to the surface. Finally, the entire GNGsample yields and the gradient stress is fully developedwithout much further change as the applied strain isincreased, e.g., the red curve at a strain of 1%. This stagecorresponds to the plateau of the stress–strain curve inFig. 2.

Our CPFE simulation also reveals the gradient plasticstrains in GNG Cu. In Fig. 3(b), we plot the distributionsof axial plastic strains in the sample cross section at vari-ous applied strains; here the axial plastic strains are evalu-ated using the same averaging scheme as the axial stresses.As the gradient stress, the gradient plastic strain is devel-oped due to progressive yielding of grains with gradientsizes. We note that the gradient plastic strains exhibit themaximum in the centerwhere large grains first attain yield

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Fig. 3. Distributions of (a) gradient axial stresses and (b) gradient axial plastic strains in the cross section at various applied strains as marked in Fig. 2.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Contours of axial stresses at the applied strain load of (a) 0.2% and (b) 1%. The color map is in the unit of MPa. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of this article.)

points and the minimum at the surface where progres-sive yielding arrives the latest. Hence, the sign of the plas-tic strain gradient is opposite to that of the stress gradi-ent, as seen from Fig. 3(a) and (b). More interestingly, thegradient plastic strains arise under an overall uniform de-formation of the GNG sample, and they stand in contrastto the widely studied strain-gradient plasticity induced byimposing a non-uniform deformation such as torsion [30],bending [31], and indentation [32]. The plastic strain gra-dient signifies the plastically inhomogeneous deformationthat can provide a non-local effect of material strengthen-ing [33], which is additional to the strengthening gener-

ated by local plastic strains. In this work, we will not fur-ther quantitatively evaluate the non-local strengtheningeffect of plastic strain gradients, due to the lack of relatedexperimental characterization and data. Nonetheless, thegradient plastic strain arising fromgradient grain sizes rep-resents a new material strengthening mechanism due tonon-homogeneous plastic deformation and warrant a sys-tematic study in the future.

To further understand the spatial distributions of gra-dient stresses and gradient plastic strains in GNG Cu,we plot in Fig. 4 the contours of axial stresses at twodifferent applied strains, showing the evolving spatial

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Fig. 5. Contours of axial plastic strains at the applied strain load of (a) 0.33% and (b) 0.5%.

distribution of axial stresses. When the applied strain islow, the deformation in gradient grains is elastic. At a rep-resentative axial strain of 0.2%, the spatial stress distri-bution in Fig. 4(a) is nearly uniform, consistent with theplot of the cross-sectional stress distribution (e.g., the pinkcurve) in Fig. 3(a). As the applied strain is increased, thegradient stress develops due to progressive yielding of gra-dient grains. At an applied strain of 1%, the spatial stressdistribution in Fig. 4(b) is clearly gradient, consistent withthe plot of the average stress distribution (e.g., the redcurve) in Fig. 3(a). Both Figs. 4(a) and (b) also show the in-homogeneous spatial distribution of axial stresses in grainswith similar sizes, i.e., located at a similar distance to thetop/bottom free surface. Such inhomogeneities can be pri-marily attributed to the random distribution of grain ori-entations and partly to the small variation of grain sizes.In addition, we plot in Fig. 5 the contours of axial plasticstrains at two intermediate applied strains, which exhibitthe progressive yielding responses: the larger, inhomoge-neous plastic strains have been accumulated in the centralregion, as opposed to the smaller, inhomogeneous plasticstrains in the top/bottom surface layers.

4. Conclusions and outlook

In this paper, we investigate the mechanics of GNGCu using a CPFE model. A two-dimensional structure ofcolumnar nano-grains with gradient grain sizes is built viaan adapted Voronoi tessellationmethod. The classical crys-tal plasticity theory is extended to incorporate the grain-size-dependent yield strengths. CPFE simulations are per-formed to study the tensile responses of GNG Cu with gra-dient grain sizes in the classical Hall–Petch regime. CPFEresults reveal the gradient stress and gradient plastic strainin the cross section of GNGCu. These spatial gradients arisedue to progressive attainment of yield points in grainswithgradient sizes and accordingly gradient slip resistances.

CPFE results also reveal the heterogeneous spatial distri-butions of gradient stresses and gradient plastic strains,which result from the combined effects of random grainorientations and gradient grain sizes.

The most significant results of the present study arethe gradient stress and gradient plastic strain revealed byCPFE simulations. Major implications and outlook fromthis work are discussed as follows.

Stress and plastic strain gradients. GNG metals have aunique microstructure with gradient grain sizes. As a re-sult, the gradient stress and gradient plastic strain developin plastically deformed GNG samples. These spatial gradi-ents arise due to the grain-size-dependent yield strengths.Importantly, the gradient stress and plastic strain are gen-erated under an overall uniform deformation, and theystand in contrast to the widely studied strain-gradientplasticity induced by imposing a non-uniform deformationsuch as torsion [30], bending [31], and indentation [32]. Tounderstand the microscopic mechanisms underlying thegradient plastic strains in a GNG structure, we note that thegeometrically-necessary and statistically-stored disloca-tions are generally the plastic deformation carriers in crys-tals [33]. The density of geometrically-necessary disloca-tions, ρG, depends on grain size, while that of statistically-stored dislocations, ρS , does not. Hence, ρG is usually re-sponsible for the grain size dependence of yield strength.Ashby showed that ρG is inversely proportional to grainsize, but is proportional to plastic strain [33]. On thisbasis, we show in Fig. 6 a schematic of the spatial dis-tribution of gradient geometrically-necessary dislocationsin a plastically deformed GNG structure with grain sizesin the classical Hall–Petch regime—our CPFE study ad-dresses this case. Under axial tension, big grains in thecentral region have lower yield strengths and thus pro-duce larger plastic strains and accordingly higher ρG, com-pared to small grains in the surface layer. Such a gradi-ent distribution of ρG underlies the gradient plastic strain.Our molecular dynamics simulations, to be reported in

218 Z. Zeng et al. / Extreme Mechanics Letters 8 (2016) 213–219

Fig. 6. Schematic illustration of a gradient variation of geometrically necessary dislocations (represented by ⊥) in GNG metals in the Hall–Petch regime:ρG gradually increases from the surface to the bulk.

a separate manuscript. [34], show that ρG is gradientin plastically deformed GNG Cu. Therefore, the gradientgeometrically-necessary dislocations furnish a plausiblemicroscopic mechanism of gradient plastic strains in GNGmetals.

Non-local effect of strengthening. Importantly, the plasticstrain gradient is expected to produce a non-local, higher-order strengthening effect and thereby elevate the overallyield strength. Such a strengthening effect has been wellrecognized in previous studies of strain gradient plasticityinduced by applying non-uniform deformations [30–32].In the future, it is appealing to perform a systematicexperimental study using samples with controlled GNGstructures, so as to provide a firm quantitative basis of thestrengthening effect of plastic strain gradients arising fromgrain size gradients. The present crystal plasticity modelaccounts for the grain size dependence of yield strengths,but does not include the non-local strengthening effect ofplastic strain gradients. Hence, future study also requiresthe development of a non-local plasticity model andassociated numerical procedure, so as to quantitativelyevaluate the effect of plastic strain gradients throughcomparison with experimental measurements.

Grain size gradient. The spatial distribution of gradientgrain sizes is expected to critically affect the stress andplastic strain gradients and accordingly the strengtheningeffects on the GNGmaterials. In this work, we study an ex-ample of GNG Cu having a linear variation of grain sizes,with the corresponding grain size gradient of |dD/dy| ≈

0.1. From the material design standpoint, the characteris-tics of grain size gradient, including itsmagnitude and spa-tial variation (e.g., constant gradient vs. varying gradient),can serve as important design parameters for achieving anoptimal strength enhancement with retained ductility inGNG materials. To this end, it is essential to develop novelprocessing methods for achieving the controlled grain sizegradients. Meanwhile, the non-local CPFEmodels and sim-ulations that include the strengthening effect of plasticstrain gradients can be used to guide the design of the op-timal grain size gradient.

Random gradient microstructures. Heterogeneous mate-rials can possess a random distribution of spatial gradi-ents in grain size. For example, such heterogeneous mi-crostructures are the prominent feature in metals and

alloys processed by additive manufacturing [35] but theireffects on the material mechanical responses remain littleunderstood. In addition, Wu et al. have recently demon-strated the effects of a non-uniform distribution of grainsizes on the strength-ductility synergy in titanium [36].The present work is focused on a special class of GNGmet-alswith awell-defined one-dimensional spatial gradient ingrain size from the surface to the bulk. It is relatively easyto achieve the controlled processing of such gradient mi-crostructures, thereby facilitating comparison between ex-periment andmodeling. A systematic study of such kind ofGNG metals will pave the way towards a quantitative un-derstanding of effects of random gradient microstructureson the mechanical behavior of a broader class of heteroge-neous materials.

Competing deformation mechanisms. Our crystal plastic-ity model accounts for the grain size dependent strengthsthat are controlled by dislocation plasticity within grains.In GNG materials, other competing deformation mecha-nisms, such as coupled shear andmigration of grain bound-aries, can play an important role in the gradient mechan-ical behavior of gradient microstructures. Particularly, ex-periments have shown that the mechanically driven graingrowth becomes pronounced at large tensile strains, lead-ing tomechanical softening that competeswith dislocationhardening [4]. The continuum finite element modeling ofthe concurrent grain growth and grain plasticity in poly-crystals, however, is challenging due to difficulties of track-ing the moving grain boundaries. Hence, development ofadvanced continuum modeling techniques is necessary tosimulate the concurrent processes of grain growth andgrain plasticity in gradient microstructures.

Acknowledgments

We acknowledge support by the NSFC grant51420105001 and the Shenyang Supercomputing Centerof CAS. TZ acknowledges support by the National ScienceFoundation grant DMR-1410331. XL acknowledges sup-port by the NSFC grant 11372152. XDS acknowledges sup-port by the National Basic Research Program of China (973Program, 2011CB606404). LL acknowledges support by theNational Basic Research Program of China (973 Program,2012CB932202) and the ‘‘Hundreds of Talents Project’’ bythe Chinese Academy of Sciences.

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