Mechanical properties and multi-scale modeling of nanocrystalline materials

Acta Materialia 55 (2007) 3563–3572 www.elsevier.com/locate/actamat

Mechanical properties and multi-scale modeling of nanocrystalline materials S. Benkassem

a,*

, L. Capolungo b, M. Cherkaoui

a,b

a b

LPMM-CNRS Universite´ Paul Verlaine, Ile du Saulcy, 57045 Metz, Cedex 1, France George W. Woodruff School of Mechanical Engineering, Atlanta, GA 30332-0405, USA

Received 31 July 2006; received in revised form 2 February 2007; accepted 5 February 2007 Available online 28 March 2007

Abstract A generalized self-consistent scheme based on the coated inclusion method and using interfacial operators is developed and used to describe the grain-size-dependent viscoplastic behavior of pure fcc nanocrystalline materials. The material is represented by an equivalent three-phase material composed of coated inclusions embedded into an equivalent homogeneous medium. Inclusions represent grain cores and behave viscoplastically via dislocation glide while the coating represents both grain boundaries and triple junctions. A recently introduced constitutive law accounting for grain boundary dislocation emission and penetration is used to model the behavior of the coating. The model is applied to pure copper and enables the quantification of the macroscopic effect of interface dislocation emission. The analysis is completed with a set of finite element simulations revealing high stress concentrations at triple junctions.  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocrystalline materials; Micromechanics; Dislocation; Hall–Petch

1. Introduction Over the past decade, numerous studies have been dedicated to the size effect in the behavior of nanocrystalline (NC) materials [1–3], best exhibited by the ‘‘breakdown’’ of the Hall–Petch law [2,4] occurring at small grain sizes (d  20 nm) [5–7]. Experimental studies as well as molecular dynamics (MD) simulations have revealed that the volume fraction of grain boundaries and triple junctions becomes significant when the grain size is smaller than 50 nm [8] and that dislocation activity within grain cores is significantly reduced and cannot account for the total plastic deformation of NC materials [9–11]. Hence, the viscoplastic behavior of NC materials was suggested to be a consequence of grain boundary related phenomena (e.g., vacancy diffusion, dislocation emission, grain rotation, etc.).

*

Corresponding author. Tel.: +33 404 271 2755. E-mail address: [email protected] (L. Capolungo).

Modeling of NC materials relies on two key aspects: the identification and description of active mechanisms and the application of adapted scale transition techniques. The latter avers to be a critical issue since most data available are extracted from MD simulations which are performed at the atomistic scale and with applied boundary conditions several orders of magnitude higher than those in real quasistatic tests. While coarse graining techniques have been successful in predicting texture evolution in metals [12], no such success has been reached in the prediction of size effects in the response and properties of metals. None the less, a recent model based on an extension of the continuous theory of dislocations offers promising results [13]. In particular, LeSar and Rickman calculated the net interaction energy, as a function of space and time parameters, of a system subdivided in cells containing dislocations [13]. These parameters could then be used in a higher-order scale model. As will be discussed, statistical mechanics appears to enable a direct scale transition from the atomic scale to the mesoscopic scale at which dislocations are modeled. Regarding the possible active deformation mech-

1359-6454/$30.00  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.02.010

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anisms, three routes have been considered: (1) creep phenomena, (2) grain boundary dislocation emission and (3) solid motion of grains. A relatively large number of models, including mechanisms typically observed in the case of superplastic behavior (e.g., Coble creep, Nabarro–Herring creep, Lifschitz sliding), have been introduced to describe the size effect in the viscoplastic behavior of NC materials [9–11,14–18]. In the case of diffusion-based mechanisms, several models based on composite approaches have been successful in predicting the ‘‘breakdown’’ of the Hall–Petch law [14,17,19,20]. Among others, it was revealed that if diffusion phenomena are indeed involved in the plastic deformation process, the preferred diffusion route is located at the grain boundary/grain core interface [15,16]. Although, these models overestimate the effect of diffusion due to the steady-state description of the process [19,21]. Moreover, recent experiments by Li et al. have clearly shown that defect-free nanocrystalline materials do not creep at room temperature [22]. A second school of thought has emerged from recent MD simulations (both two-dimensional (2-D) columnar and fully three-dimensional (3-D)) which showed that grain boundaries could actively participate in the deformation process in NC materials [23–30]. As predicted in Gutkin et al.’s theoretical work [31], it was shown via MD that grain boundaries can emit dislocations into grain cores, although some controversy related to the critical factor limiting the emission of the trailing partial dislocation still exists [26]. None the less, important information can already be extracted from MD simulations on bicrystal interfaces [32], 3-D structures [25] and 2-D columnar structures [33]. For example, simulations on pure tilt symmetric bicrystal interfaces clearly show that grain boundary dislocation emission is a thermally activated process and that dislocations are emitted on primary slip systems [32,34]. Although let us note here that solely the emission of the leading partial dislocation can be predicted from atomistic simulations. Markmann et al. [35] observed an increase in the stacking fault density of NC Pd during cold rolling, showing the activity of dislocations, which could be perfect or incomplete, in NC materials. Also, these experiments suggest that grain boundary sliding and grain rotation could be concurring processes to the dislocation emission process. Sansoz and Molinari’s [36] quasicontinuum study on sigma tilt grain boundaries revealed that, in the case of grain boundaries containing E structural units, atomic rearrangement can occur prior to the emission of the leading partial dislocation. Also, recent MD simulations on R5(1 1 0) pure tilt bicrystal interfaces have shown that for this particular geometry grain boundary ledges: (1) are more prone to emit dislocations than perfect planar interfaces and (2) are dislocation sources and not dislocation donors [37]. This study also shows that the emission of the trailing partial dislocation does not lead to a net strain localized in the neighbourhood of the dislocation

source. However, these simulations shall be extended to more geometries to lead to conclusive results. Although the growing number of molecular simulations has tremendously improved the current understanding of the grain boundary dislocation emission process, models at the continuum coupled with an appropriate scale transition from the mesoscopic scale to the macroscopic scale (e.g., finite elements, strain gradient models or micromechanics) are the only methodologies available to quantify the effect of grain boundary dislocation emission in NC materials. This issue could be critical since the emission and penetration process could promote grain boundary sliding [35]. Finally, grain boundary sliding and grain rotation have recently been subjected to in-depth studies. Wei and Anand [38] have recently successfully predicted the relatively low ductility of NC materials as resulting from interface decohesion. Moreover, a recent quasicontinuum study by Warner and Molinari suggests that the grain boundary sliding mechanism could be appropriately described as a stick–slip mechanism [39]. Note that this study also suggests that grain boundary sliding alone cannot account for the ‘‘anomalous’’ behavior of NC materials. Also, it was recently suggested that grain boundary dislocation emission could be assisted by interface decohesion localized at triple junctions [40]. As mentioned above, one of the most critical issues stands in the use of scale transition techniques, enabling the extraction of the macroscopic behavior of the material from the local constitutive laws. Typically, models describing the behavior of NC materials are based on rules of mixture [15,16,18]. Although these techniques lead to simple analytical expressions, model predictions must be considered carefully, for the local viscoplastic strain rates or the local stress fields are ineluctably overestimated. Also, these approaches cannot predict stress concentrations since the predicted stress and strain fields are homogeneous. Micromechanical schemes have recently been used and proven to lead to more realistic predictions [20,21,41]. However, to date there are no rigorous techniques enabling scale transition in the case of viscoplastic behaviors [42]. An extension of Eshelby’s solution to the inclusion problem [43], based on the decomposition of both the local stiffness tensors and the local strain rate fields as the sum of a space invariant field and local fluctuation part, was used in previous studies [17,19,43]. However, this micromechanical scheme does not account for the time/space coupling of the local behavior. In more recent work [21], a self-consistent scheme, in which the time/space coupling is accounted for [44,45], was used by the same authors. Although the model leads to good results, the constitutive laws introduced are based on mass diffusion engendered by dislocation penetration events and a three-phase model, as used in Jiand and Weng’s contribution [20], could be more appropriate. In this paper, a three-phase self-consistent scheme based on Christensen and Lo three-phase model [46] and on early work by Cherkaoui et al. [47], is developed and used to

S. Benkassem et al. / Acta Materialia 55 (2007) 3563–3572

model the viscoplastic behavior of pure fcc polycrystalline materials and its sensitivity to the grain size. The micromechanical scheme developed in this work accounts for stress jumps and velocity gradients at the grain boundary/grain core interface and should consequently lead to more accurate predictions. Among others, the primary objective of this work is to quantify the sole contribution of the grain boundary dislocation emission and penetration mechanism, for which a constitutive law was recently introduced [21], to the viscoplastic response of pure fcc NC materials. Hence, the contribution of grain boundary sliding is not accounted for in the present study. Future work will focus on the incorporation of grain boundary sliding in a generalized secant self-consistent scheme. 2. Micromechanics 2.1. Topology of the three-phase model The topology of the three-phase model relies on the assumption that the real material can be represented by a volume element including a single coated inclusion embedded in a homogeneous equivalent medium. The inclusion phase represents grain cores while the coating phase represents both grain boundaries and triple junctions (Fig. 1). In what follows, the superscripts ‘‘I’’, ‘‘C’’ and ‘‘eff’’ will refer to the inclusion, to the coating and to effective properties, respectively. Bold letters represent fourth-order tensors and Greek letters denote second-order tensors. mr, with r = I, C, eff denotes the fourth-order viscoplastic tensor and br its inverse. The macroscopic behavior of the material is purely viscoplastic and can be written as follows: E_ ¼ meff : R

ð1Þ

where E_ and R denote the macroscopic strain rate tensor and the macroscopic stress tensor, respectively. Similarly the behavior of the local constituents is written as follows: e_ r ¼ mr : rr

with r ¼ I; C

ð2Þ

Here e_ r and rr denote the local strain rate tensor and the local stress tensor, respectively. The local strain rate tensors are related to their macroscopic equivalent via a concentration equation given by e_ ¼ Ar : E_

with r ¼ I; C

ð3Þ

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Here Ar represents the fourth-order concentration tensor which expression will be derived in the following. The macroscopic stress and strain rate tensors must be equal to the volume average of their local equivalents Z Z 1 1 E_ ¼ h_ei ¼ rðrÞ dr ð4Þ e_ ðrÞ dr and R ¼ hri ¼ V V V V Here V denotes the representative element’s volume, ÆÆæ terms denote volume averages and r denotes the position. The inverse of the macroscopic viscosity tensor is obtained by using the local constitutive laws and introducing Eq. (3) into Eq. (4). One obtains after some algebra beff ¼ bC þ f I ðb1  bC Þ : AI

ð5Þ

I

Here f denotes the volume fraction of inclusion. The concentration tensors are determined via the simultaneous use of the integral equation [48–50] of interfacial operators [50,51]. Traditionally, the integral equation is given by Z _ ð6Þ e_ ¼ E  C0 ðr  r0 Þ : dbðr0 Þ : e_ ðr0 Þ dr0 V

where r is an arbitrary point, C0(r  r 0 ) denotes Green’s modified tensor and db(r) is a fluctuating part of the local viscosity tensors engendered by the viscoplastic heterogeneity introduced by the inclusion and its coating. Locally, the viscoplastic tensor bijkl is decomposed into a uniform part, beff ijkl , corresponding to the viscoplastic tensor of a homogeneous reference medium and a heterogeneous part, dbijkl(r) bðrÞ ¼ beff þ dbðrÞ

ð7Þ

Heavyside step functions are introduced to describe the fluctuating part of the inverse of the viscosity tensors ( 1 if r 2 V I I h ðrÞ ¼ and 0 if r 62 V I ( 1 if r 2 V IC ¼ V I þ V C IC ð8Þ h ðrÞ ¼ 0 if r 62 V IC where V IC, V I, V C denote the composite inclusion’s volume, the inclusion’s volume and the coating’s volume, respectively. Accordingly, the volume of the coating is described by the following Heavyside function: hC ðrÞ ¼ hIC ðrÞ  hI ðrÞ

ð9Þ

Hence, the fluctuating term db(r) in Eq. (7) can be written as

Fig. 1. Topology of the three-phase model.

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S. Benkassem et al. / Acta Materialia 55 (2007) 3563–3572

dbðrÞ ¼ DbI hI ðrÞ þ DbC hC ðrÞ I

I

eff

ð10Þ C

C

eff

where Db = b  b and Db = b  b . Introducing Eq. (11) into the integral Eq. (6) leads to: Z C0 ðr  r0 Þ : DbI ðr0 Þ : e_ ðr0 Þ dr0 e_ ¼ E_  VI Z  C0 ðr  r0 Þ : DbC ðr0 Þ : e_ ðr0 Þ dr0 ð11Þ VC

placement along the boundary imposes the following condition on the jump of the gradient of the displacement: ½ui;j dxj ¼ ðu1i;j  u2i;j Þdxj ¼ 0 :

or ½ui;j  ¼ u1i;j  u2i;j ¼ ki nj ð18Þ

From the symmetry of the strain tensor, the jump in the strain rate ½_eij  across the interface is given by

Also, Eshelby’s tensor is related to the interaction tensor by

1 ½_e ¼ e_ 1  e_ 2 ¼ ðki nj þ kj ni Þ ð19Þ 2 Here ki is a proportionality factor corresponding to the magnitude of the jump. From the equilibrium condition and with the use of the local constitutive laws one obtains

S ¼ T I ðbC Þ : bC

b1 : e_ 1 nj ¼ b2 : e_ 2 : nj

Let us introduce the interaction tensor defined as follows: Z Tðbeff Þ ¼ C0 ðr  r0 Þ dr0 ð12Þ V IC

ð13Þ

Introducing (12) into (11) and averaging the resulting equation on the coated inclusion’s volume one obtains I

C

ð20Þ

Introducing (19) into (20) leads to: ½b1  b2  : e_ 1 nj ¼ b2 : nl nj kk

ð21Þ

V V E_ ¼ IC fI þ Tðbeff Þ : DbI g : E_ I þ IC fI þ Tðbeff Þ : DbC g : E_ C V V ð14Þ

The strain rate jump across the interface is obtained from Eq. (21) and given by

For Eq. (14) to be complete, a relationship between the local averaged strain rates in the inclusion and in the effective medium must be introduced via the use of interfacial operators which are presented below.

Here Christofells’s matrix is denoted by K ik ¼ b2ijkl nl nj . Introducing (22) into (19) one obtains the expression of strain rate jump across the interface

2.2. Interfacial operators

u2i  u1i ¼ 0 with i ¼ 1; 3

ð15Þ

where [Æ] denotes a jump across the inclusion/coating interface and u denotes the displacement vector. From the continuity of the traction vector one obtains   ½rij nj ¼ rIij  r2ij nj ¼ 0 ð16Þ where ni is the outward unit normal on the interface. Hadamard’s compatibility conditions [52] for a moving interface R are given by ½vi  ¼ ½ui;j nj wa na

ð22Þ

e_ 2  e_ 1 ¼ P2 : ½b1  b2  : e_ 1

ð23Þ

Prijkl

In this section, interface relations between two phases are presented in a general case. The superscripts 1 and 2 refer to phase 1 and phase 2, respectively. The relatively small thickness of the coating (1 nm) justifies the use of the thin layer assumption. Moreover, perfect bonding is assumed between the inclusion and its coating. Hence, the displacement across an interface must be continuous and must consequently respect the following equation: ½ui  ¼ 0;

1 2 1 ki ¼ K 1 ik nl ½b  b  : e_

ð17Þ

where vi is the velocity of particles at S, wa denotes the propagation velocity of the interface R, na is the normal on the surface R, ui,j is the displacement gradient and nj the normal to the element of surface dS. Moreover, since the interface is stationary one obtains wa = 0 and [vi] = 0. At an arbitrary point r(xi) of the interface, the compatibility condition dui = ui,j dxj added to the continuity of dis-

with r = 1, 2, denotes the interfacial operator where depending of the unit normal ~ n. 2.3. Concentration tensors In what follows, the interface relations developed in Section 2.2 are applied to the topology presented in 2.2 in order to derive the expressions of the local concentration tensors introduced in Eq. (3). Applying Eq. (23) to the case of the coated inclusion problem leads to: e_ C ðrÞ ¼ e_ I ðr0 Þ þ P : DbIC : e_ I ðr0 Þ C

0

I

IC

ð24Þ I

C

where r 2 V and r 2 V ; Db = b  b . Let us suppose the local strain rate in the inclusion to be homogeneous and evaluate the average strain rate within the coating, obtaining E_ C ¼ E_ I þ T C ðbC Þ : E_ I where the interaction tensor is written as follows: Z 1 C C T ðb Þ ¼ C P dr V VC

ð25Þ

ð26Þ

The localisation tensors are simply obtained by introducing (25) into (14); one obtains  AI ¼ f I ðI þ Tðbeff Þ : DbI Þ þ f C ðI þ Tðbeff Þ : DbC Þ   1 i1 ð27Þ : I þ T C ðbC Þ : DbIC and

S. Benkassem et al. / Acta Materialia 55 (2007) 3563–3572

AC ¼ ½I þ T C ðbC Þ : DbIC  : AI I

I

eff

C

ð28Þ C

with Db = b  b , Db = b  b and with VI fI ¼ I ¼ fI IC f þ fC V

and

eff

and Db

IC

=b

IC

b

C

VC fC ¼ I ¼ fC IC f þ fC V

The volume fraction of the inclusion phase is given by  3 d I ð29Þ f ¼ d þw where d denotes the grain size and w denotes the coatings thickness. Here we use w = 1 nm in agreement with existing dislocation and disclination unit models and experiments [53]. 3. Constitutive law of the inclusions In what follows, an extension of Estrin et al.’s constitutive law for fcc materials accounting for the effect of grain boundaries on the flow stress at 0 K is used to describe the behavior of the inclusions phase [21]. Inclusions, representing grain cores, behave viscoplastically via the motion of dislocations and their constitutive law is given by e_ I ¼ mI : rI

ð30Þ

Also, since elasticity is ignored, inclusions are supposed to be not compressible and behave isotropically. Hence, using the Prandtl–Reuss flow rule, the viscosity tensor is written as follows: mI ¼

1 K 2gI

ð31Þ

Here K is the fourth-order tensor defined as 1 1 K ¼ ðdik djl þ dil djk Þ  dij dkl 2 3 and dij denotes Kronecker’s symbol. The local viscosity coefficient gI is given by gI ¼

rIeq 3_eIeq

ð32Þ

where rIeq denotes Von Mises equivalent’s stress. e_ Ieq denotes the local equivalent strain rate and is given by the following power law hardening relation: !m I r eq e_ Ieq ¼ e_ 0 ð33Þ rf Here e_ 0 is a numerical constant, m is the hardening coefficient and is inversely proportional to the temperature and rf denotes the flow stress at 0 K and accounts for the effect of dislocation forests and for the stress field resulting from the presence of by grain boundaries pffiffiffi b rf ¼ rdis þ rGB ¼ aGMb q þ pffiffiffi d

ð34Þ

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Here a, G, M and q denote a numerical constant, the shear modulus, the Taylor factor and the dislocation density, respectively; b is the Hall–Petch slope; p and ffiffiffi d is the grain size. The introduction of the term b= d in Eq. (34) is motivated by the fact that, in coarse grained fcc metals, the Hall–Petch law shall be respected at 0 K which is not predicted by existing models [54–56]. Also, as discussed in Ref. [21], existing models based on work by Kocks and Mecking [57,58] and Nes [56] lead to Hall–Petch slopes considerably lower than those observed experimentally. This can also be observed in a Haasen plot which reveals that the hybrid model of Estrin and Kocks cannot predict the offset in the maximum of the plot engendered by a decrease in grain size [21]. Let us note though that the drawback of the present model is that it introduces an instability in the Haasen plot. Physically, this term accounts for the effect of grain boundaries on the critical dislocation glide stress at 0 K and does not imply that grain boundaries are impenetrable obstacles. The sessile dislocation density evolves via the occurrence of two mechanisms: (1) an athermal storage mechanism and (2) dislocation annihilation. While dislocation storage is treated here in a purely statistical manner, the dislocation annihilation mechanism is a thermally activated mechanism   oq k pffiffiffi þ k ¼ M q ð35Þ q  k 1 2 eIeq d where q and eIeq denote the dislocation density and the equivalent strain in the inclusion, respectively and k and k1 are numerical constants describing the effect of grain boundaries and dislocation forest on the dislocation’s mean free path; k2 describes the recovery mechanism and is given by !1=n e_ Ieq k 2 ¼ k 20 ð36Þ e_  where k20 and e_  are numerical constants and n is the recovery exponent which is inversely proportional to the temperature. The first two terms in parentheses on the right-hand side of Eq. (35) account for the storage of dislocations engendered by a decrease in the mean free path of dislocations due to the presence of stored dislocations and of grain boundaries. The former contribution is related to the dislocation density via the principle of similitude. It is noted here that the term k/d in Eq. (35) results from the fact that, in coarse grain fcc metals, grain boundaries are impenetrable obstacles. The contribution of this term is relatively small, which is demonstrated by the fact that models based on the Kocks and Mecking constitutive law exhibit little size dependence (e.g., the predicted Hall–Petch slope is 1/10 of that measured experimentally). Moreover, when decreasing the grain size the contribution of the inclusion phase will decrease to the benefit of that of the coating phase in which dislocations are allowed to penetrate. Hence, the model predictions shall not be penalized by the fact that in the nanocrystalline regime (e.g., grain

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S. Benkassem et al. / Acta Materialia 55 (2007) 3563–3572

size < 50 nm) the effect of grain boundaries as impenetrable obstacles is still accounted for. Let us recall that grain cores in NC materials exhibit an almost dislocation free structure and, as mentioned in Section 1, dislocation activity emerges from the activation of grain boundary dislocation sources. Qin et al.’s model [9,10], accounting for the size effect in the excess volume of grain boundaries (e.g., a decrease in the grain size leads to an increase in the excess volume in grain boundaries due to the presence of nanovoids and void clusters), shows that a decrease in the grain size engenders a decrease in the region of stability of a dislocation within the grain cores. Hence, in the nanocrystalline range, the penetration of an emitted dislocation becomes energetically favorable. 4. Constitutive law of the coating phase Plastic deformation in the coating phase, which represents both grain boundaries and triple junctions, is expected to occur via grain boundary ledge dislocation emission and penetration. Let us recall here that the grain boundary sliding mechanism is not accounted for and that solely the viscoplastic part of the deformation is taken into account. Hence, the constitutive law of the coating can be written as follows: e_ C ¼ mC : rC

ð37Þ

The coating is not compressible and behaves isotropically. Assuming the Prandtl–Reuss flow rule, the viscosity tensor is written as follows: mC ¼

1 Ki 2gC

ð38Þ

The viscosity coefficient gC is given by gC ¼

rCeq 3_eCeq

ð39Þ

where rCeq denotes Von Mise’s equivalent’s stress. Disregarding the effect of grain boundary sliding, plastic deformation within the coating phase is driven by the simultaneous effect of grain boundary dislocation emission and penetration. Recall that these mechanisms are activated solely in the NC regime. As discussed above, grain cores exhibit an initial microstructure that is virtually dislocation free. As discussed in respect of the model by Qin et al. [9], the region of stability of a mobile dislocation severely decreases with a decrease in the grain size. Moreover, except for the occasional presence of twins within grain cores, crystallites are defect free. Hence, an emitted dislocation cannot meet pinning points and will not be stored within the grain cores. Post-mortem experiments have revealed the presence of only a few items of dislocation debris within the grain cores, which concurs with previous discussion. Let us note that recent MD simulations have shown that twin boundaries can act in a manner similar to that of grain boundaries in the sense that disloca-

tions can both be created by and penetrate into twins. Hence, the effect of twin boundaries can be modeled as a decrease in the effective grain size. It can thus be conjectured that every emitted dislocation leads a penetration event and it is proposed here to describe the combined effect of dislocation emission and penetration in a single law. Hence, it is implicitly supposed that the dislocation penetration event can be treated as statistically similarly to the treatment of dislocation storage in grain cores in typical dislocation density based hardening laws. MD simulations on bicrystal interfaces did not reveal any net strain following the emission of dislocations from grain boundaries containing a ledge, which are here supposed to be the primary interface dislocation sources. It was also shown that, in the case of a R5 grain boundary, these structures are more prone to emit dislocations than perfect planar interfaces. Again, note here that these results shall be extended to more geometries. Finally, recalling that the emission process is thermally activated, one can write the local equivalent strain rate, e_ Ceq , as the product of the average effect of a dislocation penetration event by the activation rate. As a first approach, the average strain rate is modeled as a soft collision. If denotes the effective mass of a dislocation traveling at velocity m0 ðrIeq =rIc Þm , then from momentum conservation one has [21] !m rIeq C sinðhÞ ð40Þ ðmGB þ mdis Þ_eeq L ¼ mdis v0 rIc Here mGB, L, h, m0 and e_ ave GB denote the mass of grain boundary affected by the penetration event, the critical interface length affected by the penetration, the average dislocation velocity and the angle at which the dislocation penetrates the grain boundary, respectively. It is also assumed that the critical interface length is proportional to the grain size. This is motivated by the fact that dislocations are not expected to engender mass transfer through triple junctions. The activation rate is typically written as the product of the frequency of attempt of the event by the probability of successful emission and described phenomologically via an exponential term. The frequency of attempt is supposed here to be inversely proportional to the number of ledges which can be considered inversely proportional to the square root of the grain size. Hence, e_ Ceq is given by e_ Ceq

v ¼ 3 d

rIeq rIc

!m

DG0 ðhmis Þ 1 exp kB T

rCeq rCc ðhmis Þ

!p !q !

ð41Þ where the parameters kB, T, rCeq and rCc represent the Boltzmann constant, the temperature in Kelvin, the equivalent stress in the matrix phase defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

  rCeq ¼ 3 2 rCdev : rCdev where rCdev denotes the deviatoric stress tensor, and the critical-emission stress, respectively. Let us note here that the free enthalpy of activation DG0(hmis) and the critical emission stress rCc ðhmis Þ are both

S. Benkassem et al. / Acta Materialia 55 (2007) 3563–3572

dependent on the grain boundary geometry (e.g., misfit angle) denoted by hmis. The non-dimensional parameters p and q describe the evolution of the free enthalpy of activation with the local stress. Also, we have [21] n¼

mdis mn d sinðhÞ ðmdis þ mGB Þi

ð42Þ

where d, m0, i and h denote a numerical constant, the dislocation average velocity, another numerical constant and the dislocation emission angle with respect to the grain boundary axis (only one degree of freedom is considered not null), respectively. 5. Results The model was applied to pure copper for which a relatively large number of experimental data are available. The following parameters, describing the behavior of the inclusion phase, were mostly extracted from recent work by the author [21]: a = .33, Taylor’s factor M = 3.06, Burgers vector b = .256E9 m, lI = 38,400 MPa [1], e_ 0 ¼ 0:005=s and n = 21.25 at room temperature and is inversely proportional to the temperature. b = 0.11 MPa m1/2, k ¼ 1b ¼ 3:5E9 m1 , k1 = 1.E10 m1and k20 = 330. The coating is described by the following parameters: n = 3.507E4; an order of magnitude of the critical emission stress, rM c , was estimated from MD simulations; using Spearot et al.’s simulation [27,28] on an aluminum R5 interface, we obtain rM c ¼ 2:2 GPa, DG0 = 1.4E17 N m, ˚ 2 [21]; v0 = p = 1 and q = 1.5 [21]; mdis = 3.15 pN ps2/A 0.03, 1 = 1/2; and the parameter is set to 40. Theoretical estimates of the grain boundary density typically lead to predictions of the order of 90% of the density of a perfect crystal [9,10]. On the contrary, recent experiments show much higher grain boundary densities of the order of 98% of that of a perfect crystal [59]. In the present model, the coating phase represents both grain boundaries and triple junctions. Also, depending on the fabrication process, sample densities typically lead to densities ranging from 96% to 99% of that of a perfect crystal lattice. This is due to the presence of vacancies in the interphase region (Fig. 2). Using a simple rule of mixture on 20 nm grain size NC materials with densities ranging from 96% to 99% of the theoretical density, one obtains a coating density ranging from 72% to 92% of the theoretical density. In the present model, the coating phase represents

Fig. 2. Interface modeling.

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both grain boundaries and triple junctions. In order to account for the presence of vacancies and nanovoids in the interphase region (e.g., grain boundaries and triple junctions), the grain boundary density is chosen as 85% of the theoretical density. Also, let us note that the current model exhibits little dependence on the grain boundary density. The size effect in the response of the inclusions is presented in Fig. 3. The macroscopic strain rate is 1.0E–3/s and corresponds to a state of pure traction. Three grain sizes are simulated: 1 lm, 100 nm and 10 nm. A decrease of the grain size naturally leads to an increase in the stress level of the inclusions phase. Indeed, decreasing the grain size will have two effects: (1) decrease of the mean free path of dislocation which increases the stress level required for dislocation glide and (2) increase of the long range stress created by grain boundaries. The simulation also reveals that when the grain size is very small (10 nm in this case), inclusions sustain less deformation although the final macroscopic deformation of the material is set to be independent of grain size and equal to 0.005. Hence, the coating phase must sustain a greater deformation at small grain sizes. The size effect in the response of the coating is presented in Fig. 4. Again, the macroscopic strain rate is 1.0E–3/s and corresponds to a state of pure traction. Three grain sizes are simulated: 1 lm, 100 nm and 10 nm. The response of the coating can be considered as opposite to that of the inclusion phase in the sense that a decrease in the grain size leads to a softer response of the coating phase. This is due to the fact that the activity of the dislocation emission and penetration mechanism increases with a decrease of the grain size. Also, the total deformation of the coating increases with a decrease in the grain size. A comparison of Figs. 3 and 4 indicates that up to 100 nm the inclusion’s phase has a harder behavior than that of the coating’s phase, while at 10 nm grain size the coating is much softer than the inclusions. Hence, the coating will sustain larger deformations at smaller grain sizes.

Fig. 3. Tensile response of the inclusion phase at a macroscopic strain rate of 1.0E3/s for three grain sizes.

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Fig. 4. Tensile response of the coating phase at a macroscopic strain rate of 1.0E3/s for three grain sizes.

The macroscopic response of the material under pure traction at 1.0E–3/s is presented in Fig. 5. Down to 100 nm grain size, the macroscopic response of the material is very close to that of the inclusion phase, which is due to the high volume fraction of inclusions relative to that of the coating. Considering the fact that the coating phase becomes non-negligible when d = 10 nm and also becomes the softer phase, the macroscopic response of 10 nm grain copper is expected to be softer than that of the inclusion. The size effect in the yield stress is presented in Fig. 6. Model predictions are represented with solid and dashed lines, while experimental data are represented with symbols [60–62]. One can observe that down to d = 25 nm the yield stress increases almost linearly with the inverse of the square root of the grain size. This result is consistent with the Hall–Petch law and the predicted slope is equal to 0.102 which is close to the 0.11 slope reported experimentally. The Hall–Petch slope decreases to 0.064 when the grain size is <25 nm. This grain size is consistent with the critical grain size measured experimentally. While the change in

Fig. 5. Macroscopic tensile response with an applied strain rate of 1.0E3/s for three grain sizes.

Fig. 6. Size effect in the yield stress at different strain rates.

the Hall–Petch slope is successfully predicted, a fit of the experimental data is not obtained. First, the yield stress is dependent on the microstructure of the material, which clearly depends on the fabrication process. Also, as can be observed in Fig. 6, the amount of experimental data is very limited and presents large discrepancies due to the quality of the samples. These simulations reveal that the evolution of yield stress with grain size law is affected by the dislocation emission and penetration mechanism which leads to a softer behavior of the grain boundaries relative to that of inclusions (which becomes harder as the grain size is decreased). However, the negative Hall–Petch slope measured experimentally cannot be predicted with the current model. Moreover, recalling a recent model by Warner et al. [39] suggesting that grain boundary sliding alone cannot account for the breakdown of the Hall–Petch law, the present results suggests that the breakdown of the ‘‘abnormal’’ behavior of NC materials could result from the combined effect of grain boundary dislocation emission and penetration and grain boundary sliding. Grain boundary dislocation penetration could also serve as an accommodation mechanism to grain boundary sliding. In order to identify the mechanism leading to a negative slope in the Hall–Petch plot, a finite-element simulation is computed. Fig. 7 presents a plot of the Von Mise’s equivalent stress at 5% deformation in the NC aggregate submitted to a 0.001/s tensile strain rate. The finite-element simulation aims at predicting the stress level in a pure copper polycrystalline aggregate with 20 nm grain size. In order to provide more realistic predictions, the grain cores are assumed hexagonal. The constitutive law of the inclusion’s phase, presented above, is introduced in a UMAT subroutine to describe the behavior of grain cores. The

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Clearly, the micromechanical scheme presented in Section 2 cannot account for the stress concentrations predicted via finite-element simulations. However, stress concentrations at triple junctions will ineluctably enhance dislocation emission within the grain cores. Also, at small grain sizes (d < 10 nm), where dislocation activity ceases, plastic deformation would result from grain boundary sliding or rotation since no other plastic deformation mechanism can be activated. 6. Conclusion

Fig. 7. Finite-element simulation of a 20 nm grained copper crystal aggregate.

In this paper a three-phase micromechanical scheme accounting for stress and strain rate jump at the grain core/grain boundary interface is developed to describe the size effect in the viscoplastic behavior of pure fcc polycrystalline materials. Grain cores deform viscoplastically via the glide of dislocations for which a formalism describing the effect of grain boundaries on the yield stress at 0 K is used. Grain boundaries deform via mass transfer generated by the penetration of grain boundary ledge emitted dislocations. Predictions of the model are completed with a set of finite-element simulations and the mechanisms responsible for the presence of three regions in the Hall–Petch plot are identified (see Fig. 8). Region I, corresponding to the Hall–Petch law, holds down to 25 nm and the increase in the yield stress with a decrease in the grain size is due to both the size-dependent stresses created by grain boundaries on dislocations and the size-dependent mean free path of mobile dislocations. Region II, where the Hall–Petch slope decreases, would be characterized by the combined activity of grain boundary dislocation emission and grain boundary sliding which lead to softer behaviors, while region III, where a negative Hall–Petch slope can be observed, is more than likely due to a failure mode, such as unaccomodated grain boundary sliding, engendered by the high stress concentrations present at triple junctions. References

Fig. 8. Schematic of the Hall–Petch plot.

behavior of the grain boundaries and triple junction is described by the set of Eqs. (37)–(41), where the ratio m ðrIeq =rf Þ in Eq. (40) is assumed constant and equal to 0.8. Hence, the behavior of grain boundaries in the finiteelement simulation is softer than that described in the micromechanical model. The simulation reveals high stress concentration at triple junctions – a maximum factor of 3 is predicted. However, let us note that these 2-D simulations will clearly lead to overestimation of the stress concentrations.

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