Mechanics of very fine-grained nanocrystalline materials with contributions from grain interior, GB zone, and grain-boundary sliding

International Journal of Plasticity 25 (2009) 2410–2434

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Mechanics of very fine-grained nanocrystalline materials with contributions from grain interior, GB zone, and grain-boundary sliding Pallab Barai, George J. Weng * Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, United States

a r t i c l e

i n f o

Article history: Received 19 December 2008 Received in final revised form 9 April 2009 Available online 22 April 2009 Keywords: Nanocrystalline materials Grain-boundary sliding Inverse Hall–Petch effect Strain-rate sensitivity Viscoplastic response

a b s t r a c t In this paper, we formulated an atomically-equivalent continuum model to study the viscoplastic behavior of nanocrystalline materials with special reference to the low end of grain size that is typically examined by molecular dynamic (MD) simulations. Based on the morphology disclosed in MD simulations, a twophase composite model is construed, in which three distinct inelastic deformation mechanisms disclosed from MD simulations are incorporated to build a general micromechanics-based homogenization scheme. These three mechanisms include the dislocationrelated plastic flow inside the grain interior, the uncorrelated atomic motions inside the grain-boundary region (the GB zone), and the grain-boundary sliding at the interface between the grain and GB zone. The viscoplastic behavior of the grain interior is modeled by a grain-size dependent unified constitutive equation whereas the GB zone is modeled by a size-independent unified law. The GB sliding at the interface is represented by the Newtonian flow. The development of the rate-dependent, work-hardening homogenization scheme is based on a unified approach starting from elasticity to viscoelasticity through the correspondence principle, and then from viscoelasticity to viscoplasticity through replacement of the Maxwell viscosity of the constituent phases by their respective secant viscosity. The developed theory is then applied to examine the grain size- and strain rate-dependent behavior of nanocrystalline Cu over a wide range of grain size. Within the grain-size range from 5.21 to 3.28 nm, and the strain rate range from 2.5  108 to 1.0  109/s, the calculated results show significant grain-size softening as well as strain-rate hardening that are in quantitative accord with MD simulations

* Corresponding author. E-mail address: [email protected] (G.J. Weng). 0749-6419/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2009.04.001

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[Schiotz, J., Vegge, T., Di Tolla, F.D., Jacobsen, K.W., 1999. Atomicscale simulations of the mechanical deformation of nanocrystalline metals. Phys. Rev. B 60, 11971–11983]. We have also applied the theory to investigate the flow stress, strain-rate sensitivity, and activation volume over the wider grain size range from 40 nm to as low as 2 nm under these high strain rate loading, and found that the flow stress initially displays a positive slope and then a negative one in the Hall–Petch plot, that the strain-rate sensitivity first increases and then decreases, and that the activation volume first decreases and then increases. This suggests that the maximum strain rate sensitivity and the lowest activation volume do not occur at the smallest grain size but, like the maximum yield strength (or hardness), they occur at a finite grain size. These calculated results also confirm the theoretical prediction of Rodriguez and Armstrong [Rodriguez, P., Armstrong, R.W., 2006. Strength and strain rate sensitivity for hcp and fcc nanopolycrystal metals. Bull. Mater. Sci. 29, 717–720] on the basis of grain boundary weakening and the report of Trelewicz and Schuh [Trelewicz, J.R., Schuh, C.A., 2007. The Hall–Petch breakdown in nanocrystalline metals: a crossover to glass-like deformation. Acta Mater. 55, 5948–5958] on the basis of hardness tests. In general the higher yield strength, higher strain rate sensitivity, and lower activation volume on the positive side of the Hall–Petch plot are associated with the improved yield strength of the grain interior, but the opposite trends displayed on the negative side of the plot are associated with the characteristics of the GB zone which is close to the amorphous state. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The exact mechanism behind the drop of yield strength of nanocrystalline materials at the very fine grain-size range is still a subject of intensive study. Over the past two decades a number of experiments have been conducted to explore the behavior of materials as the grain size decreases from the ultrafine (diameter > 100 nm) to the nanometer scale (diameter < 100 nm). While the Hall–Petch relation generally holds for coarse-grained materials, the yield strength tends to deviate from it as the grain size enters into the ultrafine regime (Nieman et al., 1989; El-Sherik et al., 1992; Gertman et al., 1994; Sanders et al., 1997a,b) and eventually decline as the grain size reduces to the very fine scale (Chokshi et al., 1989; Lu et al., 1990; Fougere et al., 1992; Khan et al., 2000). Other tests conducted by Khan and Zhang (2000), Jia et al. (2003), Schwaiger et al. (2003), Wei et al. (2004), Khan et al. (2006), and Farrokh and Khan (2009), among others, also showed significant grain-size effect on the strain-rate sensitivity of nanocrystalline materials. But because of the limitations in fabricating fully compact, nanocrystalline specimens with grain size less than 20 nm, not many such experiments have been conducted. However, a softening in terms of Vicker’s hardness was reported for zinc at the grain size of about 11 nm (Conrad and Narayan, 2002) and, more recently, a comprehensive investigation on the variation of hardness and strain-rate sensitivity of a Ni–W alloy system over the grain-size range from 100 nm all the way down to 2 nm was carried out by Trelewicz and Schuh (2007). Their results showed the complete transition from the positive to the negative slope of the Hall–Petch plot, and revealed a significant finding that the maximum strain-rate sensitivity of a nanocrystalline material does not occur at the smallest grain size, but rather at a finite grain size. In a separate front, molecular dynamic simulations (MD) have been widely used to uncover the deformation mechanisms and the behaviors of nanocrystalline materials. Because of the ease with which a nanocrystalline material with very small grain size (d < 5 nm) could be constructed, MD simulation has proven to be a very powerful tool for the low end of grain size. For some materials (such as Ni, Cu) reliable and simple interatomic potentials exist, which have been applied to provide an

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atomistic view of the microstructural evolution through the mean-field approximation of atomic interactions (Schiotz et al., 1998, 1999; Swygenhoven et al., 2001; Schiotz and Jacobsen, 2003). More complicated ‘complex interatomic potentials’ have also been used to simulate the deformation mechanism of a-Fe2O3 + fcc-Al alloy with electrostatic effect between the constituent atoms (Tomar and Zhou, 2007). These MD simulations have uniformly disclosed that the volume fraction of the grain boundary zone (GB zone) is very high and its deformation has a pronounced effect on the mechanical properties of the material. From several MD simulations it has been suggested that, at very small grain size, deformation of the GB zone and the interfacial sliding between the GB zone and the grain could be the major deformation mechanisms (Swygenhoven and Caro, 1997a, 1998; Schiotz, 2004; Lund and Schuh, 2005; Pan et al., 2008), and these could contribute to the inverse Hall–Petch effect. In some simulations very small amount of intra-grain dislocation activity (only 3%) has been observed (Schiotz et al., 1999; Tomar and Zhou, 2007), while in others dislocations within the grains have been reported to be completely absent (Swygenhoven and Caro, 1997a,b) with very small grain size (d < 10 nm). In spite of being a reliable tool for analyzing the behavior of nanocrystalline materials, MD simulations have some of its own limitations. As all the atoms are considered individually in this technique, simulating a large system with comparatively larger grain size requires extensive amount of computational power which might not be easily available. For instance in Schiotz et al. (1998), it took 1,000,000 atoms to simulate the polycrystal behavior with the grain size of 6.56 nm, and the required number of atoms prohibitively increases with increasing grain size. Also, since the entire MD simulation technique is very slow (time difference between two consecutive iterations is in the order of 1015 s), a very high strain rate (108–1013/s) is usually applied to generate a desirable amount of strain in a reasonable time. These kinds of loading conditions rarely appear in real life situations and are not experimentally feasible. These limitations point to the need of atomically-equivalent continuum modeling as a complementary route. This is the concept of composite modeling. Some earlier studies making use of the composite concept include those of Carsley et al. (1995), Wang et al. (1995), Meyers et al. (1999), Kim et al. (2000), Benson et al. (2001), and others that can be found in Meyers et al. (2006). They have all realized the merit of using composite model to represent the heterogeneous nanocrystalline materials, and the reported results have shed significant insights along this line. These early studies, however, all adopted the simple mixture rule by taking the stress distribution to be uniform across all constituent phases. In essence, it is a lower-bound approach to the behavior of a heterogeneous material. In this study, we shall make use of a micromechanics-based composite model that clearly differentiates the heterogeneous state of stress and strain among the constituent phases. Our focus here is on the viscoplastic behavior of nanocrystalline materials in the very fine grain-size range that is commonly studied in MD simulations. We shall take information disclosed from these simulations to build the model. In particular, we will consider three distinct inelastic deformation mechanisms in the development of the homogenization scheme: the plastic deformation of the grain interior, the plastic deformation of the GB zone, and the grain-boundary sliding at the interface between the grain and GB zone. The first mechanism will be represented by a grain-size dependent unified constitutive equation and the second one by a size-independent unified equation. The GB sliding on the other hand will be modeled by a viscous flow. But to reflect the observed dislocation-free state inside very fine grains, the first mechanism will be taken to be zero when the grain size lies below some critical value (in later calculations this value was set at 8 nm for copper). This model grew out of the recently developed ones by Jiang and Weng (2003, 2004a,b) for the rate-independent plasticity of nanocrystalline metals, nanocrystalline ceramics, and nanocrystalline polycrystals, respectively. Subsequent developments for the rate-dependent viscoplastic behavior have been made by Li and Weng (2007) for the study of strain-rate sensitivity, by Barai and Weng (2008a,b) for the time-dependent creep and the competition of grain size and porosity under a constant strain-rate loading. The new feature introduced in this paper is the additional mechanism of grain-boundary sliding that is essential for the study of very fine-grained nanocrystalline materials. Other composite models have also been suggested in recent past, including the self-consistent formulation of Capolungo et al. (2005a,b, 2007a,b), but without GB sliding. Another line of work by Wei and Anand (2004) and Wei et al. (2006) have incorporated grain-boundary sliding and cavitation into a finite element scheme to calculate the plastic behavior of nanocrystalline materials, but no

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contribution from the GB zone was considered. In addition, Warner et al. (2006) have developed an atomic-based continuum model and Joshi et al. (2006) and Joshi and Ramesh (2007) have extended the secant-moduli approach suggested in Weng (1990) and Jiang and Weng (2003, 2004a,b) to study the plastic behavior of bimodal metals with simultaneous presence of ultrafine grains and coarse grains, and the plasticity of hierarchical composites. To treat the issue of grain-boundary sliding analytically, we recall that this issue has its root in composite elasticity with an imperfect interface. The problem was first analyzed by Benveniste (1985) with interfacial shear imperfection through displacement jump at the inclusion–matrix interface, in conjunction with Christensen and Lo’s (1979) generalized self-consistent scheme. Later Hashin (1991) considered both shear and normal imperfections across the interface, and demonstrated – also with the generalized self-consistent scheme – that imperfections in both directions have significant effect on the overall moduli. In a separate study, Qu (1993a,b) first derived an Eshelby tensor for a spherical inclusion with a slightly weakened interface and then made use of the Mori–Tanaka approach to calculate the effective moduli of a composite with such interface. More recently Duan et al. (2007) made use of energy equivalence between an inclusion with an imperfect interface and a perfectly bonded inclusion to derive the effective elastic moduli of such an ‘‘equivalent inclusion”. This approach proves to be particularly useful to our later formulation, and will be recalled again. The advantage of using the energy equivalence has also been exploited by Christensen and Lo (1979), and Shen and Li (2003, 2005) to treat various interphase problems. 2. Morphology of very fine-grained nanocrystalline materials A typical morphology of nanocrystalline materials by MD simulations is reproduced in Fig. 1 for copper, at the average grain size of about 5.2 nm. We shall represent such morphology with a twophase composite, with grains serving as inclusions and the GB zone as the matrix. Grain-boundary sliding occurs only during plastic flow, not in elastic deformation, and we will build the model accordingly. We shall further refer the grains as phase 1 and the GB zone as phase 0, with the volume concentration of the rth phase denoted by cr. To model the interface imperfection, a layer of zero thickness is assumed to exist between every grain and the corresponding GB. In the model of Duan et al. (2007), the grain and the imperfect interface constitute an ‘‘equivalent inclusion” (denoted as phase ‘e’) which has the same volume fraction as the grain interior (c1) itself and is perfectly bonded to the matrix. In terms of the grain diameter d and the thickness t of the GB zone, the volume concentrations can be written as

Fig. 1. Morphology of a nanocrystalline metal by molecular dynamic simulation (Schiotz et al., 1998).

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c1 ¼ ce ¼



d dþt

3 ;

c0 ¼ 1  c1 :

ð1Þ

The grain boundary thickness in aluminum was observed to vary between 1 and 2 nm (Tomar and Zhou, 2007) and for nickel and copper it was reported to be in the range of 0.7 nm (Swygenhoven et al., 2001). Taking t = 1 nm as an average, the volume fraction of the GB zone is approximately 27% for d = 10 nm, and about 65% for d = 3.28 nm. The presence of the GB zone as well as the GB sliding will both have significant effect on the overall deformation of a nanocrystalline material in the very fine grain-size range. 3. Rate-dependent constitutive relations for the grain interior, GB zone, and grain-boundary sliding Above some critical grain size the rate-dependent viscoplastic behavior of the grain interior will be modeled by a set of grain-size dependent unified constitutive equations, but below this size – to be consistent with MD simulations and some experimental observations – it will be taken to be free from dislocation contributions. The grain boundary zone will be modeled by a grain-size independent unified constitutive law, whereas the grain-boundary sliding will be represented by a Newtonian flow. The behavior of spherical inclusions represents the collected behavior of all randomly oriented anisotropic constituent grains in a real polycrystal and thus is isotropic. This makes it realistic for its viscoplastic behavior to be represented by an isotropic unified theory. As usual the total strain rate of the grain or the GB zone is given as the sum of the elastic and viscoplastic strain rate, but the total strain rate of the entire nanocrystalline material has contributions from the elastic and viscoplastic deformation of the constituent phases and the GB sliding. To pave the way for the development of a secant viscosity model we first briefly recapitulate a unified theory that is suitable for the viscoplastic response of the grain interior and the GB zone. The total strain rate of either the grain interior or the GB zone is the sum of its elastic and viscoplastic strain rates, as e_ ij ¼ e_ eij þ e_ vp ij . The elastic rate is linearly related to the stress rate, whereas the components of the viscoplastic rate are taken to follow the Prandtl–Reuss relation

e_ vp ij ¼

3 e_ vp e r0 ; 2 re ij

ð2Þ

where re is the effective stress and e_ vp e is the effective viscoplastic strain rate, defined as

re ¼



3 0 0 rr 2 ij ij

1=2 ;

e_ vp e ¼

 1=2 2 vp vp e_ ij e_ ij ; 3

ð3Þ

in terms of the deviatoric stress r0ij and viscoplastic strain rate e_ vp ij . The unified constitutive relation for each phase is cast in the power-law form, as

_ vp e_ vp e ¼ e0 

r n e

s

ð4Þ

;

where n is the stress exponent and s is the drag stress. The reference strain rate e_ vp 0 is a scaling factor and can be set arbitrarily. The drag stress increases with deformation, and can be taken to be controlled by the competition between strain hardening and dynamic recovery in the form

  s vp s_ ¼ h  1  e_ ; s e

ð5Þ

so that, upon integration, it depends on the current strain as

 s ¼ s  ðs  s0 Þ exp 

evp e s =h

 ;

ð6Þ

where s0 represents the initial hardening state and s* the final saturation state. So there are three key material constants for each phase: the power n, the initial strength s0, and the saturation strength s*.

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This set of constitutive relations applies to both the grain interior and GB zone, but each has its own constants. To make the symbols more indicative, we shall use the subscript (g) to stand for the grain interiors and (gb) for the GB zone. Since the drag stress represents the hardening state of the phase, it is taken to follow the Hall–Petch equation for the grain interior as (Weng, 1983) ðgÞ

s0 ¼ s1 0 þ kd

1=2

ðgÞ

and sðgÞ  ¼ a:s0 :

ð7Þ ðgbÞ

But those of the GB zone are grain-size independent and can be written simply as s0 and sðgbÞ  . Constants s1 0 and k in (7) are the Hall–Petch constants of the grain interior, and ‘a’ relates its saturation strength to the initial strength. This set of constitutive equations has also been previously adopted (Li and Weng, 2007; Barai and Weng, 2008a,b). For such an elastic–viscoplastic phase, say phase r, its secant viscosity, gsr , can be defined by

e_ vpðrÞ ¼ ij

1 0ðrÞ r 2gsr ij

or

e_ vpðrÞ ¼ e

1 ðrÞ r : 3gsr e

ð8Þ

In view of (4), it can be written specifically for the grain interior (phase 1) and GB zone (phase 0), as

sðgÞ g ¼ vpðgÞ  3e_ e s 1

e_ vpðgÞ e e_ vpðgÞ 0

!1=nðgÞ ;

sðgbÞ g ¼ vpðgbÞ  3e_ e s 0

e_ vpðgbÞ e e_ vpðgbÞ 0

!1=nðgbÞ :

ð9Þ

The secant viscosity thus continues to change due to the increase of drag stress. The dilatational behavior of each phase is taken to be plastically incompressible and is characterized solely by its elastic bulk modulus. For the grain-boundary sliding at the interface between the grain and the GB zone, it will be described by a Newtonian-type flow. In general an imperfect interface can be taken to be governed by traction continuity but with a jump in displacement. As grain-boundary sliding has been observed only during the plastic deformation of the nanocrystalline material, we shall include the sliding effect only in the viscoplastic analysis. Extending the linear spring model commonly adopted in elastic analysis, we write its inelastic constitutive relation as

_ ¼ r  n; ½r  n ¼ 0 and a  ½u

ð10Þ

where [] = (out)  (in), represents the jump across the interface, r is the stress tensor, n is the unit outward normal at the interface, u_ is the velocity vector at the interface, and a is a 2-dimensional viscosity tensor which can be represented in terms of three components, by

a ¼ an n  n þ as s  s þ at t  t:

ð11Þ

Here s and t are the two tangential directions, and an, as and at represent the interface viscosity parameters. If the values of ai are made very small (ai ? 0), then the interface is completely lubricant. On the other hand if the values are very large (ai ? 1), it becomes a perfect interface with no discontinuity in particle velocity or displacement (here, i = n, s or t). For grain-boundary sliding, the two tangential directions have identical properties resulting in as = at, whereas it is perfectly bonded in the normal direction (an ? 1). 4. A homogenization scheme for the viscoplastic response with contributions from the grain interior, GB zone, and the interfacial grain-boundary sliding An explicit, nonlinear and rate-dependent homogenization scheme that is capable of capturing the contributions from the plastic deformation of grain interior, GB zone, and grain-boundary sliding at the interface of grain and GB zone, will be developed here. We shall start out with the elastic response, then transform it to a linear viscoelastic one through the correspondence principle, and then replace the Maxwell shear viscosity of the constituent phase with its secant viscosity to extend the scheme into the viscoplastic regime. The basis of the homogenization theory to be presented here is the concept of a linear composite. This concept was originally introduced by Talbot and Willis (1985) for a general nonlinear composite.

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Subsequent developments through the secant moduli have been carried out by Tandon and Weng (1988), Ponte Castañeda (1991), Qiu and Weng (1992), Suquet (1995) and Hu (1996), for the rate-independent plasticity. For the rate-dependent viscoplastic problems it was first formulated by Li and Weng, (1997a,b, 1998) through the secant viscosity. There are of course other nonlinear approaches for the heterogeneous solids (e.g. Hill, 1965; Hutchinson, 1976; Berveiller and Zaoui, 1979; Weng, 1982; Dvorak, 1992; Fotiu and Nemat-Nasser, 1996; Masson et al., 2000; Doghri and Tinel, 2005; Pierard and Doghri, 2006; Love and Batra, 2006; Berbenni et al., 2007; Mercier and Molinari, 2009). We shall not dwell upon the merit of each of these approaches and instead will focus on the problem at hand. For the method to remain explicit, and to make the partial derivatives involved in the field-fluctuation method in Section 5 manageable, the Mori–Tanaka (M–T) approach as developed in Weng (1984) will be used to start the calculation for the elastic response. Christensen and Lo’s (1979) generalized self-consistent approach can also be applied as the starting point, but it will greatly add complexities in the expressions of the Laplace transform and inversion, and in the evaluation of the partial derivatives, especially when interfacial sliding is also present. The advantage of using the M–T approach was also recognized by Tan et al. (2005) in their study on the effect of nonlinear interface debonding. 4.1. The initial elastic state To start the computation we shall make use of the two-phase theory developed in Weng (1984, 1990). With the bulk and shear moduli of the rth phase denoted by jr and lr, respectively, the effective bulk and shear moduli of the composite are given by



j ¼ j0 1 þ

 c1 ðj1  j0 Þ ; c0 a0 ðj1  j0 Þ þ j0



l ¼ l0 1 þ

 c1 ðl1  l0 Þ ; c0 b0 ðl1  l0 Þ þ l0

ð12Þ

where

a0 ¼

ð1 þ m0 Þ 3j0 ; ¼ 3ð1  m0 Þ 3j0 þ 4l0

b0 ¼

2ð4  5m0 Þ 6 j0 þ 2l0 ; ¼ 15ð1  m0 Þ 5 3j0 þ 4l0

ð13Þ

and m0 is the Poisson’s ratio of the matrix (GB zone). Since the elastic moduli of the grain boundary may be softer or harder than the grain interior (Kluge et al., 1990; Wolf and Kluge, 1990; Bassani et al., 1992; Alber et al., 1992), the effective moduli in (12) can provide the overall moduli of the nanocrystalline material at a given grain size. For later calculations it is useful to record the average hydrostatic and deviatoric strain of the inclusions in terms of the overall strain of the composite, eij , as

eð1Þ kk ¼

j0 c0 a0 ðj1  j0 Þ þ j0

ekk ;

eij0ð1Þ ¼

l0 c0 b0 ðl1  l0 Þ þ l0

e0ij :

ð14Þ

The overall stress is related to the applied strain through

r kk ¼ 3jekk and r 0ij ¼ 2le0ij :

ð15Þ

4.2. Viscoelastic state with GB sliding To construct our scheme for the viscoplastic response with grain-boundary sliding, we first develop a viscoelastic theory as a linear comparison composite. This will be done by implementing the elastic theory of Duan et al. (2007) into (12), and then transform it into the Laplace space. The elastic interface was modeled by a linear spring model

½r  n ¼ 0 and aðeÞ  ½u ¼ r  n; ðeÞ ðeÞ aðeÞ ¼ aðeÞ n n  n þ as s  s þ at t  t;

ð16Þ

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after which the linear dashpot model of (10) and (11) was conceived. Based on the energy ðeÞ ðeÞ equivalence they found that, with at ¼ as , the effective bulk and shear moduli, je and le, of the ‘‘equivalent inclusion” – which combines the real inclusion and the imperfect interface – could be written as

je mr l0 ¼ ; j1 3j1 þ mr l0 le 24Mmh þ mr ð16M þ mh NÞ ¼ : l1 80g 3 M þ 4g 3 mh ½10ð7  m1 Þ þ M þ mr ½2g 3 ð140  80m1 þ 3MÞ þ mh N ðeÞ

ð17Þ

ðeÞ

Here M = g3(7 + 5m1), N = 5(28  40m1 + M), g3 = l1/l0, mr ¼ an R1 =l0 ; mh ¼ as R1 =l0 , and R1 = d/2, the radius of the grain size. For the present GB sliding problem there is no normal debonding and thus an ? 1 and mr ? 1. As a consequence

je ¼ j1 ; le ¼ l1

ð16M þ mh NÞ : ½2g 3 ð140  80m1 þ 3MÞ þ mh N

ð18Þ

With this pair of moduli for the inclusions the overall moduli (12) can be rewritten as



j ¼ j0 1 þ

 c1 ðje  j0 Þ ; c0 a_ 0 ðje  j0 Þ þ j0



l ¼ l0 1 þ

 c1 ðle  l0 Þ : c0 b0 ðle  l0 Þ þ l0

ð19Þ

To make this pair of moduli useful in the viscoelastic context, the correspondence principle can be invoked (Hashin, 1965). The elastic moduli then become moduli in the ‘‘transformed domain” (TD), and the stress and strain are replaced by their Laplace transforms. This approach has been adopted to examine the influence of inclusion shape on the time-dependent creep, strain-rate sensitivity, and complex moduli of several composites (Wang and Weng, 1992; Li and Weng, 1994). The shear behavior of the rth viscoelastic phase is marked by its shear modulus lr and shear viscosity gr, whereas that of the interface by the shear viscosity as alone. In the Laplace transformed domain (TD), these shear moduli can be written as

lTD r ¼

lr s ; s þ Tr

with T r ¼

lr ; r ¼ 1; 0; gr

and aTD s ¼ as s;

ð20Þ

where s is the usual Laplace parameter. The dilatational behavior is simply marked by its bulk modulus jr (i.e. jTD r ¼ jr ) due to plastic incompressibility in both phases. We shall denote the stress and strain in the transformed domain by a hat ^, so that for the composite

r^ kk ¼ 3jTD ^ekk and r^ 0ij ¼ 2lTD ^e0ij ;

ð21Þ

where with the GB sliding

"

jTD ¼ jTD 1þ 0

c0 a0





#

"





#

j1  jTD c1 lTD  lTD 0 0 ; lTD ¼ lTD ; 1þ  e 0 TD TD TD TD j1  j0 þ j0 þ lTD c0 b0 lTD e  l0 0

c1  TD

ð22Þ

with

aTD 0 ¼

3j0 3j0 þ 4lTD 0

and bTD 0 ¼

6 j0 þ 2lTD 0 : 5 3j0 þ 4lTD 0

ð23Þ

TD The transformed moduli of the equivalent inclusion, jTD e and le , follow from (18), with all the constituent moduli replaced by their corresponding transformed moduli in (20). eij ¼ ð1=s2 Þ  e_ ij , we can Making use of (21) and noting that, under a constant strain-rate loading ^ write

r kk ðtÞ ¼ 3gj ðtÞe_ kk ; r 0ij ðtÞ ¼ 2gl e_ 0ij ;

ð24Þ

where gj = L1(jTD/s2), gl = L1(lTD/s2), with the symbol L1 standing for the Laplace inverse. After some lengthy algebra, one finds the effective bulk and shear viscosity as

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gk ðtÞ ¼ j0 fð1 þ A2 Þt þ A1 ½1  expðs1 tÞg; gl ðtÞ ¼ g0 ð1  expðT 0 tÞÞ þ A þ B expðT 0 tÞ þ C expðs1 tÞ

ð25Þ

þ D expðs2 tÞ þ þE expðs3 tÞ þ F expðs4 tÞ; in terms of time ‘t’, where the constants involved are listed in Appendix A.1. These constants depend on the Maxwell viscosities g1 and g0, the interfacial parameter as, and the volume concentration c1. 4.3. Replacement of the Maxwell viscosity by the secant viscosity for the viscoplastic response The extension from linear viscoelasticity to the nonlinear viscoplasticity will be carried out with the replacement of Maxwell viscosities, g1 and g0, by the secant viscosities gs1 and gs0 in (9). These secant viscosities then enter into the parameters involved in (25). The overall stress versus strain-rate relation are then replaced by

r kk ðtÞ ¼ 3gsj ðtÞe_ kk ; r 0ij ðtÞ ¼ 2gsl ðtÞe_ 0ij ;

ð26Þ

in terms of the effective secant viscosities gsj and gsl . Its uniaxial counterpart can be written as r 11 ðtÞ ¼ gsE ðtÞe_ 11 . 5. A field-fluctuation approach for the secant viscosity of the constituent phases In this computational procedure the secant viscosities of individual phases, gs1 and gs0 , are involved. These quantities need to be evaluated in order to make use of (26). Since these quantities depend on the drag stress s and the effective viscoplastic strain rate e_ vp e through (9), and the drag stress in turn also depends on the effective viscoplastic strain evp e through (6) which can be integrated from its rate _ vp e_ vp e , the main task boils down to the determination of the rate e e for the grain interior and the GB zone. We found that this could be most conveniently achieved through the application of the field-fluctuation method. The basic idea behind this technique is that, under the same boundary condition, a change in a material parameter of a constituent phase will result in a field fluctuation that gives rise to a new overall energy. Thus by taking a partial derivative of this energy with respect to that material parameter, a relation between the stress (or strain, or strain rate) of the individual phase and the overall stress (or strain, or strain rate) of the composite can be established. This concept was originally introduced by Bobeth and Diener (1986) and Kreher and Pompe (1989) for an elastic problem, and later extended to the rate-independent plasticity by Suquet (1995) and Hu (1996). In the context of rate-dependent viscoplasticity this approach was developed by Li and Weng (1997b, 2007) for the problems of timedependent creep and strain-rate sensitivity, respectively, without GB sliding. With grain-boundary sliding there is an additional work rate in the consideration of work-rate balance. In terms of the effective bulk and shear secant viscosities, the overall work rate under a constant strain-rate loading, e_ ij ¼ const:, can be written as

 ij e_ ij ¼ gsj ðe_ kk Þ2 þ 2gsl e_ 0ij e_ 0ij : U_ ¼ r

ð27Þ

On the other hand it can also be considered as the sum of the elastic and viscoplastic components of the equivalent inclusions and the matrix as

_e _ vp U_ ¼ c1 ðU_ e1 þ U_ vp e Þ þ c 0 ðU 0 þ U 0 Þ;

ð28Þ

where the superscript e stands for the elastic component and the subscript e for the equivalent inclusion. The viscoplastic work rate of the equivalent inclusion (U_ vp e ) has contribution from both _ vp the grain interior (U_ vp 1 ) and the interfacial sliding between the grain and GB zone (U f )

_ vp _ vp U_ vp e ¼ U1 þ Uf :

ð29Þ

The elastic term can be written using the elastic moduli of the corresponding phase. The viscoplastic term for each phase is defined through its secant viscosity as

P. Barai, G.J. Weng / International Journal of Plasticity 25 (2009) 2410–2434

 2 s _ vpð0Þ ; U_ vp 0 ¼ 3g0  ee

 2 s _ vpð1Þ ; U_ vp 1 ¼ 3g1  ee

2419

ð30Þ

whereas the viscoplastic work rate of the interface (U_ vp f ) depends on the interfacial viscosity, as in (10). Setting (27) equal to (28), we have

   2   2   2 s e s vpð0Þ _ _ vpð1Þ _   U c1 U_ e1 þ U_ vp þ 3 g  e þ 3 g  e þ c ¼ gsj e_ kk þ 2gsl e_ 0ij e_ 0ij : 0 1 e 0 0 e f

ð31Þ

Since this equality holds for any e_ ij , jr, lr, gs1 and gs0 , we may keep the boundary condition and the elastic constants fixed and vary only one secant viscosity of the constituent phase. Taking derivative with respect to gs1 and gs0 separately and, after some rearrangements, we arrive at the effective viscoplastic strain rate of the constituent phases in terms of the applied strain rate, as

  1 1 e_ vpð1Þ ¼ e c1 3   1 1 e_ vpð0Þ ¼ e c0 3

s @ gsj _ 2 @ gl _ 2 ðekk Þ þ s ee s @ g1 @ g1 s @ gsj _ 2 @ gl _ 2 ðekk Þ þ s ee s @ g0 @ g0

1=2 ;

ð32Þ

;

ð33Þ

1=2

here e_ 2e ¼ ð2=3Þ e_ 0ij e_ 0ij . These effective rates serve to determine the secant viscosities gs1 and gs0 through Eq. (9). Explicit forms of these four partial derivatives are given in Appendix A.2. 6. An incremental scheme for the calculation of overall stress–strain relation The overall stress–strain relation under a constant strain-rate loading can be obtained using this secant viscosity approach. Since the secant viscosity of the constituent phases continue to change with time, the rate form of (26) should be adopted so that such a change can be continuously updated. This leads to

r_ kk ðtÞ ¼ 3g_ sj ðtÞe_ kk ; r_ 0ij ðtÞ ¼ 2g_ sl ðtÞe_ 0ij :

ð34Þ _s

_s

Expressions for the rate of the secant bulk and shear viscosities (3gj ðtÞ; 2gl ) are also given Appendix A.1. The overall stress of the nanocrystalline material is then calculated through

r ij ðt þ DtÞ ¼ r ij ðtÞ þ r_ ij Dt;

ð35Þ

and this is accompanied by the change of overall strain through

eij ðt þ DtÞ ¼ eij ðtÞ þ e_ ij  Dt:

ð36Þ

7. Individual contributions from the equivalent grains and GB zone While direct experimental measurements or MD simulations can provide the overall stress–strain relation for the nanocrystalline solid, it remains a challenge for these techniques to quantify the individual contributions from various deformation sources that give rise to the overall strain. In this section, we outline a procedure to calculate the individual contributions of the equivalent inclusions (grains plus GB sliding) and the GB zone toward the total strain of the material. ð0Þ The total strain (eij ) is the weighted mean of the strains from the GB zone (eij ) and the equivalent ðeÞ inclusion (eij ); that is

eij ¼ c0 eð0Þ ðeÞ ij þ c 1 eij

ð0Þ ðeÞ and e_ ij ¼ c0 e_ ij þ c1 e_ ij ;

ð37Þ

where

1 0ðeÞ eðeÞ ðeÞ ij ¼ eij þ dij ekk ; 3

1 _ ðeÞ e_ ðeÞ _ 0ðeÞ  ij ¼ eij þ dij ekk : 3

ð38Þ

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In elastic context we have from (14)

eðeÞ kk ¼

j0 c0 a0 ðj1  j0 Þ þ j0

ekk ;

e0ðeÞ ij ¼

l0 c0 b0 ðle  l0 Þ þ l0

e0ij :

ð39Þ

To obtain its strain rate during viscoplastic deformation, we again apply the correspondence principle to (39) first, to convert the linear elastic relation to the linear viscoelastic one, and then replace the Maxwell viscosities of each phase by its secant viscosity. Going through such a process, the final expression for its strain can be established in terms of the applied strain rate. This derivation is provided in Appendix A.3. Taking the time derivative of the dilatational and deviatoric strains of the equivalent inclusion, we can write

"  # P2 P4 P4 e_ kk ; e_ ðeÞ ¼ P  exp  t 1 kk P3 P23

ð40Þ

e_ ij0ðeÞ ¼ ½R1 expðr 1 tÞ þ R2 expðr2 tÞ þ R3 expðr 3 tÞe_ 0ij ; where parameters P1, P2, P3, P4, R1, R2, R3, r1, r2, r3 can also be found there. A time-increment approach is used to compute the evolution of strain in the equivalent inclusion from (40), and that of the GB ð0Þ zone, e_ ij ,follows from (37), such that

ðeÞ _ ðeÞ eðeÞ ij ðt þ DtÞ ¼ eij ðtÞ þ Dt  eij ;

ð0Þ ð0Þ ð0Þ ; eij ðt þ DtÞ ¼ eij ðtÞ þ Dt  e_ ij :

ð41Þ

8. Results and discussions In this final section, we apply the developed theory to compute the stress–strain relations of very fine-grained nanocrystalline copper, and compare the results with MD simulations. Before we proceed to study the viscoplastic behavior, let us first examine the elastic case for the model of Duan et al. (2007) in (17) in conjunction with the MT moduli in (19). The results for the effective shear modulus with perfect normal bonding (p = 0) and imperfect normal and shear bonding (p = q/5) are shown in Fig. 2, along with the results of Hashin (1991) who used Christensen and Lo’s (1979) generalized self-consistent model to compute the overall shear modulus. (The superscript (e) in an and as has been omitted in the figure for brevity.) The calculated shear modulus by the present

Fig. 2. Comparison of the theory with Hashin (1991) for the effective shear modulus of a two-phase composite with imperfect interface.

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approach is seen to be somewhat lower in the initial stage. With perfect bonding (q = 0), the difference between the two is the widest, with the commonly known facts that the MT modulus lies on the Hashin–Shtrikman (1963) lower bound and the GSC lies above it. The discrepancy, however, gradually diminished as the extent of slip parameter, q, continues to increase. In the later stage both the MT and the GSC give essentially the same results. The results for the overall bulk modulus, not shown here, are identical. The nature of this agreement provides another reason to adopt the MT method in the present study. Now we calculate the viscoplastic response of very fine-grained nanocrystalline copper, and make comparison with MD simulations. Based on the information given in MD simulations we took the critical grain size below which dislocations cease to be active at 8 nm. The unified constitutive equation there applies to condition when d > 8 nm. The material constants used in the calculations are listed in Table 1. We further took t = 1 nm in all calculations. From MD simulations the GB thickness was found to be relatively insensitive to grain size (Schiotz, 2003). We first calculated the stress–strain relations at three different grain sizes, at d = 5.21, 4.13, and 3.28 nm, under a constant strain rate of 5.0  108/s. The results are shown in Fig. 3, along with the MD simulations of Schiotz et al. (1999). It is seen that both MD simulations and the present continuum modeling give rise to the grain-size softening of nanocrystalline materials within this very fine range of grain size. Since all sizes considered here are lower than 8 nm, grain interior essentially has only elastic deformation, and all plastic flow comes from the GB sliding and the plastic deformation of the GB zone. Grain-size softening arises from the increased contributions from these two mechanisms without the benefit of Hall–Petch effect that is embedded in the unified constitutive equation of the grain-interior phase. We also observe that the theory is capable of capturing the decrease of Young’s modulus with decreasing grain size as well. (The incremental theory has been developed based on the small-strain assumption, so comparison in the larger strain range must be read with such understanding.) We then made calculations for the stress–strain relations at the grain size of 4.13 nm under three different strain rates: 1  109/s, 5  108/s, and 2.5  108/s. The calculated results along with the MD simulations are shown in Fig. 4. The usual strain-rate sensitivities are observed in both calculations. Reasonable agreement between the two is observed. It must be pointed out that these high strain-rate calculations were made by taking the gain interiors to be dislocation-free, with no plastic deformation. At first sight it is natural to think that, to achieve such a high strain rate, dislocation mechanisms must be operative inside the grains and it would be difficult to have the grain-size softening. But as observed by Schiotz et al. (1999), ‘‘Little dislocation activity is seen in the grain interiors.” This high strain rate actually comes from the high speed atomic motion ‘‘through a large number of uncorrelated events” inside the GB zone. Since the GB zone is plastically softer than the grain interiors, its dominance is the main source of grain-size softening. Since grain-boundary sliding is taken to be active, it is also natural to think that the associated diffusion process in the Coble creep could not possibly lead to the high strain rate at the stress level of 2– 3 GPa in Figs. 3 and 4. However, it must be kept in mind that there are three independent deformation

Table 1 Material constants used in calculations. Parameters

Grain interior

Grain boundary zone

E(GPa)

145 0.3 5.03 7.9 1.5 – – 5 50 5 12

50 0.3 – – – 1300 5000 9 220 5 –

m s1 0 ðMPaÞ pffiffiffiffiffiffiffiffi kðGPa nmÞ a s0 (MPa) s* (MPa) h (MPa) n e_ 0 ð104 ; =hÞ as ðGPa=; nm sÞ

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Fig. 3. Comparison with the molecular dynamic simulations of Schiotz et al. (1999) for different grain sizes at a constant strain rate.

Fig. 4. Comparison with the molecular dynamic simulations of Schiotz et al. (1999) for different strain rates at a particular grain size.

mechanisms – grain interior, GB zone, and GB sliding – and it is the deformation of the GB zone that is the major source of the high strain rate. While the interfacial sliding is active, its generated strain is not the major source of the overall strain. Had the three deformation mechanisms been interdependent, then the slowest one would be the rate-controlling mechanism and, in that case, the high strain rate encountered in these figures would not have been able to be realized. We now explore the influence of the interfacial shear viscosity parameter, as, on the flow stress of nanocrystalline Cu at d = 5.21 nm under the strain rate of 5  108/s. The results are shown in Fig. 5, for the flow stress at 6% strain (frequently adopted in MD simulations). It is evident from this figure that for smaller values of as, the yield strength is low. As this interfacial viscosity increases the yield strength also increases, eventually reaching an asymptotic state that corresponds to the condition of perfect bonding without GB sliding.

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Fig. 5. Variation in the flow stress at 6% strain with varying slip parameter (as) at a constant grain size and strain rate.

We then extrapolated our calculations to grain sizes larger than 8 nm, so that the plasticity of grain interior would also contribute to the overall deformation of the nanocrystalline material. The results with grain sizes ranging from 10 to 25 nm under the same strain rate of 5  108/s are shown in Fig. 6. There is now evidence of initial grain size hardening from 25 to 15 nm, but from 15 to 10 nm the results display the grain-size softening of the material. The maximum yield strength appears around 15 nm. To illustrate how the interfacial viscosity, as, affects the transition from grain-size hardening to grain-size softening in the Hall–Petch plot, the yield strength at 6% strain is shown in Fig. 7, at three different levels of interfacial sliding viscosity. From this plot it is observed that:

Fig. 6. Hardening and softening in the nanocrystalline material due to decrease in the grain diameter at a particular strain rate.

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Fig. 7. Hall–Petch plot for different values of the slip parameter. The maximum strength depends on the slip parameter, but the grain size at which the maximum strength is obtained is almost independent of it.

(i) For grain size ‘d’ larger than 25–30 nm, the influence of GB sliding is not significant. In this range all three curves converge into a single one. (ii) The Hall–Petch plots with different as began to deviate from each other, showing the contribution of the GB sliding. The yield strength drops notably with decreasing as. (iii) For different values of as, the critical grain size at which the maximum yield strength occurs remains almost unaltered. (iv) The magnitude of the maximum yield strength, however, decreases with decreasing as value. An extended plot of Fig. 7 to the right toward vanishing grain size is shown in Fig. 8, from d = 4.9 to 2 nm. The four curves with four different orders of GB sliding viscosity are seen to converge into a

Fig. 8. Hall–Petch plot extended to very small grain size (d = 2 nm). Relative contribution of GB sliding to the overall strain diminishes as grain size decreases.

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single one. At d = 2 nm, the surface area of the grains is extremely small, and thus GB sliding has little contribution toward the overall strain of the material. In this extreme case it is the GB zone that dictates the overall behavior of the material. The model predictions of the strain-rate sensitivity as a function of grain size are shown in Fig. 9, for grain sizes ranging from 40 to 2 nm. Three important features are observed: (i) At any given grain size, there is monotonic increase of flow stress as a function of strain rate. (ii) At a given strain rate, the flow stress initially increases and then decreases, giving rise to the Hall–Petch and the inverse Hall–Petch effects. (iii) The strain-rate sensitivity – measured by the slope of the curve – initially increases with decreasing grain size from 40 to 20 nm, but then decreases from 10 to 2 nm. The slope at 2 nm is conspicuously lower than the others. Thus maximum strain-rate sensitivity thus does not occur at the smallest grain size, but at an intermediate size. The third feature appears to be first reported by Trelewicz and Schuh (2007) for a Ni–W alloy during their indentation tests. They attributed their observation to the possibility that at the very low end of grain size the material was very close to the amorphous state and, at this limit, the material has almost no strain-rate sensitivity. In our model, this low end corresponds to the condition that the major part of the material is occupied by the GB zone, which has a very high stress exponent n (at 220; see Table 1) and thus very low strain-rate sensitivity (which is represented by the inverse of n). The initial increase of the strain-rate sensitivity from 40 to 20 nm was due to the fact that the GB zone is plastically softer than the grain interior. Fig. 10 illustrates the transition of activation volume as the grain size decreases from d = 40 nm down to 2 nm, where (a) was taken from Trelewicz and Schuh (2007) and (b) was calculated from V ¼ kB T=½@ r=@ðln e_ Þ using the data in Fig. 9 and the Burgers vector b = 0.2556 nm. It is seen that there is the well known decrease of the activation volume as grain size decreases from 40 to 20 nm, but that the calculated result also indicates an increase from d = 10 nm down to 2 nm. This trend is consistent with the increase and then decrease of the Hall–Petch plot in Fig. 7, and is also in line with the theoretical prediction of Rodriguez and Armstrong (2006) on the basis of grain boundary weakening. It is noted that the activation volume in (b) is an order of magnitude lower than that in (a), for it is under

Fig. 9. Model calculations for the strain-rate sensitivity of the nanocrystalline Cu from 40 nm down to 2 nm.

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Fig. 10. Activation volume: (a) taken from Trelewicz and Schuh (2007) under low strain-rate loading, and (b) calculated from Fig. 9 under very high rate loading. The activation volume in (b) is about one order of magnitude lower than in (a). Armstrong and Walley (2008) and Armstrong et al. (2009) concluded that, for high strain rate greater than 105/s, the activation volume could reduce to the order of b3 even for coarse-grained metals.

an extremely high strain rate. Based on the high-rate test data of Follansbee et al. (1984) and, especially on the shock-loading tests of Swegle and Grady (1985), Armstrong and Walley (2008) and Armstrong et al. (2009) have concluded that there is a transition from the drag-controlled dislocation motion to dislocation generation as the rate-controlled mechanism, giving rise to the very high flow stress and very low activation volume – on the order of b3 – as the strain rate goes beyond 105/s (see their Fig. 24 and Fig. 2, respectively).

Fig. 11. Comparison between results calculated by setting the grain interior to be free from dislocation activities (elastic) and by allowing it to continue with dislocation activities governed by the Hall–Petch relation.

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Fig. 12. Test of the model for the nonlinearity of the dilatational response at a particular grain size and strain rate.

We recall that all the preceding calculations have been made by taking the grain interior to be free from dislocation activities once the grain size decreases below 8 nm, so it is of interest to see the difference between results calculated with and without such an assumption. The comparison is given in Fig. 11, for the grain size of 8 nm. It turns out that, at such a small grain size, the grain interior with dislocation hardening has become so hard that it behaves virtually like an elastic grain. For grain size lower than 8 nm, the difference is even less. So for the entire results presented above, the major features do not change even without making such an assumption. We now demonstrate the capability of the model to provide nonlinear dilatational response of the material in spite of plastic incompressibility of the grain interior and GB zone. It turns out that the extent of nonlinearity greatly depends on the elastic stiffness ratio between the grain interior and the GB region. Calculated results with four E1/E0 ratios are shown in Fig. 12. It is observed that the dilatational response remains almost linear for E1 = E0 and E1 > E0, but with E1 < E0, the response becomes increasingly nonlinear Since E1 P E0 for most nanocrystalline materials, their dilatational behavior is not expected to exhibit any significant nonlinearity. We finally illustrate the contributions of the equivalent inclusions (which include the grain interior and GB sliding) and the GB zone (phase 0) toward the total strain of the nanocrystalline material. Fig. 13(a) and (b) give the calculated results that correspond to d = 5.21 nm and 2 nm, respectively. The strain in the effective inclusions contains the elastic deformation of the grain interior and the viscous flow of GB sliding that is marked by as, whereas the strain from the GB zone contains both elastic and viscoplastic strains. In the first case, contribution from the GB zone is about twice as much as that of the other two combined, and in the second case with such a diminishing grain size the total strain of the nanocrystalline material is effectively contributed by the GB zone alone. 9. Conclusions In this paper, we have developed an analytical homogenization scheme that accounts for the contributions of grain interior, GB zone, and GB sliding at the interface between the two, to calculate the viscoplastic behavior of nanocrystalline materials, and used it to illustrate the grain size dependence of flow stress and strain-rate sensitivity from 40 nm down to 2 nm. The viscoplastic behavior of the grain interior is represented by a set of grain-size dependent unified theory, whereas that of the GB zone by a size-independent law. The interfacial GB sliding is modeled by a Newtonian flow. In the

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Fig. 13. Contributions to the total strain from the (grain interior + GB sliding) and the GB zone.

development of the homogenization scheme we started from the elastic solution of a two-phase composite with such three deformation mechanisms, extended it to the viscoelastic domain through the correspondence principle, and then replaced the Maxwell viscosity of the constituent phases with their secant viscosity. A field-fluctuation approach was further introduced to calculate the local effective strain rate of the constituent phases with the applied external strain rate. Several applications of the developed model have been made to uncover the effects of grain size dependence of the viscoplastic stress–strain relations of nanocrystalline copper under constant strain-rate loading. First, it was applied to calculate the stress–strain relations of Cu at the grain sizes of 5.21, 4.13 and 3.28 nm under the constant strain-rate loading of 5.0  108/s, and then under three strain rates at d = 4.13 nm, and the results are compared with the MD simulations of Schiotz et al. (1999). Both grain-size softening and strain-rate hardening were observed, and the agreement

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between the two is reasonable. We then applied the theory to study the effect of interfacial sliding viscosity on the overall yield strength of the material. The results showed that there is a transition from the low yield strength to saturation strength as the sliding viscosity increases toward the perfect bonding condition. We subsequently extended the calculations over a wider range of grain size to explore the possibility of Hall–Petch and the inverse Hall–Petch effects. Such a transition was demonstrated. The critical grain size at which the maximum yield strength occurs was found to be close to 15 nm. This critical grain size was found to be insensitive to the interfacial sliding viscosity, but the magnitude of the maximum strength does decrease with increasing interfacial imperfection. Moreover, the effect of GB sliding was found to be weak when the Hall–Petch slope is positive, and pronounced when the slope is negative. As the grain size reduces to a diminishing value, the contribution from the combined grain interior and GB sliding also diminish; in this case the deformation is effectively dominated by the plasticity of the GB zone alone. Finally we made significant contact with the test results of Trelewicz and Schuh (2007). Among some of the important findings shared by both investigations are: (i) there is a monotonic strain-rate hardening at a given grain size, (ii) at a give strain rate the yield strength first increases and then decreases as the grain size decreases, (iii) the maximum strain-rate sensitivity of a nanocrystalline material does not occur at the smallest grain size, but rather at a finite, but very low end, of the grain size, (iv) the activation volume first decreases with decreasing grain size, and then increases again, opposite to the trend of strain-rate sensitivity, and (v) the limiting condition is the nearly amorphous state of the GB zone. These common trends are also consistent with the theoretical predictions of Armstrong and Rodriguez (2006), and Rodriguez and Armstrong (2006) on the basis of grain-boundary weakening. Acknowledgments This work was supported by the NSF Division of Civil, Mechanical and Manufacturing Innovation, Mechanics and Structure of Materials Program, under Grant CMS-0510409. Appendix A A.1. Derivation of the effective secant viscosities gsj and gsl , and their rates g_ sj and g_ sl The operations involve the standard Laplace transform and inversion. The entire procedure is lengthy and unenlightening but the parameters listed below are needed to write a computer program. We start from jTD and lTD in (22), which involves lTD e . From (18) we can write

lTD e ¼

l1 sðd1 s2 þ d2 s þ d3 Þ d4 s3 þ d5 s2 þ d6 s þ d7

ðA:1Þ

;

with

d1 ¼ as R1 b1 ;

d2 ¼ 16g 3 l0 a3 þ as R1 b2 ;

d5 ¼ 2g 3 l0 b4 þ as R1 ðb1 T 1 þ b2 Þ; d7 ¼ 2g 3 l0 b6 þ as R1 b3 T 1 ;

d4 ¼ as R1 b1 ;

d6 ¼ 2g 3 l0 b5 þ as R1 ðb2 T 1 þ b3 Þ;

a1 ¼ 6j1 þ 2l1 ;

a4 ¼ 57j1 T 1 ;

a5 ¼ 48j1 þ 136l1 ;

a8 ¼ 440j1 T 1 ;

b1 ¼ 5ða5 þ g 3 a3 Þ;

b3 ¼ 5ða6 T 1 þ g 3 a4 T 0 Þ;

d3 ¼ 16g 3 l0 a4 þ as R1 b3 ; a2 ¼ 6j1 T 1 ;

a6 ¼ 48j1 T 1 ;

a3 ¼ 57j1 þ 4l1 ;

a7 ¼ 600j1 þ 440l1 ;

b2 ¼ 5½a5 T 1 þ a6 þ g 3 ða3 T 0 þ a4 Þ;

b4 ¼ a7 þ 3g 3 a3 ;

b5 ¼ a7 T 1 þ a8 þ 3g 3 ða3 T 0 þ a4 Þ;

b6 ¼ a8 T 1 þ 3g 3 a4 T 0 :

ðA:2Þ

After some algebra, we then arrive at



 p1 s þ p2 ; p3 s þ p4 l0 s l0 sðf1 s4 þ f2 s3 þ f3 s2 þ f4 s þ f5 Þ þ ; ¼ ðs þ T 0 Þ ðs þ T 0 Þðf6 s4 þ f7 s3 þ f8 s2 þ f9 s þ f10 Þ

jTD ¼ j0 1 þ

lTD

ðA:3Þ

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where

p1 ¼ c1 ðj1  j0 Þð3j0 þ 4l0 Þ;

p2 ¼ 3j0 T 0 c1 ðj1  j0 Þ;

p3 ¼ 3j0 ðc0 j1 þ c1 j0 Þ þ 4l0 j0 ;

p4 ¼ 3j0 T 0 ðc0 j1 þ c1 j0 Þ;

f 1 ¼ c1 ge1 ;

f 2 ¼ c1 ðge2 þ he1 Þ;

f 3 ¼ c1 ðge3 þ he2 Þ;

f 5 ¼ c1 he4 ;

f 6 ¼ c0 ee1 þ l0 gd4 ;

f 8 ¼ c0 ðee3 þ fe2 Þ þ l0 ðgd6 þ hd5 Þ; f 10 ¼ c0 fe4 þ l0 hd7 ; h ¼ 15j0 T 0 ;

f 7 ¼ c0 ðee2 þ fe1 Þ þ l0 ðgd5 þ hd4 Þ; f 9 ¼ c0 ðee4 þ fe3 Þ þ l0 ðgd7 þ hd6 Þ;

e ¼ 6j0 þ 12l0 ;

e1 ¼ l1 d1  l0 d4 ;

e3 ¼ l1 ðd2 T 0 þ d3 Þ  l0 d6 ;

f 4 ¼ c1 ðge4 þ he3 Þ;

g ¼ 15j0 þ 20l0 ;

f ¼ 6j0 T 0 ;

e2 ¼ l1 ðd1 T 0 þ d2 Þ  l0 d5 ;

e4 ¼ l1 d3 T 0  l0 d7 :

ðA:4Þ

eij ¼ 1=s2  e_ ij , and, with (A.3), the Laplace inversion leads to the Under constant strain-rate loading, ^ effective bulk secant and shear viscosities, gsk and gsl

gsj ¼ j0 fð1 þ A2 Þt þ A1 ½1  expðs1 tÞg; gsl ¼ gs0 ½1  expðT 0 tÞ þ A þ B expðT 0 tÞ þ C expðs1 tÞ

ðA:5Þ

þ D expðs2 tÞE expðs3 tÞ þ F expðs4 tÞ: Here t is time, and

A1 ¼

4l0 c1 c0 ðj1  j0 Þ2 3j0 T 0 ðc0 j1 þ c1 j0 Þ2

;

A2 ¼

c1 ðj1  j0 Þ ; c0 j1 þ c1 j0

s1 ¼

3j0 T 0 ðc1 j0 þ c0 j1 Þ ; 3j0 ðc1 j0 þ c0 j1 Þ þ 4l0 j0

15j0 c1 ðl1 T 0  l0 ðd7 =d3 ÞÞ f1 T 40  f2 T 30 þ f3 T 20  f4 T 0 þ f5 ; ; B ¼ gs0 15j0 l0 ðd7 =d3 Þ þ 6j0 c0 ðl1 T 0  l0 ðd7 =d3 ÞÞ f6 T 40  f7 T 30 þ f8 T 20  f9 T 0 þ f10   l0 f1 s41 þ f2 s31 þ f3 s21 þ f4 s1 þ f5 l0 f1 s42 þ f2 s32 þ f3 s22 þ f4 s2 þ f5 C¼ ; D¼ ; s1 f6 ðs1 þ T 0 Þðs1  s2 Þðs1  s3 Þðs1  s4 Þ s2 f6 ðs2 þ T 0 Þðs2  s1 Þðs2  s3 Þðs2  s4 Þ   l0 f1 s43 þ f2 s33 þ f3 s23 þ f4 s3 þ f5 l0 f1 s44 þ f2 s34 þ f3 s24 þ f4 s4 þ f5 ; F¼ ; ðA:6Þ E¼ s3 f6 ðs3 þ T 0 Þðs3  s1 Þðs3  s2 Þðs3  s4 Þ s4 f6 ðs4 þ T 0 Þðs4  s1 Þðs4  s2 Þðs4  s3 Þ A ¼ gs0

and si are the roots of the equation

s4 þ

f7 3 f8 2 f9 f10 s þ s þ sþ ¼ ðs  s1 Þðs  s2 Þðs  s3 Þðs  s4 Þ ¼ 0: f6 f6 f6 f6

ðA:7Þ

The rates of the secant viscosities, g_ sk and g_ sl , needed in (34) can be written as

g_ sj ¼

j1 j0

þ A4 expðs1 tÞ; c0 j1 þ c1 j0 s g_ l ¼ ðl0 þ B1 Þ expðT 0 tÞ þ B2 expðs1 tÞ

ðA:8Þ

þ B3 expðs2 tÞ þ B4 expðs3 tÞ þ B5 expðs4 tÞ; where

4l0 c1 c0 ðj1  j0 Þ2 ; ðc0 j1 þ c1 j0 Þ½4l0 þ ðc0 j1 þ c1 j0 Þ  l0 f1 s41 þ f2 s31 þ f3 s21 þ f4 s1 þ f5 B2 ¼ ; f6 ðs1 þ T 0 Þðs1  s2 Þðs1  s3 Þðs1  s4 Þ  4 l0 f1 s3 þ f2 s33 þ f3 s23 þ f4 s3 þ f5 ; B4 ¼ f6 ðs3 þ T 0 Þðs3  s1 Þðs3  s2 Þðs3  s4 Þ A4 ¼

B1 ¼ l0

f1 T 40  f2 T 30 þ f3 T 20  f4 T 0 þ f5

; f6 T 40  f7 T 30 þ f8 T 20  f9 T 0 þ f10  l0 f1 s42 þ f2 s32 þ f3 s22 þ f4 s2 þ f5 B3 ¼ ; f6 ðs2 þ T 0 Þðs2  s1 Þðs2  s3 Þðs2  s4 Þ  4 l0 f1 s4 þ f2 s34 þ f3 s24 þ f4 s4 þ f5 B5 ¼ : f6 ðs4 þ T 0 Þðs4  s1 Þðs4  s2 Þðs4  s3 Þ

ðA:9Þ

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A.2. Derivatives of gsl and gsj with respect to gs0 and gs1 in the field-fluctuation approach

@ gsj 4c1 c0 ðj1  j0 Þ2 1  expðs1 tÞ T 0 t expðs1 tÞ ¼ f  g; @ gs0 ðc0 j1 þ c1 j0 Þ 3ðc0 j1 þ c1 j0 Þ 4l0 þ 3ðc0 j1 þ c1 j0 Þ @ gsl @A @B T0 ¼ ½1  expðT 0 tÞ  fT 0 t expðT 0 tÞg þ s þ s expðT 0 tÞ þ Bt s expðT 0 tÞ @ g0 @ g0 @ gs0 g0 @C @s1 @D @s2 þ s expðs1 tÞ þ Ct s expðs1 tÞ þ s expðs2 tÞ þ Dt s expðs2 tÞ @ g0 @ g0 @ g0 @ g0 @E @s3 @F @s4 þ s expðs3 tÞ þ Et s expðs3 tÞ þ s expðs4 tÞ þ Ft s expðs4 tÞ: @ g0 @ g0 @ g0 @ g0

ðA:10Þ

Here the derivatives of the different parameters are calculated as follows:

@A AðtÞ  Aðt  DtÞ ðt þ DtÞ ¼ s ; @ gs0 g0 ðtÞ  gs0 ðt  DtÞ

@s1 s1 ðtÞ  s1 ðt  DtÞ ðt þ DtÞ ¼ s ; @ gs0 g0 ðtÞ  gs0 ðt  DtÞ

etc:

ðA:11Þ

On the other hand, @ gsj =@ gs1 ¼ 0, and

@ gsl @ gs1

¼

@A @B @C @s1 @D @s2 þ expðT 0 tÞ þ s expðs1 tÞ þ Ct s expðs1 tÞ þ s expðs2 tÞ þ Dt s @ gs1 @ gs1 @ g1 @ g1 @ g1 @ g1  expðs2 tÞ þ

@E @s3 @F @s4 expðs3 tÞ þ Et s expðs3 tÞ þ s expðs4 tÞ þ Ft s expðs4 tÞ @ gs1 @ g1 @ g1 @ g1

ðA:12Þ

The derivatives of the parameters can be evaluated as in (A.11). ðeÞ A.3. Parameters in Eq. (40) for the strain rate e_ ij of the effective inclusions In Eq. (40), the different parameters are given by

j0 P3 ; P2 ¼  ½j0 ð3j0 þ 4l0 Þ  P1 P 3 ; c1 j0 þ c0 j1 P4 P3 ¼ j0 ð3j0 þ 4l0 Þ þ 3j0 c0 ðj1  j0 Þ; P4 ¼ 3j0 Tðc1 j0 þ c0 j1 Þ;

P1 ¼

R1 ¼

q1 r21 þ q2 r1 þ q3 ; q4 ðr 1  r2 Þðr 1  r 3 Þ

R2 ¼

q1 r 22 þ q2 r 2 þ q3 ; q4 ðr 2  r 1 Þðr2  r3 Þ

R3 ¼

q1 r 23 þ q2 r 3 þ q3 ; q4 ðr3  r1 Þðr 3  r 2 Þ

ðA:13Þ

where parameters r i are the roots of the equation

s3 þ

q5 2 q6 q s þ s þ 7 ¼ ðs  r1 Þðs  r 2 Þðs  r 3 Þ ¼ 0; q4 q4 q4

ðA:14Þ

with

q1 ¼ l0 h3 g;

q2 ¼ l0 ðh3 h þ h4 gÞ;

q3 ¼ l0 h4 h;

q4 ¼ l0 h3 g þ c0 ½eðl1 h1  l0 h3 Þ;

q5 ¼ l0 ðh3 h þ h4 gÞ þ c0 fe½l1 ðh1 T 0 þ h2 Þ  l0 h4  þ f ðl1 h1  l0 h3 Þg; q6 ¼ l0 h4 h þ c0 fel1 h2 T 0 þ f ½l1 ðh1 T 0 þ a2 Þ  l0 h4 g; h2 ¼ c1 fhðl1  l0 Þ þ gðl1 T 0  l0 T 1 Þg;

q7 ¼ c0 f l1 h2 T 0 ;

h3 ¼ c1 hðl1 T 0  l0 T 1 Þ;

h5 ¼ c0 feðl1 T 0  l0 T 1 Þ þ f ðl1  l0 Þg þ l0 ðgT 1 þ hÞ;

h1 ¼ c1 gðl1  l0 Þ;

h4 ¼ c0 eðl1  l0 Þ þ l0 g;

h6 ¼ c0 f ðl1 T 0  l0 T 1 Þ þ l0 hT 1 :

ðA:15Þ

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