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Multi-scale modeling of elastoplastic deformation and strengthening mechanisms in aluminum-based amorphous nanocomposites
Acta Materialia 53 (2005) 2693–2701 www.actamat-journals.com
Multi-scale modeling of elastoplastic deformation and strengthening mechanisms in aluminum-based amorphous nanocomposites H.T. Liu a, L.Z. Sun b
b,*
a Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095-1593, USA Department of Civil and Environmental Engineering and Center for Computer-Aided Design, The University of Iowa, Iowa City, IA 52242-1527, USA
Received 20 January 2005; received in revised form 21 February 2005; accepted 22 February 2005 Available online 2 April 2005
Abstract Aluminum-based amorphous nanocomposites exhibit exceptional strength while preserving reasonable ductility. Although it is believed that the enhanced mechanical properties are due to the existence of defect-free nanometer-scale face-centered cubic a-Al nanoparticles in the amorphous aluminum matrix, the underlying strengthening mechanism remains to be determined so that a fundamental understanding of the material nanostructure–property relationship can be obtained. In this paper we focus on theoretical exploration of the mechanical constitutive behavior of amorphous nanocomposites in terms of a multi-scale approach starting from the nanostructure. A local heterogeneous stress field and deformation are formulated based on the concept of eigenstrain and equivalent inclusion method. An overall elastoplastic constitutive model for amorphous nanocomposites is developed through homogenization averaging procedures. Explicit expressions of the effective elastic stiffness and yield strength of amorphous nanocomposites in terms of the constituent properties and nanostructures are obtained. An experimentally observed interlayer phase between nanoparticles and the amorphous matrix is incorporated in the current model. The interlayer thickness is treated as a characteristic length scale. The nanoparticle size effect on material properties is specifically investigated within the continuum mechanics framework. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocomposites; Micromechanical modeling; Plastic deformation; Yield phenomena; Mean field analysis; Particle size effect
1. Introduction Aluminum-based amorphous nanocomposites were first developed in the early 1990s by Kim et al. in Japan [1] and Chen et al. in the United States [2] using a partial crystallization process in which the cooling rate in a certain temperature range for amorphous alloys was stringently controlled. The dispersion of nanoparticles in the
*
Corresponding author. Tel.: +1 319 384 0830; fax: +1 319 335 5660. E-mail addresses: [email protected], [email protected] (L.Z. Sun).
amorphous matrix leads to a remarkable increase of mechanical strength while preserving reasonable ductility [3]. It has been reported that the yield strength of the amorphous aluminum alloys is as high as 800 MPa in an amorphous state, and can be increased to 1500 MPa by partial crystallization [4–6]. Gogebakan [7] observed that the crystallized structure of Al–Y–Ni amorphous alloys with a nanoparticle size of 5–25 nm increased material strength to about 40% greater than corresponding single-phase amorphous alloys. This extraordinarily high mechanical strength makes Al-based amorphous nanocomposites a promising material for future industrial applications.
1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.02.029
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The typical composition of Al-based amorphous nanocomposites is Al–TM–RE, where TM is a transition metal such as Ni, Fe, Co, Cr; and RE is a rare earth element such as Y, La, Ce, Nd [8]. The microstructure of amorphous nanocomposites constitutes nanometer-scale face-centered cubic (fcc) a-Al particles dispersed in the amorphous aluminum matrix. The nanoparticle size and inter-particle distance are in the ranges of 5– 50 nm and 7–100 nm, respectively [3,8]. The total particle volume fraction is preferably in the range of 10–30% to preserve ductility. Further experiment showed that a higher annealing temperature and longer annealing time resulted in larger crystallized particle size and total particle volume fraction [9]. High-resolution electron microscopy examinations revealed that fcc a-Al nanoparticles exhibit a nearly spherical or an ellipsoidal morphology with no internal defects observed inside the nanoparticles [3]. Atomic probe field ion microscopy results of partially crystallized Al87Ni10Ce3 showed that Ce atoms are enriched within a distance of less than 3 nm at fcc a-Al particle interfaces [8]. This interesting observation suggests that a rare earth enriched interlayer phase exists between Al nanoparticles and the amorphous matrix. Many experimental results have demonstrated a remarkable increase in mechanical properties of Al-based amorphous nanocomposites. Inoue [3] measured the hardness and YoungÕs modulus of Al88Ni9Ce2Fe1 nanocomposites. The results showed that the YoungÕs modulus and hardness increased monotonously proportional to the particle volume fraction, achieving values of about 73 GPa and 420 Hv, respectively, at the volume fraction of 30%. Choi et al. [11] conducted experiments on Al88Ni10Nd2 nanocomposites where the results exhibited an almost linear increase in micro-Vickers hardness from 220 for amorphous single phase alloys to 400 for amorphous nanocomposites with particle volume fraction of 32%. Gogebakan [7] also tested Al85Y10Ni5 amorphous nanocomposites, resulting in a dramatic increase in the composite hardness. Studies of the amorphous nanocomposite strengthening mechanisms have been explored. Inoue [3] indicated that the Al nanoparticles were composed of an fcc structure without internal defects. The production of defectfree nanoparticles is due to the fact that the Al-based amorphous matrix has a high tendency for the precipitation of Al nanoparticles to relax. The absence of defects in Al nanoparticles greatly increases the critical resolved stress; and thus the yield strength determined by slips will be dramatically increased. In addition to defect-free Al nanoparticles, Zhong et al. [10] observed that the rare earth components amassed around the Al nanoparticles to form an interlayer. Hono et al. [8] suggested the occurrence of this phenomenon may be due to the slow diffusivity of rare earth atoms. During the crystallization
process, both TM and RE atoms are rejected from the a-Al phase. The atomic radius of the RE is usually much larger than the other solution elements, which may then result in an RE diffusivity that is slower than that of either Al or TM by orders of magnitude. Hence, during the growth of a-Al nanoparticles, the rejected RE atoms are enriched at the interface and form a heterogeneous interlayer. Although the extraordinarily high strength of nanocomposites has been experimentally observed and qualitatively explained by the existence of nanoparticles and nanostructures, the underlying strengthening mechanisms remain to be quantitatively revealed. Kim et al. [12] proposed a phenomenological composite model that applied the mixture rule based on the volume fraction of each constituent to estimate the hardness of Al–Ni–Y nanocomposites. Kim and Hong [13] further proposed a three-phase composite model comprising Al particles, a rare earth enriched interlayer, and an amorphous aluminum matrix. While these models provide a straightforward means to estimate the mechanical properties of the nanocomposites, no detailed local stress field and plastic deformation were considered and the local matrix–particle interaction was neglected. In the current study, theoretical exploration is applied to the nanostructure of nanocomposites. A multiscale approach that satisfies both equilibrium and compatibility is employed to model the overall elastoplastic behavior of amorphous nanocomposites. The nanocomposite is modeled as a heterogeneous material containing three distinguishing phases: defect-free fcc a-Al nanoparticles of spherical or spheroidal shape with a radius of 5–50 nm, an amorphous aluminum alloy, and a rare earth enriched interlayer surrounding the nanoparticles. At the nanometer-scale, the equivalent inclusion method [14] and the concept of eigenstrain [15] are adopted to analyze the heterogeneous stress field and local deformation. The overall mechanical properties of amorphous nanocomposites in terms of elastic stiffness and yield strength are derived via a homogenization averaging process. It is noted that the mean-field principles (homogenization procedures) can be directly applied to estimate the effective yield strength of composites since the initial yielding and plastic hardening of composites should be attributed to the collective responses of particle–matrix interactions [15–17]. However, homogenization theory may be no longer valid when the material softening and fracture occur due to the localized damage mechanisms. It is also noted that the particle size has a significant effect on the macroscopic mechanical properties for composites. Conventional continuum theory fails to incorporate the size effect given the absence of internal material length. Huang and co-workers [18,19] and Kouzeli and Mortensen [20] investigated the particle
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
size effect in alloys and composites via the theory of mechanism-based strain gradient plasticity [21]. In the current study, the interlayer thickness is treated as a characteristic length scale and the particle size effect on the overall mechanical properties of amorphous nanocomposites is investigated.
2. Heterogeneous stress field To characterize the typical microstructure of nanocomposites, a three-phase representative volume element is considered (Fig. 1), comprising spheroidal shaped Al nanoparticle domain X, rare earth enriched interlayer domain C, and amorphous matrix domain R. Furthermore, let R = X + C denote the generalized inclusion domain that contains both the particle and its corresponding interlayer. When subject to a far-field external loading r0, the heterogeneous local stress field can be expressed as 8 0 0 > < C : ½e0 þ e ðxÞ; x 2 R; rðxÞ ¼ CX : ½e0 þ e0 ðxÞ; x 2 X; ð1Þ > : C 0 C : ½e0 þ e ðxÞ; x 2 C;
2695
where the double dot symbol ‘‘:’’ indicates tensor contractions between a fourth-rank tensor and a secondrank tensor, e0 is the far-field strain corresponding to the far-field stress r0 with r0 = C0: e0, and C0 is the elastic stiffness tensor of the amorphous matrix. e 0 (x) represents the disturbance strain due to the heterogeneities. CX and CC denote the elastic stiffness tensors of nanoparticles and the interlayer, respectively. For isotropic materials, these fourth-rank elastic stiffness tensors can be expressed as C gijkl ¼ kg dij dkl þ lg ðdik djl þ dil djk Þ;
g ¼ 0; X; C;
ð2Þ
where kg and lg are LameÕs constants for the corresponding phases. Eshelby [14] and Mura [15] proposed the concept of eigenstrain (non-stress strain) and the equivalent inclusion method, in which a homogeneous material with equivalent eigenstrains in corresponding domains is used as a substitute for the heterogeneous material. For the problem described in Eq. (1), eigenstrains e*(x) are assumed to be inside the domains X and C, whereupon the stress field can be rewritten as 8 0 0 > x 2 R; < C : ½e0 þ e ðxÞ; 0 0 rðxÞ ¼ C : ½e0 þ e ðxÞ þ e ðxÞ; x 2 X; ð3Þ > : 0 0 C : ½e0 þ e ðxÞ þ e ðxÞ; x 2 C: Generally speaking, eigenstrains are a function of the material properties of the constituents, nanostructures, particle shape, and far-field loading conditions. Eshelby [14] showed that, for spheroidal particles, the eigenstrain is constant and can be expressed explicitly. However, for the current problem the eigenstrain is not constant due to the influence of existing interlayer. To simplify the derivation processes, Hori and NematNasserÕs double-inclusion method [22] is employed, in which volume averaging processes are conducted over the particle domain X and the interlayer domain C, respectively. Thus, the volume-averaged eigenstrains can be expressed as ( eX ¼ UX : e0 for particle domain X; e ¼ ð4Þ eC ¼ UC : e0 for interlayer domain C:
Fig. 1. (a) Schematic representation of the microstructure of Al-based amorphous nanocomposites: spheroidal Al nanoparticle domain X and rare-earth element enriched interlayer domain C embedded in the amorphous matrix domain R; (b) sketch of a spheroid.
Here, eX and eC denote the averaged eigenstrains inside domain X and C, respectively. The fourth-rank tensor UX and UC can be expressed explicitly for spheroidal particles as " 1 f X DS þ AC U ¼ SX þ AX þ DS SX 1f # 1 f DS þ AX ð5Þ SX 1f and
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"
U ¼ DS þ ðSX þ AX Þ SX C
SX
f DS þ AC 1f
f DS þ AX 1f
1
#1 ;
ð6Þ
respectively. In Eqs. (5) and (6), f denotes the volume fraction of the particle domain X over the generalized entire inclusion domain R as f = X/R, and AX and AC are the mismatch material property tensors for domain X and C, which can expressed as AX ¼ ðCX C0 Þ1 C0 ;
AC ¼ ðCC C0 Þ1 C0 :
ð7Þ
Moreover, DS is defined as DS = SR SX with SX and SR being EshelbyÕs tensors for particle domain X and the entire inclusion domain R, respectively. It is noted that EshelbyÕs tensors relate to only the inclusion shape for spheroidal particles. When the nanoparticle shape is approximately spherical or the thickness of the interlayer is much less than the radius of the nanoparticle, the particle domain X and the entire inclusion domain R have similar shapes. Thus it can be assumed that SR = SX = S and subsequently DS = 0. Correspondingly, Eqs. (5) and (6) can be rewritten as 1
ð8Þ
1
ð9Þ
UX ¼ ðS þ AX Þ and
UC ¼ ðS þ AC Þ ;
respectively. When an equivalent inclusion is used to replace the nanoparticle and the corresponding interlayer, the equivalent volume-averaged eigenstrain for the entire inclusion can be expressed as e ¼ U R : e0 ;
ð10Þ
where UR ¼ f UX þ ð1 f ÞUC h i ¼ f ðS þ AX Þ1 þ ð1 f ÞðS þ AC Þ1 :
Local yielding and plastic flow are dependent on local stress field. For simplicity, the commonly used von Mises yield criterion with an isotropic hardening law is assumed for the matrix material as an illustration. Namely, the local yield function reads pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ðr; ep Þ ¼ r : Id : r Kðep Þ 6 0; ð14Þ where ep and K(ep) are the equivalent plastic strain and the isotropic hardening function of the matrix-only material, respectively. Moreover, Id denotes the deviatoric part of the fourth-rank identity tensor I. Following Ju and Sun [24], we denote by H(x j g) = r(x j g) : Id : r(x j g) the square of the ‘‘current stress norm’’ at a local point x for a given inclusion configuration (assembly) g. Furthermore, ÆHæm(x) is defined as the ensemble average of H(x j g) over all possible realizations for a matrix point x, which indicates I hH im ðxÞ ¼ H 0 þ ½H ðx j gÞ H 0 P ðgÞ dg; ð15Þ g
where P(g) is the probability density function for determining an inclusion for a given configuration g, and H0 = r0 : Id : r0 is the square of the far-field stress norm applied on the composite. To calculate ÆHæm(x), Eq. (15) can be rewritten as I hH im ðxÞ ¼ H 0 þ ½H ðx j x0 Þ H 0 P ðx0 Þ dx0 ; ð16Þ x0 62NðxÞ
ð11Þ
The disturbance strain at a position x inside the matrix due to the eigenstrain e in a spheroidal inclusion centered at x 0 can be calculated as e0 ðxÞ ¼ Gðx x0 Þ : e :
a-Al nanoparticles contain no dislocations or other imperfections [12]. The interactions among nanoparticles, solute-enriched interlayers, and amorphous matrix are expected to have a significant effect on the strengthening phenomenon in the nanocomposites. Accordingly, at any matrix material point x, the local stress r(x) can be calculated using Eqs. (1) and (12) as ð13Þ rðxÞ ¼ r0 þ C0 : Gðx x0 Þ : e :
ð12Þ
Here, Gðx x0 Þ is the exterior-point EshelbyÕs tensor, which is given explicitly for spheroidal inclusions by Ju and Sun [23]. 3. Ensemble-average procedures The overall plastic behavior of Al-based amorphous nanocomposites is attributed to the plastic deformation in the amorphous matrix since it is observed that the fcc
where N(x) is the exclusion zone of x for the center location x 0 of an inclusion in the probability space, which is identical to the shape and size of a spheroidal inclusion. Since nanoparticles are uniformly distributed in the amorphous matrix, P(x 0 ) can be simplified as N/V where N is the total number of nanoparticles uniformly dispersed in entire volume V. After a series of lengthy but straightforward derivations, the ensemble-averaged ÆHæm can be evaluated as hH im ¼ r0 : T : r0 ;
ð17Þ
where the components of the fourth-rank tensor T take the form ð1Þ
ð2Þ
T ijkl ¼ T IK dij dkl þ T IJ ðdik djl þ dil djk Þ with
ð18Þ
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
ð1Þ
T IK
ð2Þ
T IJ
3 9 82 3ð35m20 70m0 þ 36ÞDIK > > > > > > > > 6 þ7ð50m2 59m þ 8ÞðD þ D Þ 7 ð1f Þ f > > þ 4 5 > > 0 I K ð1Þ ð1Þ ð2Þ ð2Þ 0 > > B B B B > > II KK II KK > > 2 > > 2ð175m 343m þ 103Þ > > 0 0 > > > > 0 1 > > ð1Þ ð1Þ > > f ðC þC Þ > > II KK > > > > ð1Þ ð1Þ > > B BII BKK C > > > > B C > > þ21ð25m 2Þð1 2m Þ 0 0 > > @ ð1f ÞðCð2Þ þCð2Þ Þ A = < R II KK 1 2/ þ ð2Þ ð2Þ ; ¼ þ B B > 3 4725ð1 m0 Þ2 > 0 ð1Þ II KKð1Þ 1 > > > > > > f ðCII DK þCKK DI Þ > > > > ð1Þ ð1Þ > > B C > > B B > > II KK B C > > þ21ð25m 23Þð1 2m Þ > > 0 0 ð2Þ ð2Þ @ ð1f ÞðC DK þC DI Þ A > > > > II KK > > þ > > ð2Þ ð2Þ > > BII BKK > > > > > > > > ð1Þ ð1Þ ð2Þ ð2Þ > > > > f C C ð1f ÞC C 2 > > II KK II KK ; : þ1575ð1 2m0 Þ ð1Þ ð1Þ þ ð2Þ ð2Þ BII BKK BII BKK 2 3 ! ð70m20 140m0 þ 72ÞDIJ R 1 / f ð1 f Þ 6 7 JÞ ¼ þ þ ð2Þ ð2Þ 4 ð175m20 266m0 þ 75Þ ðDI þD 5: ð1Þ ð1Þ 2 2 1575ð1 m0 Þ2 BIJ BIJ BIJ BIJ 2 þð350m0 476m0 þ 164Þ
Here, m0 is PoissonÕs ratio of the amorphous matrix, and /R is the volume fraction of the whole inclusions (including nanoparticles and their corresponding interlayers). Other parameters in the above equations are given in Appendix A. In Eq. (17), ÆHæm is described in terms of the far-field stress r0. Alternatively, ÆHæm can be expressed in terms of the macroscopic (ensemble-volume averaged) stress r. The relationship between the far-field stress r0 and macroscopic stress r is given by (cf. Ref. [25]) r0 ¼ P : r; where the fourth-rank tensor P reads 1 P ¼ I þ /R ðS IÞ UR :
ð20Þ
ð21Þ
Combining Eqs. (17) and (20) then leads to an alternative expression of the ensemble-averaged square of the current stress norm expressed as hH im ¼ r : T : r
ð22Þ
with T ¼ P T P. It is observed from Eq. (22) that ÆHæm can be reduced to the form derived by Ju and Sun [24] if no interlayer is present around the nanoparticles.
4. Nanocomposite constitutive modeling Based on linear elastoplasticity principle, the total macroscopic (effective) strain tensor e consists of elastic part ee and plastic part ep . The effective elastic strain ee is related to the effective stress r via r ¼ C : ee where the effective elastic tensor C of nanocomposites can be derived with the help of general governing equations of composite materials [25]. Specifically, for nanocompos-
2697
ð19Þ
ites containing randomly distributed yet aligned spheroidal particles with interlayers, the three governing equations can be obtained as r ¼ C0 : ðee /R e Þ; ee ¼ e0 þ /R S : e ;
ð23Þ
R
e ¼ U : e0 : Therefore, the explicit expression of the effective elastic tensor C of the nanocomposites can be shown as h i 1 C ¼ C0 I þ /R UR ð/R S UR IÞ : ð24Þ Upon further loading, the nanocomposites may yield and become plastic. The ensemble-volume-averaged yield function F of the nanocomposites can be characterized directly from Eq. (14): qffiffiffiffiffiffiffiffiffiffiffi F ¼ ð1 /R Þ hH im Kðep Þ 6 0: ð25Þ It is noted that, for isotropic plastic hardening, Kðep Þ can be simplified as rffiffiffi 2 q p Kðe Þ ¼ ry þ hðep Þ ; ð26Þ 3 where ry, h and q, and ep denote the yield strength of the matrix, the linear and exponential isotropic hardening parameters of the matrix, and the effective equivalent plastic strain of the nanocomposites, respectively. It is noted that Eq. (25) represents the pressure-dependent nanocomposite yield function although the matrix is assumed to be pressure-independent von Mises type yield criterion. Based on the derived effective yield function, the p effective plastic strain rate e_ can be calculated from the associative plastic flow rule:
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
_ ¼ 0; kF
k_ F_ ¼ 0:
ð28Þ
Therefore, the effective elastoplastic constitutive model of nanocomposites is developed with the formulation given in Eqs. (24)–(27), capable of estimating the overall elastoplastic stress–strain responses of nanocomposites under general three-dimensional loading conditions.
amorphous alloy
F 6 0;
1.30
1.25
1.20
/E
k_ P 0;
1.35
composite
oF p ; ð27Þ e_ ¼ k_ or where k_ is the plastic consistency parameter which can be obtained from the following Kuhn–Tucker condition:
1.15
E
2698
1.10
α = 2.0 α = 1.5 α = 1.0
1.05
1.00 0.00
0.05
0.15
0.20
0.25
volume faction φ
Fig. 2. Overall YoungÕs modulus of the nanocomposites vs. particle volume fraction for various particle shapes.
duction of the interlayer thickness as a characteristic length scale, the particle size effect can be incorporated in the current model. It is noted that, in the present study, the particle-size effect is equivalent to the effect of particle interface-to-volume ratio (IVR), since IVR is directly related with the radius of particle a and thickness of interlayer t as IVR = 3/(a + t). In Fig. 4, the overall yield strength of the nanocomposites is presented for a large range of particle sizes, from nanometer-scale to micrometer-scale. For a fixed interlayer thickness, a clear particle size effect (or IVR effect) is observed with the particle radius ranging in the nanometer scale. When the particle size corresponds to the same length scale as the interlayer thickness, a significant increase of yield strength is exhibited, proving that the existence of an interlayer significantly affects the overall mechanical properties
1.60 1.55
composite
/σy
amorphous alloy
1.50
σy
Specific Al–Ni–Y nanocomposites are applied as the model material in the following simulation. All phases are assumed to be isotropic. The amorphous Al matrix is assumed to have a YoungÕs modulus of 70 GPa and a PoissonÕs ratio of 0.3. The yield strength of the matrix is taken as 1.29 GPa [7], and the plastic hardening parameters are h = 1.0 GPa and q = 0.3, respectively. The Al a-fcc nanoparticles have a YoungÕs modulus of 71 GPa and a PoissonÕs ratio of 0.31. Since there are no mechanical data available for the rare-earth-element enriched interlayer, it is assumed that the intermetallic compound interlayer phase has a YoungÕs modulus of 500 GPa and a PoissonÕs ratio of 0.25. Due to the much higher strength of the defect-free Al nanoparticles, plastic deformation is constrained in the amorphous matrix. Unless explicitly stated otherwise, the radius of nanoparticles is assumed to be 10 nm and the thickness of interlayers to be 3 nm. Uniaxial loading simulation is conducted to investigate the elastoplastic behavior of the nanocomposites. Fig. 2 shows that the overall YoungÕs modulus increases proportional to the particle volume fraction. In our model, the spheroidal nanoparticles are aligned in the loading direction, and therefore nanocomposites with a larger particle aspect ratio exhibit a higher overall stiffness than nanocomposites with smaller particle aspect ratio. For a nanocomposite with 20% particle volume fraction, the overall YoungÕs modulus in the particle-aligned direction can be approximately 1.3 times higher than that of amorphous-alloy counterpart. The nanocomposites also demonstrate the strong strengthening effect as shown in Fig. 3. The nanocomposite having a 25% particle volume fraction indicates a yield-strength 1.5 times higher than that of amorphous alloys. It is evident that a larger aspect ratio of particles leads to a higher overall yield strength for the nanocomposites. Experimental investigations have shown that the particle size has a significant effect on the overall mechanical properties of composites. With the intro-
0.10
Σ
5. Numerical simulation and discussion
1.45 1.40 1.35 1.30 1.25 1.20 1.15
α = 2.0 α = 1.5 α = 1.0
1.10 1.05 1.00 0.00
0.05
0.10
0.15
0.20
0.25
Σ
volume faction φ
Fig. 3. Overall yield strength of the nanocomposites vs. particle volume fraction for various particle shapes.
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
2699
1.40
1.4
Σ
α = 1.0
1.2
_ amorphous alloy σe/σy
1.35
1.30
σy
composite
/σy
amorphous alloy
φ = 20%
1.0 0.8 0.6 0.4
1.25
Σ
φ = 5%
tinterlayer = 0 nm tinterlayer = 3 nm tinterlayer = 10 nm
Σ
φ = 10%
0.2
Σ
φ = 20%
1.20
0.0 10
100
1000
10000
100000
0
Log (radius of particles) (nm)
of the nanocomposites. Since the interlayer between the particles and the matrix is usually measured in nanometers, the nanoparticle-reinforced composites demonstrate higher yield strength than microparticlereinforced composites. However, when the interlayer is non-existent (i.e., the interlayer thickness is zero), the overall yield strength does not change for the entire particle size range, implying that for a fixed volume fraction, the particle size does not affect the overall yield strength. This assertion so far has not been validated with experimental observations since Al-based amorphous nanocomposites always exhibit a solute-enriched interlayer between the nanoparticle and the matrix [8,10]. The pure IVR effect without interlayer involved should be further investigated from both experimental and modeling points of view. Onedimensional strain–stress curves for uniaxial loading tests are presented in Fig. 5 for the nanocomposites
1.8 Σ
φ = 20%
amorphous alloy
/σy
composite
σ
1.4 1.2 1.0 0.8 0.6
rparticle = 5 nm rparticle = 10 nm rparticle = 20 nm rparticle = 50 nm
0.4 0.2
0.5
1.0
1.5
2.0
2.5
_
6
8
10
12
amorphous alloy
3.0
3.5
Fig. 6. Overall initial yield surfaces of the nanocomposites under axisymmetric loading.
with spherical particles. For the same volume fraction of 20%, the nanocomposites with small particles show higher stiffening and strengthening effects than those with large particles. To investigate the strengthening effect of the nanocomposites under complex loading condition, the nanocomposite yield-surface is certainly of interest. Axisymmetric loading cases are specified here to study multi-axial strengthening effect. With the assumption of overall stresses as r11 > 0; r22 ¼ r33 > 0; and r12 ¼ r13 ¼ r23 ¼ 0, the initial yield surfaces are demarcated in the volumetric and deviatoric stress space in Fig. 6. The nanocomposite volumetric-stress rv and deviatoric-stress re can be obtained from their definitions with considering the axisymmetric property as rv ¼ ðr11 þ 2r22 Þ=3 and re ¼ r11 r22 , respectively. It is shown from Fig. 6 that the yielding response of the nanocomposites is not of von Mises type, even when the nanoparticles are spherical in shape and randomly distributed. A decrease in volume fraction of nanoparticles leads to an increase of the volumetric yield stress and a simultaneous decrease of the effective yield stress. This trend indicates that the pressure-dependence of nanocomposite yielding primarily resulted from the existence of nanoparticles. When the volume fraction of particles vanishes, the matrixonly material will be restored von Mises yielding, which is consistent with the assumption of the matrix material satisfying the von Mises J2-yield criterion.
6. Conclusions
0.0 0.0
4
σv/σy
Fig. 4. Particle size effect on the overall yield strength of the nanocomposites.
1.6
2
4.0
overall strain (%)
Fig. 5. Overall uniaxial stress–strain curves of the nanocomposites.
A multi-scale elastoplastic framework has been proposed to investigate the overall mechanical properties of amorphous nanocomposites with randomly dispersed, unidirectionally aligned spheroidal particles.
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At the nanometer-scale, the nanostructure is modeled as spheroidal particles embedded in the amorphous matrix where an interlayer between the particle and the matrix exists. The locally heterogeneous stress field is obtained from the equivalent inclusion method and the concept of eigenstrain. The von Mises yield criterion and isotropic hardening rule are assumed to control the local yielding and plastic deformation in the matrix. Ensemble and volume average procedures have been conducted in a microscopic representative volume element to formulate the overall (effective) elastic stiffness and yield function for nanocomposites. The interlayer between the nanoparticle and the matrix significant affects the overall mechanical properties of nanocomposites. The interlayer thickness is treated as a characteristic length scale, thus the particle size is incorporated into the current model. The particle size effect has been investigated for nanocomposites within the continuum mechanics framework.
D11 ¼
5½2 þ a4 3a4 f ða2 Þ 2ð1 a4 Þ
2
D12 ¼ D21 ¼ D13 ¼ D31 ¼
;
15a4 ½3 þ ð1 þ 2a4 Þf ða2 Þ 4ð1 a4 Þ
2
;
D22 ¼ D23 ¼ D32 ¼ D33 ¼ 18ð15 3D11 4D12 Þ; ðA:1Þ where a = a1/a2 is the aspect ratio of the inclusion [see Fig. 1(b)]. Furthermore,
f ðaÞ ¼
8 <
1 a cos pffiffiffiffiffiffiffi ; a 1a2
a < 1;
a : cosh pffiffiffiffiffiffiffi ; a > 1; a a2 1 1
ðgÞ
ðgÞ
BIJ ¼ 2ðV IJ þ N IJ Þ;
g ¼ 1; 2;
ðA:2Þ
ðA:3Þ
9 2 9 8 31 8 ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ > > > U 21 þ M 21 U 31 þ M 31 U I1 þ M I1 > > > > = 6 U 11 þ 2V 11 þ M 11 þ 2N 11 = < CI1 > < 7 ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ 7 ¼6 CI2 U þ M U þ 2V þ M þ 2N U þ M U þ M 12 22 22 32 5 > I2 12 22 22 32 I2 > > 4 > > > ; ; : ðgÞ > : ðgÞ > ðgÞ ðgÞ ðgÞ ðgÞ CI3 U I3 þ M I3 U 13 þ M 13 U 23 þ M 23 U 33 þ 2V 33 þ M 33 þ 2N 33 ðI ¼ 1; 2; 3Þ ðA:4Þ Under the uniaxial loading condition, the effects of particle shape and particle size on the overall elastic stiffness and yield strength of nanocomposites have been numerically simulated. One-dimensional elastoplastic strain–stress curves have been presented for various particle sizes. The overall yield surfaces are demarcated for nanocomposites under axisymmetric loading to demonstrate the multi-axial strengthening effect and pressuredependent yielding response due to the existence of nanoparticles. The proposed model satisfies the continuum mechanics rules and provides a feasible means of estimating the mechanical response of nanocomposites.
with U 11 ¼ 4m0 þ U 12 ¼ U 13 U 21 ¼ U 31
2 4 ; hðaÞ þ 4m0 þ 2 2 a 1 3ða 1Þ 2a2 þ 1 2a2 ; ¼ 4m0 2 hðaÞ þ 4m0 2 a 1 a 1 2a2 þ 1 2a2 ; ¼ 2m0 2 hðaÞ 2 a 1 a 1
U 22 ¼ U 23 ¼ U 32 ¼ U 33 4a2 1 a2 ; ¼ 2m0 þ hðaÞ þ 2 2 4ða 1Þ 2ða 1Þ ðA:5Þ
Acknowledgment This research is sponsored by the National Science Foundation (NSF) under grant number CMS-0303955. The support of the NSF is gratefully acknowledged.
Parameters in Eq. (19) are expressed as 3½1 a4 f ða2 Þ ; 1 a4
V 11
4a2 2 12a2 8 ; ¼ 4m0 þ 2 hðaÞ 4m0 þ a 1 3ða2 1Þ
V 12 ¼ V 21 ¼ V 13 ¼ V 31 a2 þ 2 2 ; ¼ m0 2 hðaÞ 2m0 2 a 1 a 1
Appendix A
D1 ¼
D2 ¼ D3 ¼ 12ð3 D1 Þ;
V 22 ¼ V 23 ¼ V 32 ¼ V 33 4a2 7 a2 ; ¼ 2m0 hðaÞ þ 4ða2 1Þ 2ða2 1Þ
ðA:6Þ
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
kð0Þ lð1Þ kð1Þ lð0Þ
ð1Þ
M IJ ¼ ð1Þ
N IJ
ðlð1Þ lð0Þ Þ½2ðlð1Þ lð0Þ Þ þ 3ðkð1Þ kð0Þ Þ lð0Þ ; ¼ ð1Þ 2ðl lð0Þ Þ
ð2Þ
M IJ ¼
kð0Þ ð1 DIK dkk Þ 2lð0Þ DIJ ð2Þ
2ðlII lð0Þ Þ lð0Þ
ð2Þ
N IJ ¼
ð2Þ 2ðlIJ
lð0Þ Þ
[3] [4] [5] [6] [7] [8]
;
;
[9] [10]
;
[11]
ðA:7Þ and hðaÞ ¼
8 <
a ð1a2 Þ3=2
½að1 a Þ
:
a ða2 1Þ3=2
½cosh1 a aða2 1Þ
2 1=2
1
cos a; 1=2
[13] [14] [15]
a < 1; ;
[12]
a > 1;
[16]
ðA:8Þ 9 2 ð2Þ 8 ð2Þ ð0Þ ð0Þ > = 6 k11 k þ 2ðl11 l Þ < DI1 > ð2Þ DI2 ¼ 6 k12 kð0Þ 4 > > ; : ð2Þ DI3 k kð0Þ 13
Inoue A. Prog Mater Sci 1998;43:365. Kim YH, Inoue A, Masumoto T. Mater Trans 1991;32:599. Inoue A, Kim YH, Masumoto T. Mater Trans 1992;33:487. He Y, Poon SJ, Shiflet GJ. Science 1988;241:1640. Gogebakan M. J Light Metals 2002;2:271. Hono K, Zhang Y, Inoue A, Sakurai T. Mater Sci Eng 1997;A226:498. Jiang XY, Zhong ZC, Greer AL. Mater Sci Eng 1997;A226:789. Zhong ZC, Jiang XY, Greer AL. Mater Sci Eng 1997;A226:531. Choi GS, Kim YH, Cho HK, Inoue A, Masumoto T. Scripta Metall Mater 1995;33:1301. Kim HS, Warren PJ, Cantor B, Lee HR. Nanostruct Mater 1999;11:241. Kim HS, Hong SL. Acta Mater 1999;47:2059. Eshelby JD. Proc Roy Soc London 1957;A241:376. Mura T. Micromechanics of defects in solids. 2nd ed.. Kluwer Academic Publishers; 1987. Liu Y, Gilormini P, Ponte Castaneda P. Acta Mater 2003;51:5425.
ð2Þ
k12 kð0Þ ð2Þ
ð2Þ
k22 kð0Þ þ 2ðl22 lð0Þ Þ ð2Þ
k23 kð0Þ
2701
9 31 8 ð2Þ > kI1 kð0Þ > > > = < 7 ð2Þ ð2Þ ð0Þ ð0Þ 7 k23 k k k 5 > I2 > > > ; : ð2Þ ð2Þ ð2Þ kI3 kð0Þ k33 kð0Þ þ 2ðl33 lð0Þ Þ ð2Þ
k13 kð0Þ
ðI ¼ 1; 2; 3Þ: ðA:9Þ In the above equations, the superscripts of LameÕs constants indicate different materials: ‘‘0’’ as the matrix, ‘‘1’’ as the nanoparticle, and ‘‘2’’ as the interlayer. References [1] Kim YH, Inoue A, Masumoto T. Mater Trans 1990;31:747. [2] Chen H, He Y, Shiflet GJ, Poon SJ. Scripta Metall Mater 1991;25:1421.
[17] Jiang B, Weng GJ. Metall Mater Trans A 2003;34A:765. [18] Xue Z, Huang Y, Li M. Acta Mater 2002;50:149. [19] Huang Y, Qu S, Hwang KC, Li M, Gao H. Int J Plast 2004;20:753. [20] Kouzeli M, Mortensen A. Acta Mater 2002;50:39. [21] Gao H, Huang Y, Nix WD, Hutchinson JW. J Mech Phys Solids 1999;47:1239. [22] Hori M, Nemat-Nasser S. Mech Mater 1993;14:189. [23] Ju JW, Sun LZ. J Appl Mech 1999;66:570. [24] Ju JW, Sun LZ. Int J Solids Struct 2001;38:183. [25] Ju JW, Chen TM. Acta Mech 1994;103:103.
Multi-scale modeling of elastoplastic deformation and strengthening mechanisms in aluminum-based amorphous nanocomposites H.T. Liu a, L.Z. Sun b
b,*
a Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095-1593, USA Department of Civil and Environmental Engineering and Center for Computer-Aided Design, The University of Iowa, Iowa City, IA 52242-1527, USA
Received 20 January 2005; received in revised form 21 February 2005; accepted 22 February 2005 Available online 2 April 2005
Abstract Aluminum-based amorphous nanocomposites exhibit exceptional strength while preserving reasonable ductility. Although it is believed that the enhanced mechanical properties are due to the existence of defect-free nanometer-scale face-centered cubic a-Al nanoparticles in the amorphous aluminum matrix, the underlying strengthening mechanism remains to be determined so that a fundamental understanding of the material nanostructure–property relationship can be obtained. In this paper we focus on theoretical exploration of the mechanical constitutive behavior of amorphous nanocomposites in terms of a multi-scale approach starting from the nanostructure. A local heterogeneous stress field and deformation are formulated based on the concept of eigenstrain and equivalent inclusion method. An overall elastoplastic constitutive model for amorphous nanocomposites is developed through homogenization averaging procedures. Explicit expressions of the effective elastic stiffness and yield strength of amorphous nanocomposites in terms of the constituent properties and nanostructures are obtained. An experimentally observed interlayer phase between nanoparticles and the amorphous matrix is incorporated in the current model. The interlayer thickness is treated as a characteristic length scale. The nanoparticle size effect on material properties is specifically investigated within the continuum mechanics framework. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocomposites; Micromechanical modeling; Plastic deformation; Yield phenomena; Mean field analysis; Particle size effect
1. Introduction Aluminum-based amorphous nanocomposites were first developed in the early 1990s by Kim et al. in Japan [1] and Chen et al. in the United States [2] using a partial crystallization process in which the cooling rate in a certain temperature range for amorphous alloys was stringently controlled. The dispersion of nanoparticles in the
*
Corresponding author. Tel.: +1 319 384 0830; fax: +1 319 335 5660. E-mail addresses: [email protected], [email protected] (L.Z. Sun).
amorphous matrix leads to a remarkable increase of mechanical strength while preserving reasonable ductility [3]. It has been reported that the yield strength of the amorphous aluminum alloys is as high as 800 MPa in an amorphous state, and can be increased to 1500 MPa by partial crystallization [4–6]. Gogebakan [7] observed that the crystallized structure of Al–Y–Ni amorphous alloys with a nanoparticle size of 5–25 nm increased material strength to about 40% greater than corresponding single-phase amorphous alloys. This extraordinarily high mechanical strength makes Al-based amorphous nanocomposites a promising material for future industrial applications.
1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.02.029
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The typical composition of Al-based amorphous nanocomposites is Al–TM–RE, where TM is a transition metal such as Ni, Fe, Co, Cr; and RE is a rare earth element such as Y, La, Ce, Nd [8]. The microstructure of amorphous nanocomposites constitutes nanometer-scale face-centered cubic (fcc) a-Al particles dispersed in the amorphous aluminum matrix. The nanoparticle size and inter-particle distance are in the ranges of 5– 50 nm and 7–100 nm, respectively [3,8]. The total particle volume fraction is preferably in the range of 10–30% to preserve ductility. Further experiment showed that a higher annealing temperature and longer annealing time resulted in larger crystallized particle size and total particle volume fraction [9]. High-resolution electron microscopy examinations revealed that fcc a-Al nanoparticles exhibit a nearly spherical or an ellipsoidal morphology with no internal defects observed inside the nanoparticles [3]. Atomic probe field ion microscopy results of partially crystallized Al87Ni10Ce3 showed that Ce atoms are enriched within a distance of less than 3 nm at fcc a-Al particle interfaces [8]. This interesting observation suggests that a rare earth enriched interlayer phase exists between Al nanoparticles and the amorphous matrix. Many experimental results have demonstrated a remarkable increase in mechanical properties of Al-based amorphous nanocomposites. Inoue [3] measured the hardness and YoungÕs modulus of Al88Ni9Ce2Fe1 nanocomposites. The results showed that the YoungÕs modulus and hardness increased monotonously proportional to the particle volume fraction, achieving values of about 73 GPa and 420 Hv, respectively, at the volume fraction of 30%. Choi et al. [11] conducted experiments on Al88Ni10Nd2 nanocomposites where the results exhibited an almost linear increase in micro-Vickers hardness from 220 for amorphous single phase alloys to 400 for amorphous nanocomposites with particle volume fraction of 32%. Gogebakan [7] also tested Al85Y10Ni5 amorphous nanocomposites, resulting in a dramatic increase in the composite hardness. Studies of the amorphous nanocomposite strengthening mechanisms have been explored. Inoue [3] indicated that the Al nanoparticles were composed of an fcc structure without internal defects. The production of defectfree nanoparticles is due to the fact that the Al-based amorphous matrix has a high tendency for the precipitation of Al nanoparticles to relax. The absence of defects in Al nanoparticles greatly increases the critical resolved stress; and thus the yield strength determined by slips will be dramatically increased. In addition to defect-free Al nanoparticles, Zhong et al. [10] observed that the rare earth components amassed around the Al nanoparticles to form an interlayer. Hono et al. [8] suggested the occurrence of this phenomenon may be due to the slow diffusivity of rare earth atoms. During the crystallization
process, both TM and RE atoms are rejected from the a-Al phase. The atomic radius of the RE is usually much larger than the other solution elements, which may then result in an RE diffusivity that is slower than that of either Al or TM by orders of magnitude. Hence, during the growth of a-Al nanoparticles, the rejected RE atoms are enriched at the interface and form a heterogeneous interlayer. Although the extraordinarily high strength of nanocomposites has been experimentally observed and qualitatively explained by the existence of nanoparticles and nanostructures, the underlying strengthening mechanisms remain to be quantitatively revealed. Kim et al. [12] proposed a phenomenological composite model that applied the mixture rule based on the volume fraction of each constituent to estimate the hardness of Al–Ni–Y nanocomposites. Kim and Hong [13] further proposed a three-phase composite model comprising Al particles, a rare earth enriched interlayer, and an amorphous aluminum matrix. While these models provide a straightforward means to estimate the mechanical properties of the nanocomposites, no detailed local stress field and plastic deformation were considered and the local matrix–particle interaction was neglected. In the current study, theoretical exploration is applied to the nanostructure of nanocomposites. A multiscale approach that satisfies both equilibrium and compatibility is employed to model the overall elastoplastic behavior of amorphous nanocomposites. The nanocomposite is modeled as a heterogeneous material containing three distinguishing phases: defect-free fcc a-Al nanoparticles of spherical or spheroidal shape with a radius of 5–50 nm, an amorphous aluminum alloy, and a rare earth enriched interlayer surrounding the nanoparticles. At the nanometer-scale, the equivalent inclusion method [14] and the concept of eigenstrain [15] are adopted to analyze the heterogeneous stress field and local deformation. The overall mechanical properties of amorphous nanocomposites in terms of elastic stiffness and yield strength are derived via a homogenization averaging process. It is noted that the mean-field principles (homogenization procedures) can be directly applied to estimate the effective yield strength of composites since the initial yielding and plastic hardening of composites should be attributed to the collective responses of particle–matrix interactions [15–17]. However, homogenization theory may be no longer valid when the material softening and fracture occur due to the localized damage mechanisms. It is also noted that the particle size has a significant effect on the macroscopic mechanical properties for composites. Conventional continuum theory fails to incorporate the size effect given the absence of internal material length. Huang and co-workers [18,19] and Kouzeli and Mortensen [20] investigated the particle
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
size effect in alloys and composites via the theory of mechanism-based strain gradient plasticity [21]. In the current study, the interlayer thickness is treated as a characteristic length scale and the particle size effect on the overall mechanical properties of amorphous nanocomposites is investigated.
2. Heterogeneous stress field To characterize the typical microstructure of nanocomposites, a three-phase representative volume element is considered (Fig. 1), comprising spheroidal shaped Al nanoparticle domain X, rare earth enriched interlayer domain C, and amorphous matrix domain R. Furthermore, let R = X + C denote the generalized inclusion domain that contains both the particle and its corresponding interlayer. When subject to a far-field external loading r0, the heterogeneous local stress field can be expressed as 8 0 0 > < C : ½e0 þ e ðxÞ; x 2 R; rðxÞ ¼ CX : ½e0 þ e0 ðxÞ; x 2 X; ð1Þ > : C 0 C : ½e0 þ e ðxÞ; x 2 C;
2695
where the double dot symbol ‘‘:’’ indicates tensor contractions between a fourth-rank tensor and a secondrank tensor, e0 is the far-field strain corresponding to the far-field stress r0 with r0 = C0: e0, and C0 is the elastic stiffness tensor of the amorphous matrix. e 0 (x) represents the disturbance strain due to the heterogeneities. CX and CC denote the elastic stiffness tensors of nanoparticles and the interlayer, respectively. For isotropic materials, these fourth-rank elastic stiffness tensors can be expressed as C gijkl ¼ kg dij dkl þ lg ðdik djl þ dil djk Þ;
g ¼ 0; X; C;
ð2Þ
where kg and lg are LameÕs constants for the corresponding phases. Eshelby [14] and Mura [15] proposed the concept of eigenstrain (non-stress strain) and the equivalent inclusion method, in which a homogeneous material with equivalent eigenstrains in corresponding domains is used as a substitute for the heterogeneous material. For the problem described in Eq. (1), eigenstrains e*(x) are assumed to be inside the domains X and C, whereupon the stress field can be rewritten as 8 0 0 > x 2 R; < C : ½e0 þ e ðxÞ; 0 0 rðxÞ ¼ C : ½e0 þ e ðxÞ þ e ðxÞ; x 2 X; ð3Þ > : 0 0 C : ½e0 þ e ðxÞ þ e ðxÞ; x 2 C: Generally speaking, eigenstrains are a function of the material properties of the constituents, nanostructures, particle shape, and far-field loading conditions. Eshelby [14] showed that, for spheroidal particles, the eigenstrain is constant and can be expressed explicitly. However, for the current problem the eigenstrain is not constant due to the influence of existing interlayer. To simplify the derivation processes, Hori and NematNasserÕs double-inclusion method [22] is employed, in which volume averaging processes are conducted over the particle domain X and the interlayer domain C, respectively. Thus, the volume-averaged eigenstrains can be expressed as ( eX ¼ UX : e0 for particle domain X; e ¼ ð4Þ eC ¼ UC : e0 for interlayer domain C:
Fig. 1. (a) Schematic representation of the microstructure of Al-based amorphous nanocomposites: spheroidal Al nanoparticle domain X and rare-earth element enriched interlayer domain C embedded in the amorphous matrix domain R; (b) sketch of a spheroid.
Here, eX and eC denote the averaged eigenstrains inside domain X and C, respectively. The fourth-rank tensor UX and UC can be expressed explicitly for spheroidal particles as " 1 f X DS þ AC U ¼ SX þ AX þ DS SX 1f # 1 f DS þ AX ð5Þ SX 1f and
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H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
"
U ¼ DS þ ðSX þ AX Þ SX C
SX
f DS þ AC 1f
f DS þ AX 1f
1
#1 ;
ð6Þ
respectively. In Eqs. (5) and (6), f denotes the volume fraction of the particle domain X over the generalized entire inclusion domain R as f = X/R, and AX and AC are the mismatch material property tensors for domain X and C, which can expressed as AX ¼ ðCX C0 Þ1 C0 ;
AC ¼ ðCC C0 Þ1 C0 :
ð7Þ
Moreover, DS is defined as DS = SR SX with SX and SR being EshelbyÕs tensors for particle domain X and the entire inclusion domain R, respectively. It is noted that EshelbyÕs tensors relate to only the inclusion shape for spheroidal particles. When the nanoparticle shape is approximately spherical or the thickness of the interlayer is much less than the radius of the nanoparticle, the particle domain X and the entire inclusion domain R have similar shapes. Thus it can be assumed that SR = SX = S and subsequently DS = 0. Correspondingly, Eqs. (5) and (6) can be rewritten as 1
ð8Þ
1
ð9Þ
UX ¼ ðS þ AX Þ and
UC ¼ ðS þ AC Þ ;
respectively. When an equivalent inclusion is used to replace the nanoparticle and the corresponding interlayer, the equivalent volume-averaged eigenstrain for the entire inclusion can be expressed as e ¼ U R : e0 ;
ð10Þ
where UR ¼ f UX þ ð1 f ÞUC h i ¼ f ðS þ AX Þ1 þ ð1 f ÞðS þ AC Þ1 :
Local yielding and plastic flow are dependent on local stress field. For simplicity, the commonly used von Mises yield criterion with an isotropic hardening law is assumed for the matrix material as an illustration. Namely, the local yield function reads pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ðr; ep Þ ¼ r : Id : r Kðep Þ 6 0; ð14Þ where ep and K(ep) are the equivalent plastic strain and the isotropic hardening function of the matrix-only material, respectively. Moreover, Id denotes the deviatoric part of the fourth-rank identity tensor I. Following Ju and Sun [24], we denote by H(x j g) = r(x j g) : Id : r(x j g) the square of the ‘‘current stress norm’’ at a local point x for a given inclusion configuration (assembly) g. Furthermore, ÆHæm(x) is defined as the ensemble average of H(x j g) over all possible realizations for a matrix point x, which indicates I hH im ðxÞ ¼ H 0 þ ½H ðx j gÞ H 0 P ðgÞ dg; ð15Þ g
where P(g) is the probability density function for determining an inclusion for a given configuration g, and H0 = r0 : Id : r0 is the square of the far-field stress norm applied on the composite. To calculate ÆHæm(x), Eq. (15) can be rewritten as I hH im ðxÞ ¼ H 0 þ ½H ðx j x0 Þ H 0 P ðx0 Þ dx0 ; ð16Þ x0 62NðxÞ
ð11Þ
The disturbance strain at a position x inside the matrix due to the eigenstrain e in a spheroidal inclusion centered at x 0 can be calculated as e0 ðxÞ ¼ Gðx x0 Þ : e :
a-Al nanoparticles contain no dislocations or other imperfections [12]. The interactions among nanoparticles, solute-enriched interlayers, and amorphous matrix are expected to have a significant effect on the strengthening phenomenon in the nanocomposites. Accordingly, at any matrix material point x, the local stress r(x) can be calculated using Eqs. (1) and (12) as ð13Þ rðxÞ ¼ r0 þ C0 : Gðx x0 Þ : e :
ð12Þ
Here, Gðx x0 Þ is the exterior-point EshelbyÕs tensor, which is given explicitly for spheroidal inclusions by Ju and Sun [23]. 3. Ensemble-average procedures The overall plastic behavior of Al-based amorphous nanocomposites is attributed to the plastic deformation in the amorphous matrix since it is observed that the fcc
where N(x) is the exclusion zone of x for the center location x 0 of an inclusion in the probability space, which is identical to the shape and size of a spheroidal inclusion. Since nanoparticles are uniformly distributed in the amorphous matrix, P(x 0 ) can be simplified as N/V where N is the total number of nanoparticles uniformly dispersed in entire volume V. After a series of lengthy but straightforward derivations, the ensemble-averaged ÆHæm can be evaluated as hH im ¼ r0 : T : r0 ;
ð17Þ
where the components of the fourth-rank tensor T take the form ð1Þ
ð2Þ
T ijkl ¼ T IK dij dkl þ T IJ ðdik djl þ dil djk Þ with
ð18Þ
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
ð1Þ
T IK
ð2Þ
T IJ
3 9 82 3ð35m20 70m0 þ 36ÞDIK > > > > > > > > 6 þ7ð50m2 59m þ 8ÞðD þ D Þ 7 ð1f Þ f > > þ 4 5 > > 0 I K ð1Þ ð1Þ ð2Þ ð2Þ 0 > > B B B B > > II KK II KK > > 2 > > 2ð175m 343m þ 103Þ > > 0 0 > > > > 0 1 > > ð1Þ ð1Þ > > f ðC þC Þ > > II KK > > > > ð1Þ ð1Þ > > B BII BKK C > > > > B C > > þ21ð25m 2Þð1 2m Þ 0 0 > > @ ð1f ÞðCð2Þ þCð2Þ Þ A = < R II KK 1 2/ þ ð2Þ ð2Þ ; ¼ þ B B > 3 4725ð1 m0 Þ2 > 0 ð1Þ II KKð1Þ 1 > > > > > > f ðCII DK þCKK DI Þ > > > > ð1Þ ð1Þ > > B C > > B B > > II KK B C > > þ21ð25m 23Þð1 2m Þ > > 0 0 ð2Þ ð2Þ @ ð1f ÞðC DK þC DI Þ A > > > > II KK > > þ > > ð2Þ ð2Þ > > BII BKK > > > > > > > > ð1Þ ð1Þ ð2Þ ð2Þ > > > > f C C ð1f ÞC C 2 > > II KK II KK ; : þ1575ð1 2m0 Þ ð1Þ ð1Þ þ ð2Þ ð2Þ BII BKK BII BKK 2 3 ! ð70m20 140m0 þ 72ÞDIJ R 1 / f ð1 f Þ 6 7 JÞ ¼ þ þ ð2Þ ð2Þ 4 ð175m20 266m0 þ 75Þ ðDI þD 5: ð1Þ ð1Þ 2 2 1575ð1 m0 Þ2 BIJ BIJ BIJ BIJ 2 þð350m0 476m0 þ 164Þ
Here, m0 is PoissonÕs ratio of the amorphous matrix, and /R is the volume fraction of the whole inclusions (including nanoparticles and their corresponding interlayers). Other parameters in the above equations are given in Appendix A. In Eq. (17), ÆHæm is described in terms of the far-field stress r0. Alternatively, ÆHæm can be expressed in terms of the macroscopic (ensemble-volume averaged) stress r. The relationship between the far-field stress r0 and macroscopic stress r is given by (cf. Ref. [25]) r0 ¼ P : r; where the fourth-rank tensor P reads 1 P ¼ I þ /R ðS IÞ UR :
ð20Þ
ð21Þ
Combining Eqs. (17) and (20) then leads to an alternative expression of the ensemble-averaged square of the current stress norm expressed as hH im ¼ r : T : r
ð22Þ
with T ¼ P T P. It is observed from Eq. (22) that ÆHæm can be reduced to the form derived by Ju and Sun [24] if no interlayer is present around the nanoparticles.
4. Nanocomposite constitutive modeling Based on linear elastoplasticity principle, the total macroscopic (effective) strain tensor e consists of elastic part ee and plastic part ep . The effective elastic strain ee is related to the effective stress r via r ¼ C : ee where the effective elastic tensor C of nanocomposites can be derived with the help of general governing equations of composite materials [25]. Specifically, for nanocompos-
2697
ð19Þ
ites containing randomly distributed yet aligned spheroidal particles with interlayers, the three governing equations can be obtained as r ¼ C0 : ðee /R e Þ; ee ¼ e0 þ /R S : e ;
ð23Þ
R
e ¼ U : e0 : Therefore, the explicit expression of the effective elastic tensor C of the nanocomposites can be shown as h i 1 C ¼ C0 I þ /R UR ð/R S UR IÞ : ð24Þ Upon further loading, the nanocomposites may yield and become plastic. The ensemble-volume-averaged yield function F of the nanocomposites can be characterized directly from Eq. (14): qffiffiffiffiffiffiffiffiffiffiffi F ¼ ð1 /R Þ hH im Kðep Þ 6 0: ð25Þ It is noted that, for isotropic plastic hardening, Kðep Þ can be simplified as rffiffiffi 2 q p Kðe Þ ¼ ry þ hðep Þ ; ð26Þ 3 where ry, h and q, and ep denote the yield strength of the matrix, the linear and exponential isotropic hardening parameters of the matrix, and the effective equivalent plastic strain of the nanocomposites, respectively. It is noted that Eq. (25) represents the pressure-dependent nanocomposite yield function although the matrix is assumed to be pressure-independent von Mises type yield criterion. Based on the derived effective yield function, the p effective plastic strain rate e_ can be calculated from the associative plastic flow rule:
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
_ ¼ 0; kF
k_ F_ ¼ 0:
ð28Þ
Therefore, the effective elastoplastic constitutive model of nanocomposites is developed with the formulation given in Eqs. (24)–(27), capable of estimating the overall elastoplastic stress–strain responses of nanocomposites under general three-dimensional loading conditions.
amorphous alloy
F 6 0;
1.30
1.25
1.20
/E
k_ P 0;
1.35
composite
oF p ; ð27Þ e_ ¼ k_ or where k_ is the plastic consistency parameter which can be obtained from the following Kuhn–Tucker condition:
1.15
E
2698
1.10
α = 2.0 α = 1.5 α = 1.0
1.05
1.00 0.00
0.05
0.15
0.20
0.25
volume faction φ
Fig. 2. Overall YoungÕs modulus of the nanocomposites vs. particle volume fraction for various particle shapes.
duction of the interlayer thickness as a characteristic length scale, the particle size effect can be incorporated in the current model. It is noted that, in the present study, the particle-size effect is equivalent to the effect of particle interface-to-volume ratio (IVR), since IVR is directly related with the radius of particle a and thickness of interlayer t as IVR = 3/(a + t). In Fig. 4, the overall yield strength of the nanocomposites is presented for a large range of particle sizes, from nanometer-scale to micrometer-scale. For a fixed interlayer thickness, a clear particle size effect (or IVR effect) is observed with the particle radius ranging in the nanometer scale. When the particle size corresponds to the same length scale as the interlayer thickness, a significant increase of yield strength is exhibited, proving that the existence of an interlayer significantly affects the overall mechanical properties
1.60 1.55
composite
/σy
amorphous alloy
1.50
σy
Specific Al–Ni–Y nanocomposites are applied as the model material in the following simulation. All phases are assumed to be isotropic. The amorphous Al matrix is assumed to have a YoungÕs modulus of 70 GPa and a PoissonÕs ratio of 0.3. The yield strength of the matrix is taken as 1.29 GPa [7], and the plastic hardening parameters are h = 1.0 GPa and q = 0.3, respectively. The Al a-fcc nanoparticles have a YoungÕs modulus of 71 GPa and a PoissonÕs ratio of 0.31. Since there are no mechanical data available for the rare-earth-element enriched interlayer, it is assumed that the intermetallic compound interlayer phase has a YoungÕs modulus of 500 GPa and a PoissonÕs ratio of 0.25. Due to the much higher strength of the defect-free Al nanoparticles, plastic deformation is constrained in the amorphous matrix. Unless explicitly stated otherwise, the radius of nanoparticles is assumed to be 10 nm and the thickness of interlayers to be 3 nm. Uniaxial loading simulation is conducted to investigate the elastoplastic behavior of the nanocomposites. Fig. 2 shows that the overall YoungÕs modulus increases proportional to the particle volume fraction. In our model, the spheroidal nanoparticles are aligned in the loading direction, and therefore nanocomposites with a larger particle aspect ratio exhibit a higher overall stiffness than nanocomposites with smaller particle aspect ratio. For a nanocomposite with 20% particle volume fraction, the overall YoungÕs modulus in the particle-aligned direction can be approximately 1.3 times higher than that of amorphous-alloy counterpart. The nanocomposites also demonstrate the strong strengthening effect as shown in Fig. 3. The nanocomposite having a 25% particle volume fraction indicates a yield-strength 1.5 times higher than that of amorphous alloys. It is evident that a larger aspect ratio of particles leads to a higher overall yield strength for the nanocomposites. Experimental investigations have shown that the particle size has a significant effect on the overall mechanical properties of composites. With the intro-
0.10
Σ
5. Numerical simulation and discussion
1.45 1.40 1.35 1.30 1.25 1.20 1.15
α = 2.0 α = 1.5 α = 1.0
1.10 1.05 1.00 0.00
0.05
0.10
0.15
0.20
0.25
Σ
volume faction φ
Fig. 3. Overall yield strength of the nanocomposites vs. particle volume fraction for various particle shapes.
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
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1.40
1.4
Σ
α = 1.0
1.2
_ amorphous alloy σe/σy
1.35
1.30
σy
composite
/σy
amorphous alloy
φ = 20%
1.0 0.8 0.6 0.4
1.25
Σ
φ = 5%
tinterlayer = 0 nm tinterlayer = 3 nm tinterlayer = 10 nm
Σ
φ = 10%
0.2
Σ
φ = 20%
1.20
0.0 10
100
1000
10000
100000
0
Log (radius of particles) (nm)
of the nanocomposites. Since the interlayer between the particles and the matrix is usually measured in nanometers, the nanoparticle-reinforced composites demonstrate higher yield strength than microparticlereinforced composites. However, when the interlayer is non-existent (i.e., the interlayer thickness is zero), the overall yield strength does not change for the entire particle size range, implying that for a fixed volume fraction, the particle size does not affect the overall yield strength. This assertion so far has not been validated with experimental observations since Al-based amorphous nanocomposites always exhibit a solute-enriched interlayer between the nanoparticle and the matrix [8,10]. The pure IVR effect without interlayer involved should be further investigated from both experimental and modeling points of view. Onedimensional strain–stress curves for uniaxial loading tests are presented in Fig. 5 for the nanocomposites
1.8 Σ
φ = 20%
amorphous alloy
/σy
composite
σ
1.4 1.2 1.0 0.8 0.6
rparticle = 5 nm rparticle = 10 nm rparticle = 20 nm rparticle = 50 nm
0.4 0.2
0.5
1.0
1.5
2.0
2.5
_
6
8
10
12
amorphous alloy
3.0
3.5
Fig. 6. Overall initial yield surfaces of the nanocomposites under axisymmetric loading.
with spherical particles. For the same volume fraction of 20%, the nanocomposites with small particles show higher stiffening and strengthening effects than those with large particles. To investigate the strengthening effect of the nanocomposites under complex loading condition, the nanocomposite yield-surface is certainly of interest. Axisymmetric loading cases are specified here to study multi-axial strengthening effect. With the assumption of overall stresses as r11 > 0; r22 ¼ r33 > 0; and r12 ¼ r13 ¼ r23 ¼ 0, the initial yield surfaces are demarcated in the volumetric and deviatoric stress space in Fig. 6. The nanocomposite volumetric-stress rv and deviatoric-stress re can be obtained from their definitions with considering the axisymmetric property as rv ¼ ðr11 þ 2r22 Þ=3 and re ¼ r11 r22 , respectively. It is shown from Fig. 6 that the yielding response of the nanocomposites is not of von Mises type, even when the nanoparticles are spherical in shape and randomly distributed. A decrease in volume fraction of nanoparticles leads to an increase of the volumetric yield stress and a simultaneous decrease of the effective yield stress. This trend indicates that the pressure-dependence of nanocomposite yielding primarily resulted from the existence of nanoparticles. When the volume fraction of particles vanishes, the matrixonly material will be restored von Mises yielding, which is consistent with the assumption of the matrix material satisfying the von Mises J2-yield criterion.
6. Conclusions
0.0 0.0
4
σv/σy
Fig. 4. Particle size effect on the overall yield strength of the nanocomposites.
1.6
2
4.0
overall strain (%)
Fig. 5. Overall uniaxial stress–strain curves of the nanocomposites.
A multi-scale elastoplastic framework has been proposed to investigate the overall mechanical properties of amorphous nanocomposites with randomly dispersed, unidirectionally aligned spheroidal particles.
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H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
At the nanometer-scale, the nanostructure is modeled as spheroidal particles embedded in the amorphous matrix where an interlayer between the particle and the matrix exists. The locally heterogeneous stress field is obtained from the equivalent inclusion method and the concept of eigenstrain. The von Mises yield criterion and isotropic hardening rule are assumed to control the local yielding and plastic deformation in the matrix. Ensemble and volume average procedures have been conducted in a microscopic representative volume element to formulate the overall (effective) elastic stiffness and yield function for nanocomposites. The interlayer between the nanoparticle and the matrix significant affects the overall mechanical properties of nanocomposites. The interlayer thickness is treated as a characteristic length scale, thus the particle size is incorporated into the current model. The particle size effect has been investigated for nanocomposites within the continuum mechanics framework.
D11 ¼
5½2 þ a4 3a4 f ða2 Þ 2ð1 a4 Þ
2
D12 ¼ D21 ¼ D13 ¼ D31 ¼
;
15a4 ½3 þ ð1 þ 2a4 Þf ða2 Þ 4ð1 a4 Þ
2
;
D22 ¼ D23 ¼ D32 ¼ D33 ¼ 18ð15 3D11 4D12 Þ; ðA:1Þ where a = a1/a2 is the aspect ratio of the inclusion [see Fig. 1(b)]. Furthermore,
f ðaÞ ¼
8 <
1 a cos pffiffiffiffiffiffiffi ; a 1a2
a < 1;
a : cosh pffiffiffiffiffiffiffi ; a > 1; a a2 1 1
ðgÞ
ðgÞ
BIJ ¼ 2ðV IJ þ N IJ Þ;
g ¼ 1; 2;
ðA:2Þ
ðA:3Þ
9 2 9 8 31 8 ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ > > > U 21 þ M 21 U 31 þ M 31 U I1 þ M I1 > > > > = 6 U 11 þ 2V 11 þ M 11 þ 2N 11 = < CI1 > < 7 ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ 7 ¼6 CI2 U þ M U þ 2V þ M þ 2N U þ M U þ M 12 22 22 32 5 > I2 12 22 22 32 I2 > > 4 > > > ; ; : ðgÞ > : ðgÞ > ðgÞ ðgÞ ðgÞ ðgÞ CI3 U I3 þ M I3 U 13 þ M 13 U 23 þ M 23 U 33 þ 2V 33 þ M 33 þ 2N 33 ðI ¼ 1; 2; 3Þ ðA:4Þ Under the uniaxial loading condition, the effects of particle shape and particle size on the overall elastic stiffness and yield strength of nanocomposites have been numerically simulated. One-dimensional elastoplastic strain–stress curves have been presented for various particle sizes. The overall yield surfaces are demarcated for nanocomposites under axisymmetric loading to demonstrate the multi-axial strengthening effect and pressuredependent yielding response due to the existence of nanoparticles. The proposed model satisfies the continuum mechanics rules and provides a feasible means of estimating the mechanical response of nanocomposites.
with U 11 ¼ 4m0 þ U 12 ¼ U 13 U 21 ¼ U 31
2 4 ; hðaÞ þ 4m0 þ 2 2 a 1 3ða 1Þ 2a2 þ 1 2a2 ; ¼ 4m0 2 hðaÞ þ 4m0 2 a 1 a 1 2a2 þ 1 2a2 ; ¼ 2m0 2 hðaÞ 2 a 1 a 1
U 22 ¼ U 23 ¼ U 32 ¼ U 33 4a2 1 a2 ; ¼ 2m0 þ hðaÞ þ 2 2 4ða 1Þ 2ða 1Þ ðA:5Þ
Acknowledgment This research is sponsored by the National Science Foundation (NSF) under grant number CMS-0303955. The support of the NSF is gratefully acknowledged.
Parameters in Eq. (19) are expressed as 3½1 a4 f ða2 Þ ; 1 a4
V 11
4a2 2 12a2 8 ; ¼ 4m0 þ 2 hðaÞ 4m0 þ a 1 3ða2 1Þ
V 12 ¼ V 21 ¼ V 13 ¼ V 31 a2 þ 2 2 ; ¼ m0 2 hðaÞ 2m0 2 a 1 a 1
Appendix A
D1 ¼
D2 ¼ D3 ¼ 12ð3 D1 Þ;
V 22 ¼ V 23 ¼ V 32 ¼ V 33 4a2 7 a2 ; ¼ 2m0 hðaÞ þ 4ða2 1Þ 2ða2 1Þ
ðA:6Þ
H.T. Liu, L.Z. Sun / Acta Materialia 53 (2005) 2693–2701
kð0Þ lð1Þ kð1Þ lð0Þ
ð1Þ
M IJ ¼ ð1Þ
N IJ
ðlð1Þ lð0Þ Þ½2ðlð1Þ lð0Þ Þ þ 3ðkð1Þ kð0Þ Þ lð0Þ ; ¼ ð1Þ 2ðl lð0Þ Þ
ð2Þ
M IJ ¼
kð0Þ ð1 DIK dkk Þ 2lð0Þ DIJ ð2Þ
2ðlII lð0Þ Þ lð0Þ
ð2Þ
N IJ ¼
ð2Þ 2ðlIJ
lð0Þ Þ
[3] [4] [5] [6] [7] [8]
;
;
[9] [10]
;
[11]
ðA:7Þ and hðaÞ ¼
8 <
a ð1a2 Þ3=2
½að1 a Þ
:
a ða2 1Þ3=2
½cosh1 a aða2 1Þ
2 1=2
1
cos a; 1=2
[13] [14] [15]
a < 1; ;
[12]
a > 1;
[16]
ðA:8Þ 9 2 ð2Þ 8 ð2Þ ð0Þ ð0Þ > = 6 k11 k þ 2ðl11 l Þ < DI1 > ð2Þ DI2 ¼ 6 k12 kð0Þ 4 > > ; : ð2Þ DI3 k kð0Þ 13
Inoue A. Prog Mater Sci 1998;43:365. Kim YH, Inoue A, Masumoto T. Mater Trans 1991;32:599. Inoue A, Kim YH, Masumoto T. Mater Trans 1992;33:487. He Y, Poon SJ, Shiflet GJ. Science 1988;241:1640. Gogebakan M. J Light Metals 2002;2:271. Hono K, Zhang Y, Inoue A, Sakurai T. Mater Sci Eng 1997;A226:498. Jiang XY, Zhong ZC, Greer AL. Mater Sci Eng 1997;A226:789. Zhong ZC, Jiang XY, Greer AL. Mater Sci Eng 1997;A226:531. Choi GS, Kim YH, Cho HK, Inoue A, Masumoto T. Scripta Metall Mater 1995;33:1301. Kim HS, Warren PJ, Cantor B, Lee HR. Nanostruct Mater 1999;11:241. Kim HS, Hong SL. Acta Mater 1999;47:2059. Eshelby JD. Proc Roy Soc London 1957;A241:376. Mura T. Micromechanics of defects in solids. 2nd ed.. Kluwer Academic Publishers; 1987. Liu Y, Gilormini P, Ponte Castaneda P. Acta Mater 2003;51:5425.
ð2Þ
k12 kð0Þ ð2Þ
ð2Þ
k22 kð0Þ þ 2ðl22 lð0Þ Þ ð2Þ
k23 kð0Þ
2701
9 31 8 ð2Þ > kI1 kð0Þ > > > = < 7 ð2Þ ð2Þ ð0Þ ð0Þ 7 k23 k k k 5 > I2 > > > ; : ð2Þ ð2Þ ð2Þ kI3 kð0Þ k33 kð0Þ þ 2ðl33 lð0Þ Þ ð2Þ
k13 kð0Þ
ðI ¼ 1; 2; 3Þ: ðA:9Þ In the above equations, the superscripts of LameÕs constants indicate different materials: ‘‘0’’ as the matrix, ‘‘1’’ as the nanoparticle, and ‘‘2’’ as the interlayer. References [1] Kim YH, Inoue A, Masumoto T. Mater Trans 1990;31:747. [2] Chen H, He Y, Shiflet GJ, Poon SJ. Scripta Metall Mater 1991;25:1421.
[17] Jiang B, Weng GJ. Metall Mater Trans A 2003;34A:765. [18] Xue Z, Huang Y, Li M. Acta Mater 2002;50:149. [19] Huang Y, Qu S, Hwang KC, Li M, Gao H. Int J Plast 2004;20:753. [20] Kouzeli M, Mortensen A. Acta Mater 2002;50:39. [21] Gao H, Huang Y, Nix WD, Hutchinson JW. J Mech Phys Solids 1999;47:1239. [22] Hori M, Nemat-Nasser S. Mech Mater 1993;14:189. [23] Ju JW, Sun LZ. J Appl Mech 1999;66:570. [24] Ju JW, Sun LZ. Int J Solids Struct 2001;38:183. [25] Ju JW, Chen TM. Acta Mech 1994;103:103.