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Constitutive modeling for nanocrystalline metals based on cooperative grain boundary mechanisms
Journal of the Mechanics and Physics of Solids 52 (2004) 1151 – 1173
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Constitutive modeling for nanocrystalline metals based on cooperative grain boundary mechanisms Hong-Tao Wang, Wei Yang∗ Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Received 1 November 2002; accepted 27 August 2003
Abstract In nanocrystalline metals, the plastic deformation is accommodated primarily at the grain boundaries. Yang and Wang (J. Mech. Phys. Solids, 2003) suggested a deformation model based on clusters consisting of nine grains and incorporating both the Ashby-Verrall mechanism and a 30◦ rotation of closely linked pairs of grains. In the present article, the insertion and rotation processes are considered together as a cooperative deformation mechanisms, and the degree to which each process contributes is determined by the application of the principle of maximum plastic work. Plane strain and three-dimensional constitutive relations based on this concept are derived for which a general stress state drives the orientation evolution of various grain clusters under the Reuss assumption. Detailed calculation shows that the strain rate depends linearly on the stress, with the values of the coe7cients in this linear relationship dictated by the microscopic energy dissipation. The deformation contributed to the overall response by the grain boundary mechanism is discussed in the spirit of the Hashin-Shtrikman bounds. ? 2003 Elsevier Ltd. All rights reserved. Keywords: A. Creep; A. Grain-boundaries; A. Microstructure; B. Crystal plasticity; Nano-grained metals
1. Introduction Grain size plays a key role in deciding the dominant mechanisms of plastic deformation. In coarse-grained metals, the movements of dislocations within individual grains carry out the plastic =ow. The grain boundaries hinder the dislocation movements, cause pileups, and consequently make the material harder to deform. Within a much smaller grain size, say below 30 nm, both the theoretical analysis (Gryaznov et al., ∗
Corresponding author. Tel.: +86-10-6278-2642; fax: +86-10-6258-1824. E-mail address: [email protected] (W. Yang).
0022-5096/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2003.08.005
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1991) and the experimental evidences (Gao and Gleiter, 1987; Thomas et al., 1990; Milligan et al., 1993) showed that dislocations seldom appeared within the grains. Nevertheless, the dense network of grain boundaries facilitates the plastic deformation via certain accommodation mechanisms. In nanocrystalline metals, the available experimental data show that the grain shape remains equal-axed before and after the deformation (Lu et al., 2000; Valiev et al., 2002). This suggests that the nano-sized grains behave much more like stiF and equalaxed particles, squeezing past each other with little distortion in the grain cores. The grain boundaries serve not only as mass-transport networks but also as viscous deformable layers. Ashby and Verrall (1973) suggested a deformation mechanism carried out by fourgrain clusters, the inter-grain distance in the elongation direction increases, and that perpendicular to the elongation decreases. The mechanism is not self-complete in the sense that further elongation can only proceeds in a direction of ±30◦ from the original stretching direction. If the grains in closely linked pairs can rotate 30◦ , on the one hand, the grain cluster can achieve a strain of certain amount in the same direction as that caused by the insertion process. Furthermore, the rotation would restore the initial conHguration. Yang and Wang (2003) exploited this idea by suggesting a model based on clusters consisting of nine grains (Fig. 1) and incorporating both the insertion mechanism of Ashby and Verrall (1973) and a 30◦ rotation of closely linked grain pairs. Though formulated under uniaxial loading, the derived creep law agrees with the testing data of Cai et al. (1999). In the present article, the insertion and rotation processes are considered together as a cooperative deformation mechanism. We extend the capability of the constitutive modeling to the plane strain and three-dimensional cases, discussed in Section 2 and Section 3, respectively. Furthermore, the deformation contributed to the overall response by the viscous grain boundary layers, now composing of a Hnite volume fraction for nanocrystalline metals, is discussed in the spirit of the Hashin-Shtrikman bounds (Hashin, 1962; Hashin and Shtrikman, 1962a, b, 1963).
2. Two-dimensional constitutive relation 2.1. Assumptions and kinematics Attention is Hrst focused on the case of plane strain deformation. The polycrystal is composed of hexagon grains. Nearby hexagon grains may deform in a cooperative manner via cluster rearrangement such as that envisaged in Fig. 1. A grain may deform with one cluster at one time and join the deformation of a neighbor cluster at later time. DiFerent grains, regardless of their locations in the cluster, are assumed to be subjected to the same stress. The elastic deformation of the grain cores is neglected. Consequently, the deformation within the plane becomes incompressible. A local deformation rate tensor D and its global counterpart D˜ are introduced. The incompressibility dictates
H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173
(a) initial configuration
(c) completion of insertion stage
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(b) critical configuration for the insertion process
(d) rotation of 30 degrees
Fig. 1. Model of 9-grain cluster.
the following representation of the deformation rate tensor D in its principle-axes: d [D] = (d ¿ 0): (1) −d From the incompressibility, the constitutive response of the material is irrelevant of the hydrostatic stress. Two separate processes describe the kinematics of a grain cluster. The insertion process from the Ashby–Verrall’s mechanism, see Fig. 1 (a) – (c) where the grain cluster changes from an “armchair” conHguration to a “zigzag” conHguration. The inter-grain distance in the tensile direction elongates to disengage the grain pairs in that direction and the inter-grain distance perpendicular to the tensile direction contracts
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to engage the grains in the lateral direction. Slight amount of grain boundary sliding is involved. The rotation process involves the grain rotations of 30◦ on close-linked grain pairs, see Fig. 1 (c) and (d) where the armchair conHguration is restored. Large amount of grain boundary sliding and viscous =ow of interlayers are anticipated. The core of a nano-grain can adapt its shape to lower the energy barrier. Without the loss of generality, one may take the loading axis as the vertical direction. Consider the central seven grains in the 9-grain cluster. The conHgurations in Fig. 1(a) and (d) can accommodate an insertion process aligned with the loading axis. On the other hand, the conHguration in Fig. 1(c) gives rise to two possible deformation mechanisms: (1) an insertion process that is either plus or minus 30◦ with respect to the loading axis; or (2) a rotation process that elongates the cluster in the same direction as the loading axis. DeHne as the angle formed between the central seven grains in a cluster with the loading axis. From the six-fold symmetry of the conHguration, it is su7cient to investigate the cluster orientation angle within the range of −=6 to =6. All functions varying with the oriented angle are deHned as periodic functions of period =3. For the illustrative purpose, the cluster aligning with the loading axis is shown as solid line in Fig. 2, while the cluster forming angle is shown as dashed line. A distribution function p() is introduced to characterize the percentage of clusters in every possible orientation relative to the principal axes of the stress.
σ
θ
Fig. 2. Orientation of grain clusters.
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2.2. Insertion process In the insertion process, the plastic dissipation within a representative grain is composed of three parts: (1) dissipation by mass diFusion from sources to sinks; (2) dissipation by grain boundary sliding; and (3) dissipation by =uctuation in grain boundary area. These dissipations are labeled as W˙ diFusion ; W˙ slide and W˙ gb , respectively. The detailed calculations can be found in Yang and Wang (2003) where the following expressions were reported: W˙ diFusion = WidiF d2i ;
W˙ slide = Wis d2i ;
W˙ gb = Wigb di :
(2)
The subscript “i” labels the insertion process and superscripts “diF ”, “s”, “gb” abbreviate the diFusion, slide and grain boundary eFects, respectively. The expressions for WidiF ; Wis and Wigb can be found in Appendix A. The same dissipation calculations can be extended to the plane strain case. Only deviatoric stress is relevant with denoting the principal deviatoric stress. Accordingly, the work done by the applied load becomes : D = 2 di () ();
(3)
under the Reuss assumption that prescribes the same stress state to every grain, where ; and () = +k (4) () = cos 2 − ¡ 6 6 6 3 for any integer k. Equating the external work (3) to the internal dissipation (2), one obtains the following expression for the strain rate: di () =
2 () − Wigb : WidiF + Wis
(5)
The global deformation rate tensor is derived by the orientation average on the local deformation rate tensor in Eq. (1) under a weighting function p() for diFerent orientations as =6 cos 2 −sin 2 3 ˜ = [D] di () p() d (6) −=6 −sin 2 −cos 2 ˜ is deHned on coordinates aligned with the principal axes of the where the matrix [D] stress tensor. Fig. 2 shows the orientation change of a cluster. The arrows along the vertical line give the direction of the principal tensile stress. Fig. 3 illustrates a typical grain in the cluster. The dashed-hexagon and the solid one represent the conHgurations before and after the insertion process. When an insertion process is completed, the orientation of the cluster rotates 30◦ , namely it re-orients from to −=6 or +=6, as indicated in Fig. 3 from the dot–dashed axis to the dashed axis. Two solid lines in Fig. 3 standing for plus and minus 30◦ bound for all these axes. The periodicity of =3 renders the choices on − =6 or + =6 immaterial. We always choose the branch of − =6 hereafter. The conservation on the orientation implies p(; ˙ t)+ p(−=6; ˙ t)=0 with the
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σ α
θ
−α = θ −π/6 Fig. 3. Orientation change of a typical grain in the cluster.
following integral form ; p(; t) + p − ; t = p0 () + p0 − 6 6
(7)
under the initial conditionp(; 0) = p0 (). This implies the invariance of p(; t) + p( − =6; t) regardless of the stress variation during the period. If the material is initially “isotropic”, i.e. p0 () = 1, then one has p(; t) + p( − =6; t) ≡ 2. The distribution function p(; t) evolves according to 2 p(; ˙ t) = di − p − ; t − di ()p(; t) : (8) ln 3 6 6 Namely the insertion along the direction reduces while the insertion along the −=6 direction enhances the orientation distribution at . For an applied load that changes slowly with time, the strain rate di does not explicitly depend on time during the evolution. Then p(; t) has the following form: 1 p(; t) = p0 () + p0 − di − 6 6 di () + di − 6 −2=ln 3[di ()+di (−=6)]t p0 − e + di ()p0 () − di − : (9) 6 6 The Hrst term in Eq. (9) represents a steady state solution. The second term is transient in nature and decays exponentially. We henceforth consider only the steady state response after the transient dies out. If the material is prepared without texture, such as in Lu et al. (2000), the initial orientation distribution should be isotropic. Thus, the
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1.4
p(θ)
1.2
1.0
0.8
0.6
-π /4 -π /6 -π/12
0
θ
π/12
π/6
π/4
Fig. 4. Distribution of cluster orientation in the insertion process.
steady state distribution p() is reduced to 2di ( − =6) p() = : di () + di ( − =6)
(10)
The ratio Wigb = is small as argued by Ashby and Verrall (1973), one may omit the term associated with Wigb . Thus, the form of p() is reduced even further as 2 ( − =6) : (11) p() = () + ( − =6) As shown in Fig. 4, the distribution function p() reaches its minimum at = 0 and its maxima at = ±=6. This coincides with the physical model. From Eq. (5), the strain rate maximizes at = 0, and consequently the percentage of clusters whose orientations are near = 0 should decrease. The conservation of orientations stated in Eq. (7) dictates that the decreasing amount near the orientation of = 0 should be compensated by the increasing amount near = ±=6. Fig. 4 clearly re=ects that tendency for the steady state limit. Substituting Eq. (11) into Eq. (6), one arrives at the following plane strain constitutive relation √ 6(4 − ln 3) gb c c ˜ √ − i = Ci D; i = Wi √ : (12) : +3 3 √ The creep coe7cient Ci = 2(WidiF + Wis )=( + 3 3) is listed in Appendix A. The threshold stress ic is negligible for the insertion case. Two important observations emerge. First, if the material is initially isotropic, its macroscopic response, as delineated by Eq. (12), would also be isotropic regardless of the amount of deformation. The non-uniform responses in various clusters are erased by orientation average. Second, the global response would be non-hardening since all material parameters in Eq. (12), such as WidiF ; Wis and Wigb , are not aFected by the extent of deformation. This non-hardening behavior does occur for well-prepared (highly pure and fully densiHed) nanocrystalline metals under extremely large deformation,
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Lu et al. (2000). It is worthwhile to mention that the conclusions above neglect the possible occurrence of microcracks or microcavities under heterogeneous local stress Held. 2.3. Rotation process In a rotation process, the grain boundary energy is unchanged but the energy dissipation in ordering–disordering transformation near the triple junctions provide a considerable threshold stress. The energy dissipation in the ordering–disordering transformation is given by W˙ disorder = Wrdis dr , where the subscript “r” labels the association with a rotation process. Appendix A lists the detailed expression for Wrdis . Equivalence of the microscopic plastic dissipation and the macroscopic power delivers: dis 2 ( − =6) − Wr ¿ Wrdis if 2
− WrdiF + Wrs 6 = dr − (13) 6 0 otherwise: Similar analysis gives the evolution equation of p(; t) in the rotation process: 1 √ p(; t) : dr ()p − ; t − dr − p(; ˙ t) = 6 6 ln(2= 3)
(14)
The steady state solution is derived as p() =
2dr () dr () + dr ( − =6)
(15)
by assuming that the material is prepared isotropic. To facilitate the subsequent discussions, a dimensionless loading amplitude parameter ≡ =Wrdis is introduced. Given diFerent values, the curves of p() are plotted in Fig. 5. The wavy amplitude in p() curves decreases as the stress increases. The curves qualitatively resemble the curve for the insertion case in Fig. 4, but oFset an angle of 2.0
λ=∞ λ=2 λ=1
p(θ)
1.5
1.0
0.5
0.0
-π/4 -π /6 - π/12 0
θ
π/12 π/6
π/4
Fig. 5. Distribution of cluster orientation in the rotation process.
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=6. The minimal and maximal values are reached at = ±=6 and = 0, respectively, contrary to the distribution in the insertion process. The elongation aligns with the tension stress. An orientation average on the local deformation rate tensor in Eq. (13) leads to the global deformation rate tensor. For relatively large stress amplitude ( ¿ 1), a fairly accurate approximation for the plane strain constitutive relation is found to be ˜ − rc = Cr D; : (16) rc ≈ 0:95Wrdis √ : The creep rate coe7cient Cr ≈ 0:75(WrdiF + Wrs ) for the rotation process is provided in Appendix A. 2.4. Cooperative insertion and rotation process Insertion and rotation processes can contribute to the deformation in a cooperative manner. Denote () as the percentage of insertion clusters over all operating ones. The deformation rate tensor under a cooperative insertion and rotation process is given by 3 =6 ˜ [D] = p − dr () ()di ()p() + 1 − − −=6 6 6 cos 2 −sin 2 × d: (17) −sin 2 −cos 2 The orientation distribution p(; t) is evolved under p(; ˙ t) = d − p − ; t − d()p(; t): 6 6
(18)
where the cooperation deformation rate, scaled by the respective logarithmic strains, is 1 2 √ (1 − ())dr − : ()di () + d() = 6 ln 3 ln(2= 3) Suppose the initial state is isotropic, the steady state asymptote is p(; t) =
2d( − =6) : d() + d( − =6)
(19)
The cooperative ratio () under a prescribed stress can be determined by the principle of maximum plastic work (PMPW). The plastic work is denoted by H (())= ˜ a functional of the cooperative ratio (). We are looking for the proper : D, distribution of () so that H (()) = : D˜ maximizes. Numerical investigation (see Appendix B) suggests that () is a step function: 1 0 6 ¡ 0 ; (20) () = 0 0 6 ¡ =6:
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θ0=π/6
ln(16/3)/ln(4/3)
10
ξ
π/12>θ0>0
ln3/ln12
10
π/6>θ0>π/12
0
θ0=π /6 θ0=π /12
θ0=0 -1
10
θ0=0
1
2
3
4
5
λ
6
7
8
9
10
Fig. 6. DiFerent regimes for the insertion and rotation process.
Dimensional analysis dictates that the transition angle 0 is a function of the stress ratio and the dissipation ratio = (WrdiF + Wrs )=(WidiF + Wis ):
(21)
˜ one reduces the plastic Substituting Eq. (20) into the expression of H (()) = : D, work functional to the following function of 0 : 0 12
cos(2)p()di ()d H (0 ) ≡ 0 +
0
=6−0
cos(2)p(=6 − )dr () d :
(22)
The principle of maximum plastic work can be engaged by either the stationary condition, @H=@0 = 0, for the transition angle 0 or considering the extreme values of 0 at 0 and =6. Based on the parameter ranges of the loading amplitude and the dissipation ratio , three distinct cases exist for the case of ¿ 1, as shown in Fig. 6. (1) Pure insertion. When ¿ ln(16=3)=ln(4=3)(1 − 1=2) the plastic dissipation required in a rotation process overwhelms that for an insertion process. Accordingly, the deformation proceeds solely by insertion process, and the transition angle 0 is Hxed at =6. (2) Pure rotation. When √ 3 ln 3(22 − 3 + 1) √ √ ¡ 6 ln(2 3)2 − 2 ln(6 3) the plastic dissipation required in an insertion process overwhelms that for a rotation process. The deformation proceeds solely by rotation of grain clusters, and the transition angle 0 is Hxed at 0.
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(3) Cooperative insertion–rotation process. The cooperative case occurs only if √ 1 3 ln 3(22 − 3 + 1) ln(16=3) √ √ 1− ¿¿ : ln(4=3) 2 6 ln(2 3)2 − 2 ln(6 3) Two sub-cases can be identiHed: (a) pro-insertion case when ln(16=3) 1 1 1− ¿¿1 − √ ; ln(4=3) 2 3 where the transition angle 0 is found to be in the range of =6 ¿ 0 ¿ =12, and is given precisely by solving the following equation: ln 2(cos(=3 − 20 ) − 1=2) cos(=3 − 20 ) √ ln(2= 3) cos(=3 − 20 ) + ln 3(cos(=3 − 20 ) − 1=2) √
= cos(20 );
(23a)
(b) pro-rotation case when √ 3 ln 3(22 − 3 + 1) 1 √ √ 6¡1 − √ ; 2 6 ln(2 3) − 2 ln(6 3) 3 where the transition angle 0 is found to be in the range of 0 6 0 6 =12, and is given precisely by solving the following equation: ln 2 cos2 2 √ √ (ln(2= 3) + (ln 3=)) cos 2 − (ln 3=)1=2 √
=
[cos(=3 − 2) + cos 2](cos(=3 − 2) − 1=2) : cos(=3 − 2) + cos 2 − 1=
For the special case of 1, the transition angle explicit form: =6 1 ln(3=2) + ln(4=3) 1=2 arctan √ ln 2 3 √ √ 0 = 1 ln(2 3) − ln 3 √ √ if 1=2 arctan √ 3 ln(2= 3) + ln 3 0
(23b)
can be expressed in the following ¿ ln(16=3)=ln(4=3) ln(16=3)=ln(4=3) ¿ ¿ 1 (24) 1 ¿ ¿ ln 3=ln 12 6 ln 3=ln 12
Fig. 7 shows that the value of 0 is mainly controlled by the ratio of energy dissipation rather than the stress amplitude . This point can be exempliHed by Eq. (24) where 0 depends solely on , in various asymptotes shown as the double dot–dash lines in Fig. 6.
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1.0
θ0/(π/6)
0.8 0.6 0.4 0.2 0.0 0
10
1
2 4
λ
10
6
0
ξ
8 10
10
-1
Fig. 7. ProHle of the transition angle 0 as functions of and .
˜ = We are now in a position to evaluate the deformation rate [D]
d˜
under −d˜ prescribed stress loading. Dimensional analysis gives rise to the following parameter ˜ dependence of d: d˜ =
Wrdis F(; ); + Wis
WidiF
where F(; ) ≡
6
0
=6
(25)
[2()p() cos 2 + −1 (1 − (=6 − ))
×p(=6 − )(2 cos 2 − 1)]cos 2 d:
(26)
Numerical results (Fig. 8) of (26) show that the strain rate is approximately linear with respect to the stress when ¿ 1. When the stress is small, namely ¡ 1, cluster rotation hardly takes place and the insertion process dictates the strain rate. When ¿ 1 and ln(16=3)=ln(4=3) ¿ ¿ ln 3=ln 12; F(; ) can be Htted by F(; ) ≈ (1:2 + 0:75−1 ) + 0:1 − 0:55−1 :
(27)
The direct computing result and the Htting result are plotted in Fig. 9, the largest Htting error is about 7%. Recasting in a dimensional form, one arrives at the following plane strain constitutive relation: 0:1 0:55 Wrdis : (28) − D˜ ≈ Cco + WrdiF + Wrs WidiF + Wis : =2 √ The creep coe7cient of the cooperative process, Cco = (2:4=( + 3 3)Ci + 9=16Cr )−1 , can be regarded as a weighted average of the insertion and rotation creep coe7cients.
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140 120
F(λ,ξ)
100 80 60 40 20
10
0 10 9 8 7 6 5 4 3 1 2 1 10 λ 0
10
-1
0
ξ
Fig. 8. ProHle of function F(; ).
30
F(λ,ξ)
25 20 15 10 5 0 10 9 8 7
λ
6
5
10
4
3
2
10
1
10
0.5
-0.5
0
ξ
Fig. 9. Comparison between the computed proHle of F(; ) and the Htting expression (27).
When ¿ ln(6=3)=ln(4=3) the pure insertion constitutive law (12) should be observed, while the pure rotation relation (16) prevails when 6 ln 3=ln 12. 3. Three dimensional constitutive relation 3.1. Basic assumptions For a compact assembly of grains, we adopt the grain shape of tetrakai-decahedral in the spirit of Hahn and Padmanabhan (1997) as shown in Fig. 10. In the insertion process, the meta-stable grain shape assumes the shape of rhombic dodecahedron. The recent experimental observation by Wang et al. (2002) supports this assumption. The shadow areas represent the grain boundaries. We restrict the occurrence of both
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Fig. 10. A grain cluster assembled by tetrakai-decahedral grains during diFerent stages of deformation (after Hahn and Padmanabhan, 1997).
insertion and rotation process in the close-packed planes. Accordingly, one may treat the deformation process in a close-packed plane in a similar manner as that in the previous plane strain case, except some modiHcation on the coe7cients as stated in Appendices A and C. We evaluate the three-dimensional deformation rate tensor from the contributions of possible close-packed planes in various orientations. Initial isotropy is assumed to simplify the discussion. 3.2. Constitutive relation Under the three-dimensional model, the global deformation rate tensor is evaluated through its microscopic counterparts as 1 ˜ D = D ≡ D d; (29) 4
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where is the solid angle. The symbol D denotes the deformation rate tensor of a close-packed plane, orientated within a unit sphere in a three-dimensional space. A numeric scheme can be constructed that projects a three-dimensional stress into a close-packed plane and, consequently delivers the deformation rate tensor of the plane by Eqs. (12), (16) or (28). The orientation average via integral (29) would predict the overall response. In order to explore explicitly the diFerence between the three-dimensional constitutive law and the plane strain one, the special case of uniaxial loading is considered. Under the initial isotropy assumption, the deformed material will remain isotropic. That statement was exempliHed in the previous section for the plane strain case, and would remain true in the three-dimensional case since it only involves orientation average of clusters that deform in a similar manner as the plane strain case. Thus, stressing the nanocrystalline metal would not destroy the isotropy of the grain assembly regardless of its loading direction or its loading sequence. Thus, without the loss of generality, one may assume the uniaxial stress is acting along the e3 direction. The deviatoric stress tensor is given by (30) = − 12 ij + 32 3i 3j ei ⊗ ej ; where ei (i = 1; 2; 3) are the orthogonal unit vectors along the principal axes of stress. The deviatoric stress components within a plane of unit norm n is (n )ij = = 34 (1 + n23 )ni nj − 32 (ni n3 3j + nj n3 3i ) − 34 (1 − n23 )ij + 32 3i 3j ;
(31)
where n denotes the in-plane stress tensor when all projections along the n direction is removed. Substituting Eqs. (31) and (12) into Eq. (29), one obtains the constitutive relation for the insertion process via direct integration =
5(WidiF + Wis ) ˜ √ D; +3 3
(32)
where WidiF and Wis share the same meanings as their plane strain counterparts, but are of diFerent coe7cients accounting for the three-dimensional features of the assembly. Appendix C lists estimates for all dissipation energies for reference. The threshold stress for the insertion process is neglected in writing down Eq. (32). Next consider the rotation process. The threshold for rotation blocks the possibility of rotation for those planes whose norm vectors form small angles with the loading direction e3 . Denote the angle between n and e3 as , the largest principle stress in the plane is
max = 34 sin2 :
(33)
If one equals max with the threshold stress rc deHned in Eq. (A.3), the critical angle to suppress rotations is determined as 4 rc : (34) c = arcsin 3
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H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173
(2)
φc
(1)
Fig. 11. Non-rotating cones. Rotation only occurs in the region labeled by (1) but not in cones labeled by (2).
Fig. 11 shows two cones bounded by Eq. (34) where the rotation process is compromised. Averaging the constitutive response (16) outside the non-rotation zones, one yields the following constitutive relation for the rotation process: c 3 3 5 3 1 3 1 4 cos c − 2 cos c + 20 cos c − cos c − 3 cos c r ˜ = 0:75(Wrd + Wrs )D:
(35)
The threshold by a uniaxial stress (30), is given by rc ≈ √ stress tensor, stipulated dis c 0:866Wr = : . When ¿ 3 r , one has c ¡ arcsin 23 ≈ 0:73 from Eq. (34). Numerical testing indicates that the following asymptote (in the absence of non-rotating zone) gives a fairly accurate description (within 5%) for the constitutive response: ˜ (36) − 5 c ≈ 15 (WrdiF + Wrs )D: 3
8
We now turn to the cooperative insertion–rotation process. The constitutive responses for two extreme cases, i.e. ¿ ln(16=3)=ln(4=3) and 6 ln 3=ln 12, are given by
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Eqs. (32) and (36), respectively. When both insertion and rotation processes are operative, the following constitutive response is derived in the case of ¿ 3 rc : −1=2 ; −1=2 D˜ ≈ d˜ (37) 1 where
1:2 5 dis 0:8
− + W WrdiF + Wrs 3 r WidiF + Wis 1:3 2 0:25 + Wrdis : + diF Wr + Wrs 3 WidiF + Wis
2 d˜ ≈ 5
(38)
4. Discussions 4.1. Creeping rates The constitutive relations derived herein for the plane strain and three-dimensional cases share the similar linear viscous structure as discussed by Yang and Wang (2003). Moreover, the simple model based on clusters of nine grains by Yang and Wang (2003) predicted the same scale dependence for the creep coe7cient as the present model, namely the strain rate under constant stress changes in inverse proportion with respect to the cubic power of the grain size. However, the plane strain and three-dimension constitutive relations predict lower creep rates. The creep rate under the plane strain formulation is about 89% of the value predicted by Yang and Wang (2003) and that under the fully three-dimensional case is about 47% of the one-dimensional creep rate. The present work considers two factors that may hinder the creep rate. First, in the one-dimensional case, all deforming clusters are aligned with the tension direction, while in the two- and three-dimensional cases their elongation directions are frequently inclined with the principle stress. Those unfavorable orientations would compromise the global creep rate through statistical averaging. Second, given the same macroscopic stress, the local driving force for a cluster with inclined orientation would be smaller than the one anticipated in Yang and Wang (2003) in the one-dimensional model. Both factors would have more impact in the three-dimensional case than in the plane strain case, resulting in a much slower creep rate in the former. The slowing down eFect on the creep rate due to various orientations of grain clusters can be balanced to a certain degree by the non-uniform distribution of grain sizes. In the present model, all grains have uniform size and shape. It is not true in a realistic material. The current assumption of uniform grain size, along with the assumption of planar packing of the grain cluster, give stringent limitation for the possible non-uniform insertion and rotation processes. If the grains are of diFerent sizes, there will be more possible insertion directions, as well as more rotation cluster combinations. Both insertion and rotation will become easier. Another factor to speed
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up the creep rate might be the elasticity of the grain. Yang and Hong (2002) reported a computational approach to take into account for the elasticity of grains. 4.2. Grain insertion versus rotation in a cooperative process In the plane strain case, the present work demonstrates that the ratio of energy dissipation of the two competing processes decides the occurrence of whether an insertion or a rotation process. If the energy dissipation due to the grain boundary sliding and bulk diFusion is negligible when comparing with the energy dissipation due to grain boundary diFusion (Yang and Wang, 2003), the dissipation ratio deHned in Eq. (21) can be approximated by 2 2 2 m c2 m c2 WrdiF ln 3 ≈ diF = ln(4=3) n n i Wi r 2 2 m c2 m c2 ; (39) ≈ 14:6 × n r n i where the values of m; n and c2 for both insertion and rotation processes are listed in Table 1 of Yang and Wang (2003) for the plane strain case, and in Table 1 of the present work for the three-dimensional case. For the plane strain case, one has m = 0:122; n = 4 and c2 = 0:235 for the insertion process, and m = 0:061; n = 6 and c2 =0:271 for the rotation process. That leads to a value of ≈ 2:80. Consequently, one has 0 ≈ 22:6◦ according to Eq. (24), indicating the insertion (|| ¡ 22:6◦ ) is favored than rotation (22:6◦ ¡ || ¡ 30◦ ). For the three-dimensional case, on the other hand, one has m = 0:167; n = 12 and c2 = 0:286 for the insertion process, and m = 0:083; n = 14 and c2 = 0:246 for the rotation process. A formula similar to Eq. (24) gives a critical
Table 1 Characteristic values for grain shape changes in the three dimensional model
True strain Volume of matter transported per grain Mean diFusion distance Grain boundary sliding Net area of boundary which slides Change in grain boundary area per grain Matter undergone ordering–disordering change Number of diFusion paths per grain
Symbol
Stage 1
Stage 2
Stage 3
Sum
T
0.131
0.418
0.144
0.693
M = mD3 l = c2 D Ts = sD
0.083 0.286 0.131
0.083 0.286 0.131
0.083 0.246 0.230
0.249 0.273
A = c4 D 2
TA = c5 D2
0.756
−0:756
0
TV = D3
0
0
0.083
n
12
12
14
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orientation angle of 0 ≈ 22:1◦ . Consequently, the critical angles for the plane strain and three-dimensional cases are roughly the same. 4.3. In9uence of viscous 9ow within inter-grain layers In the model consisting of grain clusters, the grain boundaries serve two functions: the fast diFusion channels for mass transportation and the kinematical discontinuity for relative sliding between adjacent grains. The deformation of the inter-grain layer itself is omitted. When the grain size is in a nano scale, the volume fraction of the matter in the grain boundary region in a polycrystalline aggregate is of the same order as that of grains. Other than providing the accommodation mechanism for grain movement, the viscous =ow of the inter-grain layers may also contribute to the deformation of the aggregate. When the matter in grain boundary is amorphous, its behavior is like that of the viscous =uid. We assume that its stress response has the following linear form: T = Cgb D:
(40)
where T = − c with the anticipation that the threshold for the viscous =ow of inter-grain layers being the same as the one to liquefy the triple junctions. In order to estimate its in=uence, we regard the grain boundary as the network-like matrix phase with grains embedded in it. On the other hand, the deformation due to the clusters of grains, with the incorporation of mass diFusion and grain boundary sliding can be presented as T = Cgrain D:
(41)
We model the nano-grained polycrystal as a two-phase composite with a 3-0 connection. The grains are disconnected, while the inter-grain layers are connected in all directions. The equal-axed grains can be approximated by spherical inclusions. That enables us to use the Hashin-Shtrikman bound (Hashin, 1962; Hashin and Shtrikman, 1962a, b, 1963) for the eFective creep coe7cient C of the nano-grained materials in terms of Cgb and Cgrain : (1 − Vf )(Cgrain =Cgb − 1) C =1+ ; (42) Cgb 1 + 25 Vf (Cgrain =Cgb − 1) where Vf denotes the volume fraction of the inter-grain layers. Two extreme cases are C = Cgb for Vf = 1 and C = Cgrain for Vf = 0. 5. Concluding remarks In this paper, we regard the insertion process and the rotation process as two competing plastic deformation mechanisms. The orientation of the grain clusters changes with the deformation of the nanocrystalline materials. The percentage of insertion or rotation clusters among all operating clusters is determined by the plastic dissipation = (WrdiF + Wrs )=(WidiF + Wis ). Two critical values ln(16=3)=ln(4=3) and ln 3=ln 12 separate the mechanism map (Fig. 6) in three parts. The magnitude of the stress attributes
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little to the deformation mechanism. In the plane strain case, we get the linear relation between the creep rate and the stress. In the three-dimension model, only the case of uniaxial tension is treated in detail and a similar result is derived. The critical angles for the transition from insertion to rotation are almost the same for the plane strain and the three-dimensional cases. The Hashin–Shtrikman bounds provide us an estimate for the in=uence of the deformability of grain boundaries.
Acknowledgements This work is supported by the National Science Foundation of China under Grants 10120202 and 90205023.
Appendix A A.1. Calculation of creep coe:cients The speciHcation of the constants WidiF ; Wis ; WrdiF ; Wrs ; Wrdis as deHned in the Section 2, are summarized below, Yang and Wang (2003): WidiF =
Wis =
1 3(B 2 Ds √ 2 ln 3
WrdiF =
Wrs =
1 m2 KB TD2 # "n 2 √ nDV + DB i=1 =li ln 3 !
1 m2 KB TD3 # "n √ 2 nDV + DB i=1 =li ln 2= 3 !
(A.1a)
(A.1b)
(A.1c)
1 3(B 2 Ds √ 2 ln 2= 3
(A.1d)
1 √ ln 2= 3
(A.1e)
Wrdis = )Tg
√ √ in √ which = 8=3 3 and ) = 1 − =2 3 for the hexagonal grain, and = 6=; ) = 1=2 − 3=16 for the tetrakai-decahedral grain. Other parameters have the same meaning as in Yang and Wang (2003). The characteristic values, m; n; li ; s, can be found from Table 1 in Yang and Wang (2003) for the plane strain model and listed in the table for the three dimensional model. For both the plane strain and three-dimensional cases, the creep coe7cients of the insertion process, the rotation process and the cooperative
H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173
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process are given by
3(B Ds2 m2 KB TD2 16 #+ ! √ "n ; Ci = √ (3 3 + 27)(ln 3)2 nDV + DB i=1 =li m2 KB TD2 3(B 2 2 ! #+ √ "n Cr = √ Ds 3(ln(2= 3))2 nDV + DB i=1 =li
and
(A.2b)
−1
Cco =
(A.2a)
2:4 9 √ + 16C r + 3 3 Ci
:
In these expressions, = 1 is for the plane strain case, and = dimensional case. The threshold stress is c ≈ , Tg √ ; :
(A.2c) 5 2
for the three-
(A.3)
where , = 0:61 for the plane strain case and , = 0:50 for the three-dimensional case. Appendix B B.1. Numerical scheme to determine cooperative ratio () by PMPW The speciHc form of () relies on the maximization of the functional deHned as ˜ The plastic work can be written as H (()) = : D. 6 =6 H () = cos(2)[()p()di () 0 +(1 − (=6 − ))p(=6 − )dr ()] d:
(B.1)
Eq. (B.1) can be discretized as N +1
H () =
( cos(2i )[(i )p(i )di (i ) N +1 i=0
+(1 − (=6 − i ))p(=6 − i )dr (i )];
(B.2)
where i = i=6(N + 1) and = [(1 ); : : : ; (N )]T . Apparently one has the boundary conditions of (0) = 1 and (=6) = 0. An optimization algorithm can be constructed under max H ();
∀ ∈ [0; 1]N :
(B.3)
A typical result is shown in Fig. 12 for = = 1 where an initial linear distribution of () (dashed line) is assumed. Extensive numerical tests for other stress ratios and
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H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173 1.0
Result Initial Value
α(θ)
0.8 0.6
λ=1 ξ=1
0.4 0.2 0.0 0.0
0.2
0.4
θ
0.6
0.8
1.0
Fig. 12. Optimizing result of ().
Fig. 13. Mass diFusion in the insertion process.
dissipation ratios suggest that () is approximately a step function: 1 0 6 ¡ 0 ; () = 0 0 6 ¡ =6
(B.4)
and this Hnal step-like curve is irrelevant to the initial distribution. Appendix C C.1. Estimates for dissipation processes in three-dimension model To quantify the dissipation processes under three-dimensional model, we start with a grain shape that is approximately spherical. The second graph in Fig. 10 indicates that
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the meta-stable grain shape in the insertion process can be roughly treated as a cube rotated 45◦ in its plane that reserves the volume of the sphere. The matters diFuse along the paths depicted Fig. 13 to accomplish the insertion process. In the rotation process, the atoms lying between the tetrakai-decahedral grain and its inscribed sphere change their assembling states from ordering to disordering or vice versa. In order to make up for the gap or overlap caused by the grain rotation, the disordered mass is required to diFuse along the surface of the inscribe sphere for an arc path that spans an angle of =6 with respect to the grain center. These assumptions lead to various parameters in Table 1 to characterize the dissipation processes during the deformation. References Ashby, M.F., Verrall, R.A., 1973. DiFusion-accommodated =ow and superplasticity. Acta Metall. 21, 149–163. Cai, B., Kong, Q.P., Lu, L., Lu, K., 1999. Interface controlled diFusional creep of nanocrystalline pure copper. Scr. Metall. 41, 755–759. Gao, P., Gleiter, H., 1987. High resolution electron microscope observation of small gold crystals. Acta Metall. 35, 1571–1575. Gryaznov, V.G., Polonsky, I.A., Romanov, A.E., Trusov, L.I., 1991. Size eFects of dislocation stability in nanocrystals. Phys. Rev. B 44, 42–46. Hahn, H., Padmanabhan, K.A., 1997. A model for the deformation of nanocrystalline materials. Philos. Mag. B 76, 559–571. Hashin, Z., 1962. The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150. Hashin, Z., Shtrikman, S., 1962a. On some variational principles in anisotropic and non-homogeneous elasticity. J. Mech. Phys. Solid 10, 335–342. Hashin, Z., Shtrikman, S., 1962b. A variational approach to the theory of the elastic behavior of polycrystals. J. Mech. Phys. Solid 10, 343–352. Hashin, Z., Shtrikman, S., 1963. A variational approach to the theory of elastic behavior of multiphase materials. J. Mech. Phys. Solid 11, 127–140. Lu, L., Sui, M.L., Lu, K., 2000. Superplastic extensibility of nanocrystalline copper at room temperature. Science 287, 1463–1466. Milligan, W.W., Hackney, S.A., Ke, M., Aifantis, E.C., 1993. In situ studies of deformation and fracture in nanophase materials. Nanostruct. Mater. 2, 267–276. Thomas, G.J., Siegel, R.W., Eastman, J.A., 1990. Grain boundaries in nanophase palladium: high resolution electron microscopy and image simulation. Scripta Metall. Mater. 24, 201–206. Valiev, R.Z., Alexandrov, I.V., Zhu, Y.T., Lowe, T.C., 2002. Paradox of strength and ductility in metals processed by severe plastic deformation. Mater. Res. 17, 5–8. Wang, Z.L., Zhang, Z., Liu, Y., 2002. Transmission electron microscopy and spectroscopy. In: Wang, Z.L., Liu, Y., Zhang, Z. (Eds.), Handbook of Nanophase and Nanostructured Materials—Characterization. Tsinghua University Press and Kluwer Academic/Plenum Publishers, Beijing, pp. 29–98 (Chapter 2). Yang, W., Hong, W., 2002. Numerical simulation for deformation of nano-grained metals. Acta Mechanica Sinica 18, 506–515. Yang, W., Wang, H.T., 2003. Mechanics modeling for deformation of nano-grained metals. J. Mech. Phys. Solids, doi:10.1016/j.jmps.2003.07.003
www.elsevier.com/locate/jmps
Constitutive modeling for nanocrystalline metals based on cooperative grain boundary mechanisms Hong-Tao Wang, Wei Yang∗ Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Received 1 November 2002; accepted 27 August 2003
Abstract In nanocrystalline metals, the plastic deformation is accommodated primarily at the grain boundaries. Yang and Wang (J. Mech. Phys. Solids, 2003) suggested a deformation model based on clusters consisting of nine grains and incorporating both the Ashby-Verrall mechanism and a 30◦ rotation of closely linked pairs of grains. In the present article, the insertion and rotation processes are considered together as a cooperative deformation mechanisms, and the degree to which each process contributes is determined by the application of the principle of maximum plastic work. Plane strain and three-dimensional constitutive relations based on this concept are derived for which a general stress state drives the orientation evolution of various grain clusters under the Reuss assumption. Detailed calculation shows that the strain rate depends linearly on the stress, with the values of the coe7cients in this linear relationship dictated by the microscopic energy dissipation. The deformation contributed to the overall response by the grain boundary mechanism is discussed in the spirit of the Hashin-Shtrikman bounds. ? 2003 Elsevier Ltd. All rights reserved. Keywords: A. Creep; A. Grain-boundaries; A. Microstructure; B. Crystal plasticity; Nano-grained metals
1. Introduction Grain size plays a key role in deciding the dominant mechanisms of plastic deformation. In coarse-grained metals, the movements of dislocations within individual grains carry out the plastic =ow. The grain boundaries hinder the dislocation movements, cause pileups, and consequently make the material harder to deform. Within a much smaller grain size, say below 30 nm, both the theoretical analysis (Gryaznov et al., ∗
Corresponding author. Tel.: +86-10-6278-2642; fax: +86-10-6258-1824. E-mail address: [email protected] (W. Yang).
0022-5096/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2003.08.005
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1991) and the experimental evidences (Gao and Gleiter, 1987; Thomas et al., 1990; Milligan et al., 1993) showed that dislocations seldom appeared within the grains. Nevertheless, the dense network of grain boundaries facilitates the plastic deformation via certain accommodation mechanisms. In nanocrystalline metals, the available experimental data show that the grain shape remains equal-axed before and after the deformation (Lu et al., 2000; Valiev et al., 2002). This suggests that the nano-sized grains behave much more like stiF and equalaxed particles, squeezing past each other with little distortion in the grain cores. The grain boundaries serve not only as mass-transport networks but also as viscous deformable layers. Ashby and Verrall (1973) suggested a deformation mechanism carried out by fourgrain clusters, the inter-grain distance in the elongation direction increases, and that perpendicular to the elongation decreases. The mechanism is not self-complete in the sense that further elongation can only proceeds in a direction of ±30◦ from the original stretching direction. If the grains in closely linked pairs can rotate 30◦ , on the one hand, the grain cluster can achieve a strain of certain amount in the same direction as that caused by the insertion process. Furthermore, the rotation would restore the initial conHguration. Yang and Wang (2003) exploited this idea by suggesting a model based on clusters consisting of nine grains (Fig. 1) and incorporating both the insertion mechanism of Ashby and Verrall (1973) and a 30◦ rotation of closely linked grain pairs. Though formulated under uniaxial loading, the derived creep law agrees with the testing data of Cai et al. (1999). In the present article, the insertion and rotation processes are considered together as a cooperative deformation mechanism. We extend the capability of the constitutive modeling to the plane strain and three-dimensional cases, discussed in Section 2 and Section 3, respectively. Furthermore, the deformation contributed to the overall response by the viscous grain boundary layers, now composing of a Hnite volume fraction for nanocrystalline metals, is discussed in the spirit of the Hashin-Shtrikman bounds (Hashin, 1962; Hashin and Shtrikman, 1962a, b, 1963).
2. Two-dimensional constitutive relation 2.1. Assumptions and kinematics Attention is Hrst focused on the case of plane strain deformation. The polycrystal is composed of hexagon grains. Nearby hexagon grains may deform in a cooperative manner via cluster rearrangement such as that envisaged in Fig. 1. A grain may deform with one cluster at one time and join the deformation of a neighbor cluster at later time. DiFerent grains, regardless of their locations in the cluster, are assumed to be subjected to the same stress. The elastic deformation of the grain cores is neglected. Consequently, the deformation within the plane becomes incompressible. A local deformation rate tensor D and its global counterpart D˜ are introduced. The incompressibility dictates
H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173
(a) initial configuration
(c) completion of insertion stage
1153
(b) critical configuration for the insertion process
(d) rotation of 30 degrees
Fig. 1. Model of 9-grain cluster.
the following representation of the deformation rate tensor D in its principle-axes: d [D] = (d ¿ 0): (1) −d From the incompressibility, the constitutive response of the material is irrelevant of the hydrostatic stress. Two separate processes describe the kinematics of a grain cluster. The insertion process from the Ashby–Verrall’s mechanism, see Fig. 1 (a) – (c) where the grain cluster changes from an “armchair” conHguration to a “zigzag” conHguration. The inter-grain distance in the tensile direction elongates to disengage the grain pairs in that direction and the inter-grain distance perpendicular to the tensile direction contracts
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to engage the grains in the lateral direction. Slight amount of grain boundary sliding is involved. The rotation process involves the grain rotations of 30◦ on close-linked grain pairs, see Fig. 1 (c) and (d) where the armchair conHguration is restored. Large amount of grain boundary sliding and viscous =ow of interlayers are anticipated. The core of a nano-grain can adapt its shape to lower the energy barrier. Without the loss of generality, one may take the loading axis as the vertical direction. Consider the central seven grains in the 9-grain cluster. The conHgurations in Fig. 1(a) and (d) can accommodate an insertion process aligned with the loading axis. On the other hand, the conHguration in Fig. 1(c) gives rise to two possible deformation mechanisms: (1) an insertion process that is either plus or minus 30◦ with respect to the loading axis; or (2) a rotation process that elongates the cluster in the same direction as the loading axis. DeHne as the angle formed between the central seven grains in a cluster with the loading axis. From the six-fold symmetry of the conHguration, it is su7cient to investigate the cluster orientation angle within the range of −=6 to =6. All functions varying with the oriented angle are deHned as periodic functions of period =3. For the illustrative purpose, the cluster aligning with the loading axis is shown as solid line in Fig. 2, while the cluster forming angle is shown as dashed line. A distribution function p() is introduced to characterize the percentage of clusters in every possible orientation relative to the principal axes of the stress.
σ
θ
Fig. 2. Orientation of grain clusters.
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2.2. Insertion process In the insertion process, the plastic dissipation within a representative grain is composed of three parts: (1) dissipation by mass diFusion from sources to sinks; (2) dissipation by grain boundary sliding; and (3) dissipation by =uctuation in grain boundary area. These dissipations are labeled as W˙ diFusion ; W˙ slide and W˙ gb , respectively. The detailed calculations can be found in Yang and Wang (2003) where the following expressions were reported: W˙ diFusion = WidiF d2i ;
W˙ slide = Wis d2i ;
W˙ gb = Wigb di :
(2)
The subscript “i” labels the insertion process and superscripts “diF ”, “s”, “gb” abbreviate the diFusion, slide and grain boundary eFects, respectively. The expressions for WidiF ; Wis and Wigb can be found in Appendix A. The same dissipation calculations can be extended to the plane strain case. Only deviatoric stress is relevant with denoting the principal deviatoric stress. Accordingly, the work done by the applied load becomes : D = 2 di () ();
(3)
under the Reuss assumption that prescribes the same stress state to every grain, where ; and () = +k (4) () = cos 2 − ¡ 6 6 6 3 for any integer k. Equating the external work (3) to the internal dissipation (2), one obtains the following expression for the strain rate: di () =
2 () − Wigb : WidiF + Wis
(5)
The global deformation rate tensor is derived by the orientation average on the local deformation rate tensor in Eq. (1) under a weighting function p() for diFerent orientations as =6 cos 2 −sin 2 3 ˜ = [D] di () p() d (6) −=6 −sin 2 −cos 2 ˜ is deHned on coordinates aligned with the principal axes of the where the matrix [D] stress tensor. Fig. 2 shows the orientation change of a cluster. The arrows along the vertical line give the direction of the principal tensile stress. Fig. 3 illustrates a typical grain in the cluster. The dashed-hexagon and the solid one represent the conHgurations before and after the insertion process. When an insertion process is completed, the orientation of the cluster rotates 30◦ , namely it re-orients from to −=6 or +=6, as indicated in Fig. 3 from the dot–dashed axis to the dashed axis. Two solid lines in Fig. 3 standing for plus and minus 30◦ bound for all these axes. The periodicity of =3 renders the choices on − =6 or + =6 immaterial. We always choose the branch of − =6 hereafter. The conservation on the orientation implies p(; ˙ t)+ p(−=6; ˙ t)=0 with the
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σ α
θ
−α = θ −π/6 Fig. 3. Orientation change of a typical grain in the cluster.
following integral form ; p(; t) + p − ; t = p0 () + p0 − 6 6
(7)
under the initial conditionp(; 0) = p0 (). This implies the invariance of p(; t) + p( − =6; t) regardless of the stress variation during the period. If the material is initially “isotropic”, i.e. p0 () = 1, then one has p(; t) + p( − =6; t) ≡ 2. The distribution function p(; t) evolves according to 2 p(; ˙ t) = di − p − ; t − di ()p(; t) : (8) ln 3 6 6 Namely the insertion along the direction reduces while the insertion along the −=6 direction enhances the orientation distribution at . For an applied load that changes slowly with time, the strain rate di does not explicitly depend on time during the evolution. Then p(; t) has the following form: 1 p(; t) = p0 () + p0 − di − 6 6 di () + di − 6 −2=ln 3[di ()+di (−=6)]t p0 − e + di ()p0 () − di − : (9) 6 6 The Hrst term in Eq. (9) represents a steady state solution. The second term is transient in nature and decays exponentially. We henceforth consider only the steady state response after the transient dies out. If the material is prepared without texture, such as in Lu et al. (2000), the initial orientation distribution should be isotropic. Thus, the
H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173
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1.4
p(θ)
1.2
1.0
0.8
0.6
-π /4 -π /6 -π/12
0
θ
π/12
π/6
π/4
Fig. 4. Distribution of cluster orientation in the insertion process.
steady state distribution p() is reduced to 2di ( − =6) p() = : di () + di ( − =6)
(10)
The ratio Wigb = is small as argued by Ashby and Verrall (1973), one may omit the term associated with Wigb . Thus, the form of p() is reduced even further as 2 ( − =6) : (11) p() = () + ( − =6) As shown in Fig. 4, the distribution function p() reaches its minimum at = 0 and its maxima at = ±=6. This coincides with the physical model. From Eq. (5), the strain rate maximizes at = 0, and consequently the percentage of clusters whose orientations are near = 0 should decrease. The conservation of orientations stated in Eq. (7) dictates that the decreasing amount near the orientation of = 0 should be compensated by the increasing amount near = ±=6. Fig. 4 clearly re=ects that tendency for the steady state limit. Substituting Eq. (11) into Eq. (6), one arrives at the following plane strain constitutive relation √ 6(4 − ln 3) gb c c ˜ √ − i = Ci D; i = Wi √ : (12) : +3 3 √ The creep coe7cient Ci = 2(WidiF + Wis )=( + 3 3) is listed in Appendix A. The threshold stress ic is negligible for the insertion case. Two important observations emerge. First, if the material is initially isotropic, its macroscopic response, as delineated by Eq. (12), would also be isotropic regardless of the amount of deformation. The non-uniform responses in various clusters are erased by orientation average. Second, the global response would be non-hardening since all material parameters in Eq. (12), such as WidiF ; Wis and Wigb , are not aFected by the extent of deformation. This non-hardening behavior does occur for well-prepared (highly pure and fully densiHed) nanocrystalline metals under extremely large deformation,
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Lu et al. (2000). It is worthwhile to mention that the conclusions above neglect the possible occurrence of microcracks or microcavities under heterogeneous local stress Held. 2.3. Rotation process In a rotation process, the grain boundary energy is unchanged but the energy dissipation in ordering–disordering transformation near the triple junctions provide a considerable threshold stress. The energy dissipation in the ordering–disordering transformation is given by W˙ disorder = Wrdis dr , where the subscript “r” labels the association with a rotation process. Appendix A lists the detailed expression for Wrdis . Equivalence of the microscopic plastic dissipation and the macroscopic power delivers: dis 2 ( − =6) − Wr ¿ Wrdis if 2
− WrdiF + Wrs 6 = dr − (13) 6 0 otherwise: Similar analysis gives the evolution equation of p(; t) in the rotation process: 1 √ p(; t) : dr ()p − ; t − dr − p(; ˙ t) = 6 6 ln(2= 3)
(14)
The steady state solution is derived as p() =
2dr () dr () + dr ( − =6)
(15)
by assuming that the material is prepared isotropic. To facilitate the subsequent discussions, a dimensionless loading amplitude parameter ≡ =Wrdis is introduced. Given diFerent values, the curves of p() are plotted in Fig. 5. The wavy amplitude in p() curves decreases as the stress increases. The curves qualitatively resemble the curve for the insertion case in Fig. 4, but oFset an angle of 2.0
λ=∞ λ=2 λ=1
p(θ)
1.5
1.0
0.5
0.0
-π/4 -π /6 - π/12 0
θ
π/12 π/6
π/4
Fig. 5. Distribution of cluster orientation in the rotation process.
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=6. The minimal and maximal values are reached at = ±=6 and = 0, respectively, contrary to the distribution in the insertion process. The elongation aligns with the tension stress. An orientation average on the local deformation rate tensor in Eq. (13) leads to the global deformation rate tensor. For relatively large stress amplitude ( ¿ 1), a fairly accurate approximation for the plane strain constitutive relation is found to be ˜ − rc = Cr D; : (16) rc ≈ 0:95Wrdis √ : The creep rate coe7cient Cr ≈ 0:75(WrdiF + Wrs ) for the rotation process is provided in Appendix A. 2.4. Cooperative insertion and rotation process Insertion and rotation processes can contribute to the deformation in a cooperative manner. Denote () as the percentage of insertion clusters over all operating ones. The deformation rate tensor under a cooperative insertion and rotation process is given by 3 =6 ˜ [D] = p − dr () ()di ()p() + 1 − − −=6 6 6 cos 2 −sin 2 × d: (17) −sin 2 −cos 2 The orientation distribution p(; t) is evolved under p(; ˙ t) = d − p − ; t − d()p(; t): 6 6
(18)
where the cooperation deformation rate, scaled by the respective logarithmic strains, is 1 2 √ (1 − ())dr − : ()di () + d() = 6 ln 3 ln(2= 3) Suppose the initial state is isotropic, the steady state asymptote is p(; t) =
2d( − =6) : d() + d( − =6)
(19)
The cooperative ratio () under a prescribed stress can be determined by the principle of maximum plastic work (PMPW). The plastic work is denoted by H (())= ˜ a functional of the cooperative ratio (). We are looking for the proper : D, distribution of () so that H (()) = : D˜ maximizes. Numerical investigation (see Appendix B) suggests that () is a step function: 1 0 6 ¡ 0 ; (20) () = 0 0 6 ¡ =6:
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H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173 1
θ0=π/6
ln(16/3)/ln(4/3)
10
ξ
π/12>θ0>0
ln3/ln12
10
π/6>θ0>π/12
0
θ0=π /6 θ0=π /12
θ0=0 -1
10
θ0=0
1
2
3
4
5
λ
6
7
8
9
10
Fig. 6. DiFerent regimes for the insertion and rotation process.
Dimensional analysis dictates that the transition angle 0 is a function of the stress ratio and the dissipation ratio = (WrdiF + Wrs )=(WidiF + Wis ):
(21)
˜ one reduces the plastic Substituting Eq. (20) into the expression of H (()) = : D, work functional to the following function of 0 : 0 12
cos(2)p()di ()d H (0 ) ≡ 0 +
0
=6−0
cos(2)p(=6 − )dr () d :
(22)
The principle of maximum plastic work can be engaged by either the stationary condition, @H=@0 = 0, for the transition angle 0 or considering the extreme values of 0 at 0 and =6. Based on the parameter ranges of the loading amplitude and the dissipation ratio , three distinct cases exist for the case of ¿ 1, as shown in Fig. 6. (1) Pure insertion. When ¿ ln(16=3)=ln(4=3)(1 − 1=2) the plastic dissipation required in a rotation process overwhelms that for an insertion process. Accordingly, the deformation proceeds solely by insertion process, and the transition angle 0 is Hxed at =6. (2) Pure rotation. When √ 3 ln 3(22 − 3 + 1) √ √ ¡ 6 ln(2 3)2 − 2 ln(6 3) the plastic dissipation required in an insertion process overwhelms that for a rotation process. The deformation proceeds solely by rotation of grain clusters, and the transition angle 0 is Hxed at 0.
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(3) Cooperative insertion–rotation process. The cooperative case occurs only if √ 1 3 ln 3(22 − 3 + 1) ln(16=3) √ √ 1− ¿¿ : ln(4=3) 2 6 ln(2 3)2 − 2 ln(6 3) Two sub-cases can be identiHed: (a) pro-insertion case when ln(16=3) 1 1 1− ¿¿1 − √ ; ln(4=3) 2 3 where the transition angle 0 is found to be in the range of =6 ¿ 0 ¿ =12, and is given precisely by solving the following equation: ln 2(cos(=3 − 20 ) − 1=2) cos(=3 − 20 ) √ ln(2= 3) cos(=3 − 20 ) + ln 3(cos(=3 − 20 ) − 1=2) √
= cos(20 );
(23a)
(b) pro-rotation case when √ 3 ln 3(22 − 3 + 1) 1 √ √ 6¡1 − √ ; 2 6 ln(2 3) − 2 ln(6 3) 3 where the transition angle 0 is found to be in the range of 0 6 0 6 =12, and is given precisely by solving the following equation: ln 2 cos2 2 √ √ (ln(2= 3) + (ln 3=)) cos 2 − (ln 3=)1=2 √
=
[cos(=3 − 2) + cos 2](cos(=3 − 2) − 1=2) : cos(=3 − 2) + cos 2 − 1=
For the special case of 1, the transition angle explicit form: =6 1 ln(3=2) + ln(4=3) 1=2 arctan √ ln 2 3 √ √ 0 = 1 ln(2 3) − ln 3 √ √ if 1=2 arctan √ 3 ln(2= 3) + ln 3 0
(23b)
can be expressed in the following ¿ ln(16=3)=ln(4=3) ln(16=3)=ln(4=3) ¿ ¿ 1 (24) 1 ¿ ¿ ln 3=ln 12 6 ln 3=ln 12
Fig. 7 shows that the value of 0 is mainly controlled by the ratio of energy dissipation rather than the stress amplitude . This point can be exempliHed by Eq. (24) where 0 depends solely on , in various asymptotes shown as the double dot–dash lines in Fig. 6.
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1.0
θ0/(π/6)
0.8 0.6 0.4 0.2 0.0 0
10
1
2 4
λ
10
6
0
ξ
8 10
10
-1
Fig. 7. ProHle of the transition angle 0 as functions of and .
˜ = We are now in a position to evaluate the deformation rate [D]
d˜
under −d˜ prescribed stress loading. Dimensional analysis gives rise to the following parameter ˜ dependence of d: d˜ =
Wrdis F(; ); + Wis
WidiF
where F(; ) ≡
6
0
=6
(25)
[2()p() cos 2 + −1 (1 − (=6 − ))
×p(=6 − )(2 cos 2 − 1)]cos 2 d:
(26)
Numerical results (Fig. 8) of (26) show that the strain rate is approximately linear with respect to the stress when ¿ 1. When the stress is small, namely ¡ 1, cluster rotation hardly takes place and the insertion process dictates the strain rate. When ¿ 1 and ln(16=3)=ln(4=3) ¿ ¿ ln 3=ln 12; F(; ) can be Htted by F(; ) ≈ (1:2 + 0:75−1 ) + 0:1 − 0:55−1 :
(27)
The direct computing result and the Htting result are plotted in Fig. 9, the largest Htting error is about 7%. Recasting in a dimensional form, one arrives at the following plane strain constitutive relation: 0:1 0:55 Wrdis : (28) − D˜ ≈ Cco + WrdiF + Wrs WidiF + Wis : =2 √ The creep coe7cient of the cooperative process, Cco = (2:4=( + 3 3)Ci + 9=16Cr )−1 , can be regarded as a weighted average of the insertion and rotation creep coe7cients.
H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173
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140 120
F(λ,ξ)
100 80 60 40 20
10
0 10 9 8 7 6 5 4 3 1 2 1 10 λ 0
10
-1
0
ξ
Fig. 8. ProHle of function F(; ).
30
F(λ,ξ)
25 20 15 10 5 0 10 9 8 7
λ
6
5
10
4
3
2
10
1
10
0.5
-0.5
0
ξ
Fig. 9. Comparison between the computed proHle of F(; ) and the Htting expression (27).
When ¿ ln(6=3)=ln(4=3) the pure insertion constitutive law (12) should be observed, while the pure rotation relation (16) prevails when 6 ln 3=ln 12. 3. Three dimensional constitutive relation 3.1. Basic assumptions For a compact assembly of grains, we adopt the grain shape of tetrakai-decahedral in the spirit of Hahn and Padmanabhan (1997) as shown in Fig. 10. In the insertion process, the meta-stable grain shape assumes the shape of rhombic dodecahedron. The recent experimental observation by Wang et al. (2002) supports this assumption. The shadow areas represent the grain boundaries. We restrict the occurrence of both
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Fig. 10. A grain cluster assembled by tetrakai-decahedral grains during diFerent stages of deformation (after Hahn and Padmanabhan, 1997).
insertion and rotation process in the close-packed planes. Accordingly, one may treat the deformation process in a close-packed plane in a similar manner as that in the previous plane strain case, except some modiHcation on the coe7cients as stated in Appendices A and C. We evaluate the three-dimensional deformation rate tensor from the contributions of possible close-packed planes in various orientations. Initial isotropy is assumed to simplify the discussion. 3.2. Constitutive relation Under the three-dimensional model, the global deformation rate tensor is evaluated through its microscopic counterparts as 1 ˜ D = D ≡ D d; (29) 4
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where is the solid angle. The symbol D denotes the deformation rate tensor of a close-packed plane, orientated within a unit sphere in a three-dimensional space. A numeric scheme can be constructed that projects a three-dimensional stress into a close-packed plane and, consequently delivers the deformation rate tensor of the plane by Eqs. (12), (16) or (28). The orientation average via integral (29) would predict the overall response. In order to explore explicitly the diFerence between the three-dimensional constitutive law and the plane strain one, the special case of uniaxial loading is considered. Under the initial isotropy assumption, the deformed material will remain isotropic. That statement was exempliHed in the previous section for the plane strain case, and would remain true in the three-dimensional case since it only involves orientation average of clusters that deform in a similar manner as the plane strain case. Thus, stressing the nanocrystalline metal would not destroy the isotropy of the grain assembly regardless of its loading direction or its loading sequence. Thus, without the loss of generality, one may assume the uniaxial stress is acting along the e3 direction. The deviatoric stress tensor is given by (30) = − 12 ij + 32 3i 3j ei ⊗ ej ; where ei (i = 1; 2; 3) are the orthogonal unit vectors along the principal axes of stress. The deviatoric stress components within a plane of unit norm n is (n )ij = = 34 (1 + n23 )ni nj − 32 (ni n3 3j + nj n3 3i ) − 34 (1 − n23 )ij + 32 3i 3j ;
(31)
where n denotes the in-plane stress tensor when all projections along the n direction is removed. Substituting Eqs. (31) and (12) into Eq. (29), one obtains the constitutive relation for the insertion process via direct integration =
5(WidiF + Wis ) ˜ √ D; +3 3
(32)
where WidiF and Wis share the same meanings as their plane strain counterparts, but are of diFerent coe7cients accounting for the three-dimensional features of the assembly. Appendix C lists estimates for all dissipation energies for reference. The threshold stress for the insertion process is neglected in writing down Eq. (32). Next consider the rotation process. The threshold for rotation blocks the possibility of rotation for those planes whose norm vectors form small angles with the loading direction e3 . Denote the angle between n and e3 as , the largest principle stress in the plane is
max = 34 sin2 :
(33)
If one equals max with the threshold stress rc deHned in Eq. (A.3), the critical angle to suppress rotations is determined as 4 rc : (34) c = arcsin 3
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(2)
φc
(1)
Fig. 11. Non-rotating cones. Rotation only occurs in the region labeled by (1) but not in cones labeled by (2).
Fig. 11 shows two cones bounded by Eq. (34) where the rotation process is compromised. Averaging the constitutive response (16) outside the non-rotation zones, one yields the following constitutive relation for the rotation process: c 3 3 5 3 1 3 1 4 cos c − 2 cos c + 20 cos c − cos c − 3 cos c r ˜ = 0:75(Wrd + Wrs )D:
(35)
The threshold by a uniaxial stress (30), is given by rc ≈ √ stress tensor, stipulated dis c 0:866Wr = : . When ¿ 3 r , one has c ¡ arcsin 23 ≈ 0:73 from Eq. (34). Numerical testing indicates that the following asymptote (in the absence of non-rotating zone) gives a fairly accurate description (within 5%) for the constitutive response: ˜ (36) − 5 c ≈ 15 (WrdiF + Wrs )D: 3
8
We now turn to the cooperative insertion–rotation process. The constitutive responses for two extreme cases, i.e. ¿ ln(16=3)=ln(4=3) and 6 ln 3=ln 12, are given by
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Eqs. (32) and (36), respectively. When both insertion and rotation processes are operative, the following constitutive response is derived in the case of ¿ 3 rc : −1=2 ; −1=2 D˜ ≈ d˜ (37) 1 where
1:2 5 dis 0:8
− + W WrdiF + Wrs 3 r WidiF + Wis 1:3 2 0:25 + Wrdis : + diF Wr + Wrs 3 WidiF + Wis
2 d˜ ≈ 5
(38)
4. Discussions 4.1. Creeping rates The constitutive relations derived herein for the plane strain and three-dimensional cases share the similar linear viscous structure as discussed by Yang and Wang (2003). Moreover, the simple model based on clusters of nine grains by Yang and Wang (2003) predicted the same scale dependence for the creep coe7cient as the present model, namely the strain rate under constant stress changes in inverse proportion with respect to the cubic power of the grain size. However, the plane strain and three-dimension constitutive relations predict lower creep rates. The creep rate under the plane strain formulation is about 89% of the value predicted by Yang and Wang (2003) and that under the fully three-dimensional case is about 47% of the one-dimensional creep rate. The present work considers two factors that may hinder the creep rate. First, in the one-dimensional case, all deforming clusters are aligned with the tension direction, while in the two- and three-dimensional cases their elongation directions are frequently inclined with the principle stress. Those unfavorable orientations would compromise the global creep rate through statistical averaging. Second, given the same macroscopic stress, the local driving force for a cluster with inclined orientation would be smaller than the one anticipated in Yang and Wang (2003) in the one-dimensional model. Both factors would have more impact in the three-dimensional case than in the plane strain case, resulting in a much slower creep rate in the former. The slowing down eFect on the creep rate due to various orientations of grain clusters can be balanced to a certain degree by the non-uniform distribution of grain sizes. In the present model, all grains have uniform size and shape. It is not true in a realistic material. The current assumption of uniform grain size, along with the assumption of planar packing of the grain cluster, give stringent limitation for the possible non-uniform insertion and rotation processes. If the grains are of diFerent sizes, there will be more possible insertion directions, as well as more rotation cluster combinations. Both insertion and rotation will become easier. Another factor to speed
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up the creep rate might be the elasticity of the grain. Yang and Hong (2002) reported a computational approach to take into account for the elasticity of grains. 4.2. Grain insertion versus rotation in a cooperative process In the plane strain case, the present work demonstrates that the ratio of energy dissipation of the two competing processes decides the occurrence of whether an insertion or a rotation process. If the energy dissipation due to the grain boundary sliding and bulk diFusion is negligible when comparing with the energy dissipation due to grain boundary diFusion (Yang and Wang, 2003), the dissipation ratio deHned in Eq. (21) can be approximated by 2 2 2 m c2 m c2 WrdiF ln 3 ≈ diF = ln(4=3) n n i Wi r 2 2 m c2 m c2 ; (39) ≈ 14:6 × n r n i where the values of m; n and c2 for both insertion and rotation processes are listed in Table 1 of Yang and Wang (2003) for the plane strain case, and in Table 1 of the present work for the three-dimensional case. For the plane strain case, one has m = 0:122; n = 4 and c2 = 0:235 for the insertion process, and m = 0:061; n = 6 and c2 =0:271 for the rotation process. That leads to a value of ≈ 2:80. Consequently, one has 0 ≈ 22:6◦ according to Eq. (24), indicating the insertion (|| ¡ 22:6◦ ) is favored than rotation (22:6◦ ¡ || ¡ 30◦ ). For the three-dimensional case, on the other hand, one has m = 0:167; n = 12 and c2 = 0:286 for the insertion process, and m = 0:083; n = 14 and c2 = 0:246 for the rotation process. A formula similar to Eq. (24) gives a critical
Table 1 Characteristic values for grain shape changes in the three dimensional model
True strain Volume of matter transported per grain Mean diFusion distance Grain boundary sliding Net area of boundary which slides Change in grain boundary area per grain Matter undergone ordering–disordering change Number of diFusion paths per grain
Symbol
Stage 1
Stage 2
Stage 3
Sum
T
0.131
0.418
0.144
0.693
M = mD3 l = c2 D Ts = sD
0.083 0.286 0.131
0.083 0.286 0.131
0.083 0.246 0.230
0.249 0.273
A = c4 D 2
TA = c5 D2
0.756
−0:756
0
TV = D3
0
0
0.083
n
12
12
14
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orientation angle of 0 ≈ 22:1◦ . Consequently, the critical angles for the plane strain and three-dimensional cases are roughly the same. 4.3. In9uence of viscous 9ow within inter-grain layers In the model consisting of grain clusters, the grain boundaries serve two functions: the fast diFusion channels for mass transportation and the kinematical discontinuity for relative sliding between adjacent grains. The deformation of the inter-grain layer itself is omitted. When the grain size is in a nano scale, the volume fraction of the matter in the grain boundary region in a polycrystalline aggregate is of the same order as that of grains. Other than providing the accommodation mechanism for grain movement, the viscous =ow of the inter-grain layers may also contribute to the deformation of the aggregate. When the matter in grain boundary is amorphous, its behavior is like that of the viscous =uid. We assume that its stress response has the following linear form: T = Cgb D:
(40)
where T = − c with the anticipation that the threshold for the viscous =ow of inter-grain layers being the same as the one to liquefy the triple junctions. In order to estimate its in=uence, we regard the grain boundary as the network-like matrix phase with grains embedded in it. On the other hand, the deformation due to the clusters of grains, with the incorporation of mass diFusion and grain boundary sliding can be presented as T = Cgrain D:
(41)
We model the nano-grained polycrystal as a two-phase composite with a 3-0 connection. The grains are disconnected, while the inter-grain layers are connected in all directions. The equal-axed grains can be approximated by spherical inclusions. That enables us to use the Hashin-Shtrikman bound (Hashin, 1962; Hashin and Shtrikman, 1962a, b, 1963) for the eFective creep coe7cient C of the nano-grained materials in terms of Cgb and Cgrain : (1 − Vf )(Cgrain =Cgb − 1) C =1+ ; (42) Cgb 1 + 25 Vf (Cgrain =Cgb − 1) where Vf denotes the volume fraction of the inter-grain layers. Two extreme cases are C = Cgb for Vf = 1 and C = Cgrain for Vf = 0. 5. Concluding remarks In this paper, we regard the insertion process and the rotation process as two competing plastic deformation mechanisms. The orientation of the grain clusters changes with the deformation of the nanocrystalline materials. The percentage of insertion or rotation clusters among all operating clusters is determined by the plastic dissipation = (WrdiF + Wrs )=(WidiF + Wis ). Two critical values ln(16=3)=ln(4=3) and ln 3=ln 12 separate the mechanism map (Fig. 6) in three parts. The magnitude of the stress attributes
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little to the deformation mechanism. In the plane strain case, we get the linear relation between the creep rate and the stress. In the three-dimension model, only the case of uniaxial tension is treated in detail and a similar result is derived. The critical angles for the transition from insertion to rotation are almost the same for the plane strain and the three-dimensional cases. The Hashin–Shtrikman bounds provide us an estimate for the in=uence of the deformability of grain boundaries.
Acknowledgements This work is supported by the National Science Foundation of China under Grants 10120202 and 90205023.
Appendix A A.1. Calculation of creep coe:cients The speciHcation of the constants WidiF ; Wis ; WrdiF ; Wrs ; Wrdis as deHned in the Section 2, are summarized below, Yang and Wang (2003): WidiF =
Wis =
1 3(B 2 Ds √ 2 ln 3
WrdiF =
Wrs =
1 m2 KB TD2 # "n 2 √ nDV + DB i=1 =li ln 3 !
1 m2 KB TD3 # "n √ 2 nDV + DB i=1 =li ln 2= 3 !
(A.1a)
(A.1b)
(A.1c)
1 3(B 2 Ds √ 2 ln 2= 3
(A.1d)
1 √ ln 2= 3
(A.1e)
Wrdis = )Tg
√ √ in √ which = 8=3 3 and ) = 1 − =2 3 for the hexagonal grain, and = 6=; ) = 1=2 − 3=16 for the tetrakai-decahedral grain. Other parameters have the same meaning as in Yang and Wang (2003). The characteristic values, m; n; li ; s, can be found from Table 1 in Yang and Wang (2003) for the plane strain model and listed in the table for the three dimensional model. For both the plane strain and three-dimensional cases, the creep coe7cients of the insertion process, the rotation process and the cooperative
H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173
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process are given by
3(B Ds2 m2 KB TD2 16 #+ ! √ "n ; Ci = √ (3 3 + 27)(ln 3)2 nDV + DB i=1 =li m2 KB TD2 3(B 2 2 ! #+ √ "n Cr = √ Ds 3(ln(2= 3))2 nDV + DB i=1 =li
and
(A.2b)
−1
Cco =
(A.2a)
2:4 9 √ + 16C r + 3 3 Ci
:
In these expressions, = 1 is for the plane strain case, and = dimensional case. The threshold stress is c ≈ , Tg √ ; :
(A.2c) 5 2
for the three-
(A.3)
where , = 0:61 for the plane strain case and , = 0:50 for the three-dimensional case. Appendix B B.1. Numerical scheme to determine cooperative ratio () by PMPW The speciHc form of () relies on the maximization of the functional deHned as ˜ The plastic work can be written as H (()) = : D. 6 =6 H () = cos(2)[()p()di () 0 +(1 − (=6 − ))p(=6 − )dr ()] d:
(B.1)
Eq. (B.1) can be discretized as N +1
H () =
( cos(2i )[(i )p(i )di (i ) N +1 i=0
+(1 − (=6 − i ))p(=6 − i )dr (i )];
(B.2)
where i = i=6(N + 1) and = [(1 ); : : : ; (N )]T . Apparently one has the boundary conditions of (0) = 1 and (=6) = 0. An optimization algorithm can be constructed under max H ();
∀ ∈ [0; 1]N :
(B.3)
A typical result is shown in Fig. 12 for = = 1 where an initial linear distribution of () (dashed line) is assumed. Extensive numerical tests for other stress ratios and
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H.-T. Wang, W. Yang / J. Mech. Phys. Solids 52 (2004) 1151 – 1173 1.0
Result Initial Value
α(θ)
0.8 0.6
λ=1 ξ=1
0.4 0.2 0.0 0.0
0.2
0.4
θ
0.6
0.8
1.0
Fig. 12. Optimizing result of ().
Fig. 13. Mass diFusion in the insertion process.
dissipation ratios suggest that () is approximately a step function: 1 0 6 ¡ 0 ; () = 0 0 6 ¡ =6
(B.4)
and this Hnal step-like curve is irrelevant to the initial distribution. Appendix C C.1. Estimates for dissipation processes in three-dimension model To quantify the dissipation processes under three-dimensional model, we start with a grain shape that is approximately spherical. The second graph in Fig. 10 indicates that
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the meta-stable grain shape in the insertion process can be roughly treated as a cube rotated 45◦ in its plane that reserves the volume of the sphere. The matters diFuse along the paths depicted Fig. 13 to accomplish the insertion process. In the rotation process, the atoms lying between the tetrakai-decahedral grain and its inscribed sphere change their assembling states from ordering to disordering or vice versa. In order to make up for the gap or overlap caused by the grain rotation, the disordered mass is required to diFuse along the surface of the inscribe sphere for an arc path that spans an angle of =6 with respect to the grain center. These assumptions lead to various parameters in Table 1 to characterize the dissipation processes during the deformation. References Ashby, M.F., Verrall, R.A., 1973. DiFusion-accommodated =ow and superplasticity. Acta Metall. 21, 149–163. Cai, B., Kong, Q.P., Lu, L., Lu, K., 1999. Interface controlled diFusional creep of nanocrystalline pure copper. Scr. Metall. 41, 755–759. Gao, P., Gleiter, H., 1987. High resolution electron microscope observation of small gold crystals. Acta Metall. 35, 1571–1575. Gryaznov, V.G., Polonsky, I.A., Romanov, A.E., Trusov, L.I., 1991. Size eFects of dislocation stability in nanocrystals. Phys. Rev. B 44, 42–46. Hahn, H., Padmanabhan, K.A., 1997. A model for the deformation of nanocrystalline materials. Philos. Mag. B 76, 559–571. Hashin, Z., 1962. The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150. Hashin, Z., Shtrikman, S., 1962a. On some variational principles in anisotropic and non-homogeneous elasticity. J. Mech. Phys. Solid 10, 335–342. Hashin, Z., Shtrikman, S., 1962b. A variational approach to the theory of the elastic behavior of polycrystals. J. Mech. Phys. Solid 10, 343–352. Hashin, Z., Shtrikman, S., 1963. A variational approach to the theory of elastic behavior of multiphase materials. J. Mech. Phys. Solid 11, 127–140. Lu, L., Sui, M.L., Lu, K., 2000. Superplastic extensibility of nanocrystalline copper at room temperature. Science 287, 1463–1466. Milligan, W.W., Hackney, S.A., Ke, M., Aifantis, E.C., 1993. In situ studies of deformation and fracture in nanophase materials. Nanostruct. Mater. 2, 267–276. Thomas, G.J., Siegel, R.W., Eastman, J.A., 1990. Grain boundaries in nanophase palladium: high resolution electron microscopy and image simulation. Scripta Metall. Mater. 24, 201–206. Valiev, R.Z., Alexandrov, I.V., Zhu, Y.T., Lowe, T.C., 2002. Paradox of strength and ductility in metals processed by severe plastic deformation. Mater. Res. 17, 5–8. Wang, Z.L., Zhang, Z., Liu, Y., 2002. Transmission electron microscopy and spectroscopy. In: Wang, Z.L., Liu, Y., Zhang, Z. (Eds.), Handbook of Nanophase and Nanostructured Materials—Characterization. Tsinghua University Press and Kluwer Academic/Plenum Publishers, Beijing, pp. 29–98 (Chapter 2). Yang, W., Hong, W., 2002. Numerical simulation for deformation of nano-grained metals. Acta Mechanica Sinica 18, 506–515. Yang, W., Wang, H.T., 2003. Mechanics modeling for deformation of nano-grained metals. J. Mech. Phys. Solids, doi:10.1016/j.jmps.2003.07.003