Grain-Size Dependent Mechanical Behavior of Nanocrystalline Metals
Grain-Size Dependent Mechanical Behavior of Nanocrystalline Metals
Grain-Size Dependent Mechanical Behavior of Nanocrystalline Metals
Review article
art ic l e i nf o a b s t r a c t
Article history: Grain size has a profound effect on the mechanical response of metals. Molecular dynamics continues to
Received 18 June 2015 expand its range from a handful of atoms to grain sizes up to 50 nm, albeit commonly at strain rates
Accepted 24 July 2015 generally upwards of 106 s 1. In this review we examine the most important theories of grain size de-
Available online 31 July 2015
pendent mechanical behavior pertaining to the nanocrystalline regime. For the sake of clarity, grain sizes
Keywords: d are commonly divided into three regimes: d4 1 μm, 1 μm od o100 nm; and d o100 nm. These dif-
Nanocrystalline metals ferent regimes are dominated by different mechanisms of plastic flow initiation. We focus here in the
Molecular dynamics region d o 100 nm, aptly named the nanocrystalline region. An interesting and representative phe-
Hall Petch nomenon at this reduced spatial scale is the inverse Hall–Petch effect observed experimentally and in
Inverse Hall Petch
MD simulations in FCC, BCC, and HCP metals. Significantly, we compare the results of molecular dy-
namics simulations with analytical models and mechanisms based on the contributions of Conrad and
Narayan and Argon and Yip, who attribute the inverse Hall–Petch relationship to the increased con-
tribution of grain-boundary shear as the grain size is reduced. The occurrence of twinning, more pre-
valent at the high strain rates enabled by shock compression, is evaluated.
& 2015 Elsevier B.V. All rights reserved.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2. Physical polycrystalline models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.1. Hall–Petch relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.2. On the exponent in the Hall–Petch relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.3. Inverse Hall–Petch Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.4. Special dislocation configurations in nanocrystalline metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3. Molecular dynamics fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.1. Molecular dynamics foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2. Defect identification methods for crystalline materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.1. Centro-symmetry parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2.2. Common neighbor analysis (CNA) and parameter (CNP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2.3. Bond angle analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.2.4. Dislocation extraction algorithm (DXA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.2.5. Orientation imaging map (OIM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.3. Interatomic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4. Polycrystalline sample construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4. Molecular dynamics simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1. Face-centered cubic metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2. Body-centered cubic metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3. Hexagonal close-packed metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
☆
This article is an invited contribution for the 2014 MSEA Journal Prize. Prof Marc Meyers is the Winner of the 2014 MSEA Journal Prize, an annual prize to recognize
research excellence in the field of structural materials, with special consideration for those who have made outstanding contributions to the Journal.
n
Corresponding author.
E-mail address: mameyers@eng.ucsd.edu (M.A. Meyers).
http://dx.doi.org/10.1016/j.msea.2015.07.075
0921-5093/& 2015 Elsevier B.V. All rights reserved.
102 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
kCC ΩμbDGb ⎛⎜ δ ⎞⎟ ⎛ b ⎞ ⎛ τ ⎞
3
γ̇ = ⎜ ⎟ ⎜ ⎟
kT ⎝ b ⎠⎝ d ⎠ ⎝ μ ⎠ (2) Fig. 1. Experimentally observed decrease of the Hall–Petch slope in the nano-
crystalline domain and comparison with Meyers–Ashworth model [9]. Experi-
where δ is the grain-boundary thickness, DGb is the grain- mental results from Abrahamson [7] and Malloy and Koch [8].
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 103
Fig. 2. Plots showing the trend of yield stress with grain size for different metals as compared to the conventional Hall–Petch response: (a) copper, (b) iron, (c) nickel and
(d) titanium. From Meyers et al. [13].
deformation behavior at the nanoscale. The complementary con- grain rotation, will be made through atomic-scale simulation and
clusions of these reviews indicate that the capacity of molecular modeling. Simulations deriving from atomic and quantum models
dynamics simulations to directly visualize defects with atomic extend their reach by providing valuable input criteria for multi-
resolution provides a utility unmatched by experimental char- scale models, continuum models, and materials design [24–32]
acterization, but that conclusions must be tempered by experi- especially in iterative feedback loops [33].
mental results. As computation power climbs and cost plummets, In order to clarify and highlight the physical mechanisms re-
it is to be expected that fundamental insights into the structure sponsible for the effects of these grain sizes, we present analytical
and properties of crystalline defects, as well as physical mechan- models predicting such behavior first: Conrad and Narayan
isms ranging from atomic diffusion to interface migration and [34,35], and Argon and Yip [36]. We also discuss the equations of
Fig. 3. (a) Model of nanostructured material showing crystals only a few nanometers or less in diameter. Boundary atoms in the two-dimensional sample are white circles
(from early work of Gleiter [249]). (b) Model of grain boundary showing facets (from [250]). (For interpretation of the references to color in this figure, the reader is referred
to the web version of this article.)
104 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
Fig. 4. The increase in the volume fraction of grain boundaries and triple junctions as a function of grain size in the nanocrystalline ( o 100 nm) and ultrafine grain (100 nm
to 1 mm) regimes. These plots are based on space-filling tetrakaidecahedra grains with a grain boundary thickness of 1 nm (thick line), where the dotted lines show the
evolution for grain boundary thicknesses of 0.9–0.5 nm in increments of 0.1 nm. From Tschopp et al. [17].
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 105
πd2m/2 3m
ρ= = ·
πd3/6 d (5)
being of the same sign, the stress field resulting from a summation 2.2. On the exponent in the Hall–Petch relationship
over all such dislocations push each toward a common grain
boundary, generating a stress concentration at the “tip” of the Substantial deviations observed in the Hall–Petch relationship
pileup located adjacent to the grain boundary. The equilibrium actually suggest that other relationships are prevalent. In 1956,
positions of linear free dislocations were obtained by Eshelby et al. Baldwin [67] found that in many cases the exponent 1 or 1/3
in 1951 [56]. Fig. 5 shows a simple pileup and estimates for re- provided as good a fit as 1/2. More recent analyses, such as the
lationships between the length of the pileup and dislocation one by Dunstan and Bushby [68], point to the same. Consistent
density. Later, more complex arrangements of dislocations were with the results of Fig. 1, which show a reduction of the prefactor
considered by Lubarda et al. [57]. Fig. 6 demonstrates just such a from 0.48 to 0.21 MPa m1/2, Durstan and Bushby [68] report a
complex pile-up forming in a single grain roughly 30 nm in dia- decrease from 0.52 to 0.054–0.14 MPa m1/2 for the nanocrystalline
meter as simulated by Shiotz [58]. domain, in iron. A more general manner to plot grain size de-
An upper limit to the length of the pileup is l¼ 2nA/sb (refer to pendent strength data is in a double logarithmic form:
Sezgo [59]), where l is related to half the grain size, d/2. The sum of log
σ
σ1
stresses at the tip, stip ¼ns. Taken together and assuming yielding −m = d
⋅
occurs at a critical value we arrive at the classic Hall Petch re- log d1 (7)
lationship:
where s1 and d1 are reference points on plot and m is the slope
4σc A −1/2 of the linear fit. Adding a frictional stress expressing the resistance
σy = σ 0 + d = σ 0 + kd−1/2,
b (3) of the lattice in a single crystal:
that grains in the nanometer regime will have at least 1% porosity explanation for the Hall–Petch phenomenon; as grain size is re-
for 25 nm grains and 5% effective porosity for 5 nm grains. It has duced, the number of dislocations associated with a given grain
also been shown that elastic modulus decreases with increasing and its boundaries is reduced and the summative contribution to
porosity [13,74]. Li [4,75] related impurity content of the grain the stress field is diminished. Yet, although the stress field asso-
boundary (theorizing that impurities stabilize ledges) to porosity ciated with a single grain is diminished, the influence of stress
on the basis that equilibrium segregation decreases with increas- fields emanating across neighboring grains provides sufficient
ing porosity by decreasing internal stresses. Li's equation in- motivation for relaxation of grain boundaries and triple junctions
corporating impurity segregation and effective porosity is by grain boundary sliding [79]. A compilation of yield stress data
x − (ϕ − ϕ0 ) taken by Meyers et al. [13] for FCC Cu indicates that the critical
σ = σ0 + A ⋅ grain diameter for this transition in deformation mechanism is
d/3KNβb (9)
right under 20 nm (Fig. 7).
A is a fitting constant akin to the Hall–Petch slope, x is a fraction As mentioned earlier, the first experimental evidence of the
of possible impurity sites occupied (this term is grain size de- inverse Hall–Petch response in the nanocrystalline regime was by
pendent and defined as 0o xo1), K is the equilibrium constant, Chokshi et al. [12] when it was reported that a negative Hall–Petch
and Nβb is a concentration of impurities per unit of grain boundary, slope was observed for nanocrystalline copper (FCC) and palla-
ϕ is porosity and ϕ0 is equilibrium porosity. For ϕ0 = ϕ , and x dium (FCC).
approaching zero for small grain sizes, a Hall–Petch type plot will Conrad and Narayan [34,35], and Conrad [80] developed a
decrease in slope before plateauing in strength. This type of effect model based on independent atomic shear events and related the
has clear experimental evidence as discussed earlier and shown in shear rate at grain boundaries to the global strain rate. They ob-
FigS. 1 and 2. tained the following equation:
Using static values of excess porosity Li [4,75] demonstrated a
2δνD ⎛v τ ⎞ ⎛ ΔG 0 ⎞
turnover in strength below critical grain sizes. Carlton and Ferreira γ̇ = sinh ⎜ D e ⎟ exp ⎜ − ⎟,
d ⎝ kT ⎠ ⎝ kT ⎠ (10)
[76] produce a similar relationship including a term incorporating
increased dislocation adsorption for small grain sizes. Both models where δ is the grain boundary width, taken as 3b, and
imply that dislocations remain the primary carrier of plasticity τe = τ − τcr . The term 2δνD represents the volume of a grain bound-
d
down to the smallest grain sizes. Many studies indicate that dis-
ary and the area it sweeps over during shear.
locations continue to play an important role in small grain sizes
Alternatively, the shear stress τ is expressed as
such as strain hardening in 20 nm Ni [77] and that dislocation-
mediated grain boundary rotation occurs at smaller grain sizes ⎡ kT ⎛ δν ⎞ ΔF ⎤ kT
τ = τ0 + ⎢ ln ⎜ D ⎟ + ⎥+ ln d
[78]. The former is an example of traditional intragranular plasti- ⎣V ⎝ γ̇ ⎠ V ⎦ V (11)
city while the latter is a primary example of deformation origi-
nating from intergranular sources and alternative deformation The symbols not previously defined are V, the activation vo-
mechanisms at a reduced size. lume; ΔF, the Helmholtz free energy; νD, the vibrational frequency
of an atom (Debye frequency, 1013 s 1, a first order approx-
2.3. Inverse Hall–Petch Models imation can be taken as the time taken for sound waves to travel
one interatomic distance), and τ0, a frictional stress. Fig. 8a shows
The “negative” or “inverse” trend in yield strength behavior the prediction from the Conrad–Narayan equation together with
with decreasing grain size results from a scale-determined inter- experimental results for copper. The transition from positive to
ruption of dislocation phenomenon associated with the traditional inverse HP relationship is clear and the line, which is from Eq. (11),
Fig. 7. Compilation of yield stress as a function of grain size (plotted by d 1/2) for Cu. The plot illustrates the Hall–Petch relationship for larger grain sizes and the ambiguity
near d ¼25 nm (d 1/2 ¼ 0.2) where some results show a plateau in strength and other results show a decrease indicative of the inverse Hall–Petch relationship [13].
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 107
b ⎡ ΔG 0 ⎛ τ ⎞⎤
γḊ = νd exp ⎢ − ⎜ 1 − ⎟ ⎥,
d ⎣ kT ⎝ τ0 ⎠ ⎦ (13)
n
b ⎛ τ⎞
γḊ = νd ⎜ ⎟
d ⎝ τ0 ⎠ (15)
4πR2δ 3δ 6δ
Vf = 4
= =
πR 3 R d (16)
3
Fig. 11. (left) Thermodynamic stability map for stabilizing nanocrystalline copper at 0.6Tm; the red and black dots indicate the stabilizing and nonstabilizing solutes with
respect to the elastic enthalpy and enthalpy of mixing. (right) Molecular dynamics simulations at 1200 K indicate that nanocrystalline Cu rapidly coarsens [254] while Ta
precipitates in the Cu–5%Ta sample stabilize the surrounding nanocrystalline grain size. From Tschopp [17].
Rapaport [88] extensively covers the practical applications of the Plimpton and Gale [94] based on the well-received LAMMPS code
method and tools available towards undertaking simulations. developed in 1995 by Plimpton [95].
The focus of this review is on the mechanical behavior of nano- Finally, a number of techniques have surfaced over the years for
crystalline metals, which is directly related to the evolution of pre- simulating and evaluating nanostructured materials. A constant
existing defects and those arising from imposed loading on the ma- quest to scale up the number and diameter of grains as well as our
terial. The fundamentals of defects in solids are very well covered in ability to accurately model them through interatomic potentials
classic books on the matter [89,90] and the computer simulation of has stimulated research in this area of computational materials
dislocations and other defects is a topic treated in an number of in- science. Computational advances have principally thrust molecular
fluential papers [91,92]. Fortunately, the book by Bulatov and Cai [93] dynamics simulations to growing level of importance.
gathers many important methods into a single text.
Non-trivially, the implementation of broadly applicable com- 3.2. Defect identification methods for crystalline materials.
munity codes for materials modeling and simulation has been a
boon to the success of atomistic computational modeling as As continuously enumerated, defects play a unique role in
highlighted by a recent opinion article on community codes by materials phenomena and, therefore, their proper identification is
Fig. 12. Visualization of a dislocation traversing a nanocrystalline grain by six different analysis coloring schemes. From Swygenhoven [92].
110 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
3.2.1. Centro-symmetry parameter Fig. 14. ABC sequence of FCC {111} planes. 12 nearest neighbor atoms surrounding
The centro-symmetry deviation parameter, commonly named the a center atom, with the 3 dashed atoms belonging in plane A (red), the 7 atoms
centrosymmetry parameter (CSP), is a robust method that relies on a belong in plane B (green) and the last 3 dotted atoms belong in plane C (blue).
characteristic that is common to simple cubic (SC), FCC and BCC
structures: every atom is a center of inversion symmetry, which
means that taking an atom as a center, its neighboring atoms are The identification of defects with this method is affected by
(centro)symmetric relative to it. This property can be used to distin- elevated temperatures, as shown by Stukowski [99].
guish these atoms from other structures when the local bond sym-
metry is not verified or it deviates from an established value. 3.2.2. Common neighbor analysis (CNA) and parameter (CNP)
This metric for structural identification was developed by Albeit a higher computational cost compared to CSP, structure
Kelchner et al. [97] in equation form and practical applications can analysis algorithms that employ high-dimensional signatures to
be found in several references [93,98]. characterize atom arrangements are usually more efficient to
discern between structures. The Common Neighbor Analysis (CNA)
N /2 2
→ → is one of these methods. Proposed by Honeycutt and Andersen
CSP = ∑ Ri + Ri + N /2
i=1 (21) [100] and later further developed by Faken and Jonsson [101] and
Tsuzuki, Brancio, and Rino [102], the CNA computes a character-
Here the N nearest neighbors, specified as an input parameter, istic signature from the topology of bonds that connect an atom to
→ →
of each atom are identified such that Ri and Ri + N/2 are the vectors its surrounding neighbors.
from the central atom to a given pair of opposing neighbor atoms. The neighborhood of an atom is defined by a cutoff distance so
Opposite pairs of atoms for FCC, BCC, and HCP are indicated in that all the atoms within that distance are said to be neighbors.
Fig. 13 by atoms of the same color. For an atom sitting on an ex- Each neighbor is taken into account for in the calculation of three
pected lattice point the CSP determined by this sum will be zero. characteristic numbers that are computed, yielding a triplet that,
Thermal vibrations will not cause much fluctuation from 0, but when compared with a set of reference signatures, allows for the
defects that break symmetry will produce a larger (positive) CSP establishment of a structural type to the atom whose triplet is
value. Fig. 14 shows a schematic representation of a central atom evaluated.
in FCC surrounded by 12 closest neighbors. They are diametrically Unlike CSP, CNA can be used on non-centro-symmetrical
opposed in pairs. Note that BCC structures have 8 neighbors structures such as HCP crystals. The latter are centrosymmetric
(though the second shell is often considered totaling 14) and SC only if the c/a ratio is ideal, 1.633 (the metal that comes closest to
have 4. We would like to highlight some characteristics of CSP: ideal HCP is magnesium). To see how one of the triplets is com-
puted, a representative Common Neighborhood Parameter (CNP)
As seen in its definition, the CSP parameter is just a scalar can be defined as:
quantity and therefore, its applicability to oriented defective 2
ni nij
structures is limited. 1
The parameter is not suitable for treatment of HCP, diamond
Qi = ∑ ∑ ( Rik + Rjk )
ni j=1 k=1 (22)
cubic (DC) and some other structures that do not have the
symmetrical characteristic described before. The index j evaluates the ni nearest neighbors of atom i, and the
Fig. 13. Centro-Symmetry Parameter (CSP) distinguishes between plastically deformed regions of dislocations and stacking faults (asymmetry) from purely elastically
deformed regions (which would have symmetry). Pairs of atoms diametrically opposed from a central atom are identified by the same color; (a) FCC, 6 pairs; (b) BCC, 7 pairs;
(c) HCP, 6 pairs [102].
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 111
Fig. 17. Orientation imaging map using Euler angles as proposed by Rudd [109].
Fig. 15. Common Neighbor Analysis (CNA) provides a measure of degree of crystal-
linity and identification of crystal structures present. Analysis is derived based on the another. Two new tools by have been developed by Stukowski
number of common neighbors (k) shared by an atom pair (i-j):k1, k2, k3, and k4[102].
[99,104,105]. The ‘sharpness’ of dislocations is seen in Fig. 16 in
contrast with the CNA filtering. This enables a better determina-
index k evaluates nij common nearest neighbors of atom i and
tion of dislocation densities, since no dislocations are missed by
atom j. This is visually represented in Fig. 15 where k atoms are
superposition as shown by Ruestes et al. [106]. The voids from
common neighbors to atom i and atom j.
The Adaptive Common Neighbor Analysis (a-CNA), recently which these dislocations emanate are visible in DXA but cannot be
proposed by Stukowski [99] takes CNA as a basis, and is particu- distinguished by CNA.
larly suitable for multi-phase systems, adapting the cutoff distance
of the standard CNA to each individual atom depending on a re- 3.2.5. Orientation imaging map (OIM)
ference structure for comparison purposes. The reader is referred Typical materials exhibit some degree of texture (non-random
to the cited article for a thorough explanation and example of the grain orientation distribution) imparted by processing or past
methodology. We note that similar structures may not be ade- deformation. Twinning is one such deformation mechanism that
quately identified with respect to one another by methods utiliz- can impart texture by preferentially adjusting the orientation in
ing characteristic signatures. soft grains.
The first integration of an orientation imaging map to atomistic
3.2.3. Bond angle analysis simulations was accomplished by Rudd [107] and can be seen in
Developed by Ackland and Jones [103] and with a computa- Fig. 17. Ravelo et al. [108] also demonstrate an OIM mapping
tional cost in the order of the CNA, this method is able to distin- function in the application of the SPaSM code. The foundation of
guish FCC, BCC, and HCP structures effectively. The authors have the method lies in a centro-symmetry-like formulation where
made important efforts in showing the efficiency of this method nearest neighbors are located for each atom and is summarized
for treatment of HCP structures and its comparison with the ap- below for the case of BCC structures:
plication of CNA to HCP structures is worth reading.
(i) Find nearest 8 neighbors for each BCC atom and create op-
3.2.4. Dislocation extraction algorithm (DXA) posing pairs to form the family of o111 4 directions.
Detection of defects from crystalline structures is valuable, but (ii) Take cross products, e^022
¯ = e^111 × e^111 ^ × e^¯
¯ / e111 111 and
the current state of the art is the distinction of one defect from e^422
¯ = e^022 ^ /|e^ ¯ × e^ |.
¯ × e111 022 111
Fig. 16. Comparison of (a) a conventional atomistic visualization using CNA filtering in tantalum and (b) a geometric line visualization of the dislocations provided by the
DXA. In addition to extracting the dislocation line network, the DXA analysis also produces a geometric representation of non-dislocation defects, such as void surfaces,
visible in the center of box [106].
112 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
Table 1
Common Potentials for different Metals and Structures.
Fig. 19. (a) Different grains designated by different colors as generated by Voronoi
tessellation, not the straight lines indicative of the geometric relation between Fig. 21. Snapshot of atomic scale stresses shown along the solid black line tra-
grain centers. (b) Grain-boundary atoms in light blue identified by CSP after re- versing multiple grains after Voronoi construction but before thermalization. (top)
laxation [255]. atoms colored by von Mises stress. One fourth of the residual stress averaged over
the entire sample shown in red. Intergranular stresses averaged over each grain
shown in blue. Middle pane shows distribution of von Mises stress and bottom
3.4. Polycrystalline sample construction pane shows hydrostatic stress distribution [120].
Fig. 19a shows a box containing approximately 100 grains, configuration was generated by Voronoi tessellation and the grain-
which are identified by different colors. This structural grain boundary atoms, identified by CNA, are shown in light-blue in
Fig. 20. Vertex growth method (left) vs. Voronoi Tesselation (right). Refer to Wolf et al. [23] for alternatives to Voronoi tessellation.
114 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
Fig. 19b, whereas grain interiors are dark blue. One characteristic
of the Voronoi tessellation is that all boundaries are flat surfaces.
The differences between actual and simulated grain structures are
still significant. For instance, annealing twins and other low energy
boundaries (Σ3, Σ5, Σ7, Σ11, etc) that are prevalent in low stacking
fault energy FCC metals, do not have their higher propensity well
recreated in simulations because of how orientation distributions
are prescribed. In Voronoi tessellation, the orientations of the
generated grains are usually prescribed though a random dis-
tribution of orientation directions, then taking on a McKenzie
distribution of misorientation as prescribed by crystal symmetry.
This does not correspond to actual polycrystals, where the low-
energy grain boundaries are more stable and orientation re-
lationships between neighbors evolve accordingly.
Varying simulated fabrication methods for producing grain-
boundary networks will manifest just as varying experimental
manufacturing techniques, through processing dependent me-
chanical properties. Wolf et al. [23] provides a description of var-
Fig. 23. Snapshot of atomic structure at an asymmetric grain boundary viewed
ious in silico polycrystalline fabrication techniques such as the along the tilt axis [110] The open and filled circles indicate atoms occupying al-
vertex growth method, Voronoi tessellation, and compaction of ternating (220) planes. Arrows indicate faceting with (001)/(111) plane matching.
monocrystalline spheres by presuming assembly followed by The dashed lines indicate intrinsic stacking faults that originate from the junctions
of the nanofacets [256].
heating. A comparison between the first two methods can be seen
in Fig. 20. Nanocrystalline structures have also been fabricated by
such as polycrystalline aggregates[123]. Theirs is the principal
simulated laser irradiation [119]. Guo et al. [120] investigated in-
work that investigates the statistical representation of grain size
ternal stresses in polycrystals, namely the residual stress of as-
and structure distributions that does not draw as much attention
fabricated nanocrystalline Cu.
as deformation mechanism research. Through directed Monte
The Voronoi tessellation creates internal stresses that are of a high
Carlo studies they were able to recreate realistic grain textures.
magnitude, especially in nanocrystals below 25 nm grain sizes [120].
The ability to identify and track triple junctions, grain boundary
Fig. 21 shows the variation of von Mises and hydrostatic stresses
along a horizontal line drawn across three grains. The average von ledges, and faces is seen in Fig. 25. The triple junctions are imaged
Mises stress is 7 GPa and is indicated by the red line referred to the in red, the grain-boundary intersections are seen in yellow, and
right hand side scale (sIvm). The variation in the stress is given by the the grain boundaries in green. This work is due to Xu and Li [121],
individual dots (.) that are labeled siv and Li and Xu [122]. With such information the authors have ac-
vm. These vary from 0 to 15 GPa
(left-hand scale). The same can be seen for the hydrostatic stresses, cess to a complete picture of polycrystallinity including grain
except that they have positive (tension) as well as negative (com- boundary shape, texture, and grain size (as well as its distribution).
pression) values. This shows that annealing (thermalization) of the Readers are directed to their work for a description of accurate
Voronoi construction is essential before plastic deformation. polycrystal construction and appropriate methodology.
Xu and Li [121], and Li and Xu [122] investigated the appro- It was recognized earlier that Voronoi constructions may not
priateness of Voronoi tessellation to represent physical materials generate equilibrium structures and larger grain sizes were often
generated through long annealing processes. Farkas et al. [124]
reported three-dimensional simulations of grain growth beginning
with 5 nm grain sizes and were able to generate larger grains and
possibly more realistic distributions while also obtaining the ac-
tivation energy for annealing-twins and grain-boundary motion.
Fig. 22 shows fivefold twin formation during an annealing simu-
lation of nanocrystalline Cu as directly compared to experimental
evidence of such star shaped twins in nanocrystalline NiW [125] as
well as earlier studies by Zhu et al. [126] that found fivefold twins
formed in nanocrystalline Cu by high-pressure torsion.
After thermalization, the grain-boundary structure can become
faceted as non-equilibrium grain boundaries reach lower energy
configurations. This is a well-known fact and explains the facets
Fig. 22. High resolution transmission electron microscopy image of many-fold twin
formation in nc NiW. Solid lines are twin boundaries and dashed lines are grain
boundaries. A time sequence of formation during molecular dynamics simulation Fig. 24. Special grain boundary configuration indicated by the “E” structural unit
can be found in the original article by Bringa et al.[125]. and transition to the “C” structure after dislocation transmission [53].
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 115
commonly observed as non-coherent segments in coherent an- 4.1. Face-centered cubic metals
nealing twin boundaries. Thus, for the FCC superalloy Inconel 600
the majority of boundaries are Σ3, Σ7, and Σ13 [127]. For grain Early work by Van Swygenhoven et al. [63,130,131] detailed
boundaries to reach a special configuration, both the grains and nucleation, transmission, and absorption of a partial dislocation as
the boundary have to be properly oriented. Fig. 23 shows two shown in Fig. 26. The sequence shows the emission of partial
grains that are at an orientation close to a special coincidence site dislocations (at the upper left-hand corner) and their progression
through the grain and eventual annihilation on the bottom grain
lattice. The black and white circles designate atoms at two ad-
boundary. The stacking-fault between leading and partial dis-
joining atomic planes. The faceting of the boundary is evident
locations is indicated in red (darker gray in printed version). More
from the polygon sequences drawn along the boundary. The grain-
recent MD simulations have shown that dislocations are emitted
boundary nature is important in the generation of dislocations.
from grain-boundary ledges at a lower stress than from planar
The figure shows two dashed lines which indicate stacking faults
grain boundary regions [26]. Direct evidence of intragranular de-
that were emitted from the nanofacet junctions. These are also
formation mechanisms was observed by TEM [132,133] and Fig. 27
referred, in the older literature, as ledges. Fig. 24 shows two slip
shows that partial dislocations are preferred to full dislocations
systems in neighboring grains intersecting a boundary defined by
below a specific grain size [78]. Van Swygenhoven and co-workers
a polygon sequence designated structural unit E. Upon transmit- [79] also identified varying grain boundary structures of “typical”
ting from one grain to the other, a dislocation changes the local grain boundaries and quantified sliding mechanisms. Haslam,
character of the grain boundary. This is seen by the change of Wolf, Philpot, Gleiter and other coworkers [134] identified the role
polygonal structure from E to C and indicates that certain grain of grain rotation as contributing to plasticity and grain growth via
boundary structures are prone to dislocation transmission while grain coalescence during deformation. Fig. 32 details a visualiza-
others prevent dislocation motion. tion of grain coalescence by mapping the grain dependent [011]
It is clear that a diversity of grain-boundary configurations, or direction of adjacent grains during deformation [135]. This shows
lack there-of, can alter the resulting mechanical properties [128] the first simulated evidence of grain rotation and coalescence. The
and the rapid development of realistic grain-boundary networks is degree of relative misorientation is indicated by solid black line;
critical for well-informed molecular dynamics simulations and the grain-boundary motion rotates each grain to orient with the
continues to be a long-standing goal in the field. One such method other. Thus, a larger grain is formed by the joining of two grains.
to-be-developed is the direct reconstruction of atomistic samples Simulation results by Trautt and Mishin [136] verified predictions
from experimental imaging analogous to DREAM3D [129] for made by Cahn and Taylor [137] regarding dynamics of grain
three-dimensional characterization of microstructural inputs boundary motion. Their results indicate that grain-boundary
Fig. 26. Early work by Swygenhoven et al. detailing nucleation, transmission, and absorption of a partial dislocation [257]. Leading partial emitted from top left-handed
corner propagating towards bottom right hand corner, followed by trailing partial dislocation. Stacking fault between them in red.
116 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
Fig. 27. (a) Full dislocations (marked with ‘T’) in an 11- nm-sized grain. (b) Partial dislocations resulting in stacking faults (as noted by the arrows) in a 7- nm-sized grain.
Scale bars, 2 nm. TB indicates a twin boundary. From Wang et al. [78].
τ ∝ ρ1/2 (24)
Fig. 29. (a) MD simulations of flow and yield stress versus strain for four different grain sizes in Cu. Each data point is the average of seven simulations. The yield stress
decreases with decreasing grain size, resulting in an inverse Hall–Petch effect. The maximal flow stress is taken as the stress at the flat part of the stress–strain curves; the
yield stress is defined as the stress where the strain departs 0.2% from linearity.) [216] (b) stress-strain curves for grain sizes from d ¼7.5 to 45 nm in nanocrystalline copper;
(c) flow stresses as a function of grain size showing inverse HP region[53].
Fig. 31. Stress-strain curves for three strain rates, 107 s-1, 108 s-1, and 109 s-1. Inset in (a) shows discrete dislocation stress drops associated with individual dislocation
propagation within grains at 107 s-1. (b) Dislocations per 1% strain as a function of strain rate for all three strain rates. (c) Cumulative count of cross-slip events normalized by
the number of grains [146].
The initial misorientation between grains 8 and 14 is 18. At in- dictates that motion of the boundary occurs according to the co-
creasing time steps the angle of misorientation decreases from 18° operative displacement. This coupled motion has also been seen in
to 11°, 9°, and 4°. The decreasing areas of grains 8 and 16 indicate asymmetrical tilt grain boundaries [152]. It was also shown by
that rotation-coalescence and grain boundary migration mechan- Brandl et al. [153] that grain boundaries are not necessarily the
isms are coupled to one another [135]. Fig. 33 shows experimental static structures that we think them to be, especially under dy-
in situ TEM imaging of grain rotation and grain boundary annihi- namic loading. Frolov and co-workers [154] showed that grain
lation of 5-7 nm grains [78]. The white arrow in b indicates the boundary structure was critical in determining such coupling
loading axis. In (a), grain 1 is surrounded by high-angle GBs. With factors and the motion of such boundaries. This work builds upon
straining the grain rotates and the boundaries between grains 1-3 a recent investigation that uncovered first grain-boundary struc-
and 1-4 transition to small angle grain boundaries. With increas- tural phase transitions in metallic grain boundaries [155].
ing strain the grain boundary is annihilated and grains 1 and The main impediment to previous observation of such phase
3 coalesce into a single grain. transitions in atomic simulations was the lack of variation in
Coupled grain boundary motion as detailed by Schäfer et al. atomic density within a given boundary. As temperature increases,
[151] can be seen in Fig. 34 where compatibility between grains the grain boundary thickness increases as seen in Fig. 35. This
Fig. 32. Successive snapshots illustrating the atomic scale mechanism of a rotation-coalescence event. Solid black lines indicate o 110 4 direction and the initial mis-
orientation between grains 8 and 14 is 18°. At timesteps b-d the angle of misorientation decreases to 11°, 9°, and 4 °respectively. The decreasing areas of grains 8 and 16
indicate that rotation-coalescence and grain boundary migration mechanisms are coupled to one another [135].
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 119
Fig. 33. Experimental in situ imaging of grain rotation and grain boundary annihilation of 5-7 nm grains. The white arrow in b indicates the loading axis. In a, grain 1 is
surrounded by high-angle GBs. With straining the grain rotates and the boundaries between grains 1-3 and 1-4 transition to small angle grain boundaries. With increasing
strain the grain boundary is annihilated and grains 1 and 3 coalesce into a single grain. Scale bars, 2 nm. From Wang et al. [78].
thickening would have an effect on strength derived from the ratio The combination of strength and ductility translates into
of grain boundaries to grain interiors as introduced earlier based toughness, and this is shown in Fig. 36.
on work by Argon and Yip [36]. This serves as a reminder of the In Fig. 36a, DPD (Dynamic Plastic Deformation) is a treatment
importance of residual “porosity” within grain boundaries and by which a nanocrystalline structure with high twin density is
reemphasizes the need for imparting increasing degrees of realism produced [156]. SMAT (surface mechanical attrition treatment) is a
into simulations. technique by which the surface is hardened through the formation
The limited work hardening ability of nanocrystalline metals is of a nanocrystalline layer, while the interior has a conventional
the direct result of the availability of sinks (the grain boundaries). polycrystalline structure [157–159]. Both processes push the ten-
Thus, specimens tend to undergo necking right after plastic flow. sile strength-elongation envelope to the right, increasing the
This is corroborated by MD simulations. The decrease is ductility ductility at a fixed level of strength. Fig. 36b shows that some
with increasing strength is a characteristic of all metals, and the nanocrystalline structures can have a combination of strength and
ductility that place them significantly to the right of the blue re-
search for increased ductility has been a challenge for researchers
gion, with higher toughness. Special processing procedures can
worldwide. The development of TRIP steels which use a marten-
access this region. Shifting towards the right is also shown in
sitic transformation the decreases the stress concentrations at the
Fig. 36c, in which the combination of polycrystalline and nano-
crack tip, represents a significant advance. Steels subjected to the
crystalline structures leads to enhanced ductility, at a specified
treatment are shown in Fig. 36a as an 'island.' The DPD nano-
strength level. The gradient structure has both a high strength and
crystalline copper have a better performance than the TRIP
the ductility due to the polycrystalline core.
(TRansformation Induced Plasticity) steels. In nanocrystalline In tensile deformation, grain-boundary void formation is a
metals, several developments are worth mentioning: prevalent mechanism and perhaps a peculiarity of the high-
strain rates imparted by MD simulations. The opening of voids
The use of pulsed electroplating to generate a nano-scaled at the grain boundaries is accompanied by profuse emission of
structure consisting primarily of coherent annealing twins. This partial dislocations, which transform a sharp separation into a
work has been spearheaded by Lu and coworkers [156] at IMR, broader void. The simulation sequences are shown in Figs. 37
Shenyang, China. and 38, for uniaxial strain tension and hydrostatic tension re-
Gradient structures in which the surface is nanocrystalline and spectively. The arrows indicate places in the grain boundaries
the interior is polycrystalline are effective in retarding necking where void initiation takes place. The light-blue lines are either
[157–159]. grain boundaries or stacking faults. The simulation views
120 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
Fig. 34. Coupled grain boundary motion in Pd and Cu crystals occurring at a grain boundary of misorientation of about 33 °between grains. Top snapshot shows initial
configuration with marker atoms in red (aligned with mesoscopic slide plane) and green (perpendicular to mesoscopic slide plane), grain boundary atoms in black, and fcc
atoms in light gray. The evolution of the marker lines during loading determines an estimated coupling factor of 0.5 [151].
Fig. 35. The thickness of Σ5(310) GB increases and undergoes phase change at
increasing temperature. The Σ5(310) GB undergoes a transformation at 400 K be-
fore pre-melting near Tm. From Frolov et al. [155]. Fig. 36a. Correlations of tensile strength and ductility. Purple squares are from
Washko and Aggen [258], black square from Truman [259], black and white box
from Chen et al. [260], black triangle from Ucok et al. [261], Stars from Eskandari
et al. [262] and original data from Yan et al. [156]. DPD: dynamic plastic de-
represent slices with thickness equal to a0. The process of ten- formation; CR: cold rolling; SMAT: surface mechanical attrition treatment.
Fig. 37. Slices (a0 thick) of the nanocrystalline copper with d 20 nm subjected to uniaxial tension. Strains are: 6%, 6.5%, 7%, 7.2%, 7.3% and 8%. Lattice atoms: blue; void
surface atoms: red/green; stacking faults, partial dislocations and twin boundary atoms: light blue. Stacking faults show as two planes of atoms while twin boundaries show
as a single atomic plane. (a) Emission of partial dislocations from GBs; (b) void nucleation (marked by an arrow); (c) three voids nucleated at a grain boundary;
(d) coalescence of voids; (e and f) void opening by continued dislocation emission at its extremities. A few twin boundaries are seen in frames (c) and (d). From Bringa et al.
[263].
Setting l, the mean free path, equal to d and assuming that the Orowan equation and from MD predictions in uniaxial compres-
grain boundary contribution is zero, one arrives at: sive strain Orowan-based densities for accommodation of plastic
γT − γE strains γt γe are nearly proportional to grain size 1/d; the role of
ρ= grain-boundary shear decreases with increasing grain size d.
Mbd (25)
Work by Rudd [107,109] indicates that both dislocations and
The results of the MD simulation and Equation 25 are shown in twinning are observed in tantalum during high strain-rate de-
Fig. 44 for a strain of 0.18; they demonstrate that, for d o30 nm, formation. Pan et al. [164,168] observed twins in small grain sizes
the Orowan equation overestimates the dislocation density. The and dislocations in larger grains. Zhang et al. [169] simulated
difference in density is accommodated by grain-boundary shear. o110 4 columnar grains of molybdenum, identifying primarily
This corroborates the results attributing the deformation in the twin deformation, which agrees well with work by Tramontina
inverse Hall-Petch region to grain-boundary shear. From both et al. [170] showing an increased propensity to deform by
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 123
Fig. 38. Slices (a0 thick) of the nanocrystalline copper with d 20 nm subjected to hydrostatic tension. Same color scheme as in Fig. 37. Void nucleation and associated
dislocation strucutres are much more drastic than uniaxial tension. From Bringa et al. [263].
twinning in o1104 single crystals of Ta. Earlier simulations of showed that there is little dependence of sliding on strain
nanocrystalline molybdenum by Frederiksen et al. [171] show a rate.
propensity for both twinning and grain boundary sliding at high Fig. 45 shows peak and average flow stresses as a function of
strain rates. Simulations agree well with nanoindentation of 10- grain size for nanocrystalline Fe [160]. The maximum occurs, at 4%
30 nm nanocrystalline tantalum by Wang et al. [172]. strain for d 15 nm. This is consistent with other metals, as shown
Smith et al. [173] simulated thin films of nanocrystalline in Table 2. The curves show a maximum at 15% and then drop
demonstrating that work softening follows work hardening in
tantalum achieving strain rates as low as 105 s-1. At low strains
nanocrystalline Fe.
and low strain-rates dislocations are the primary means of de-
formation, but above 5% strain and strain rates of 106 s-1 twin- 4.3. Hexagonal close-packed metals
ning was shown to play a significant role, especially at strain
rates above 108 s-1 where twinning is the initial deformation Hexagonal close-packed metals continue to be the least in-
mechanism. They found the strain rate sensitivity to increase vestigated of the principal crystal systems despite the technolo-
markedly above 107 s-1. Analysis of grain boundary sliding gical importance of magnesium, titanium, zirconium, and cobalt.
124 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
Fig. 39. Snapshots of nanocrystalline Fe colored by structure (a,b) and strain (c,d) with white corresponding to non-bcc atoms and green corresponding to highly strained
atoms. Grain sizes are 19.7 nm and 3.7 nm and each are deformed to 15% strain. Dislocations are white clusters in the grain interiors marked by black triangles and are absent
in the small grain size [160].
The relative quantity of studies is owed to a deficiency in intera- Fig. 49 shows snapshots at 300 K, 10% strain, strain rate 5
tomic potentials for use in molecular dynamics simulations. Zheng 108 s 1, hcp Zr for: (a) d ¼13 nm, (b) d¼ 53 nm, (c) d ¼131 nm. The
et al. [174] were among the first groups to investigate nanocrys- critical diameter for transition to an inverse Hall-Petch relation-
talline cobalt of 10 nm grain size as fabricated by a kinetic Potts ship is dc ¼20 nm. Grayscale atoms are indicative of atomic dis-
model through Monte Carlo simulation. In contrast to nanocrys- placements due to dislocation motion and reveal dislocation paths.
talline FCC metals, both partial and dissociated dislocations were Circular insets illustrate grain boundary sliding through the use of
observed. Dissociated dislocations were shown to emit from both purple marker atoms (originally lying in a straight line in the
grain boundaries and from intragranular defects arising from in- undeformed state) and blue grain boundary atoms [176].
teraction of stacking faults and disordered atom segments. Inter-
action of dislocations with disordered atom segments are shown
4.4. Extreme deformation of nanocrystalline metals
in Fig. 6. Interaction between these two defects cause a split of the
stacking fault with opposite leading partial dislocations [174]. Of
The response of nanocrystalline metals to laser shock has been
further interest is the lack of twinning even at high stresses and
investigated by our group and the findings are well matched by
the deformation-induced allotropic phase transformation to FCC
Fig. 46. MD simulations because both the strain rate and grain sizes are
In 2012, Song and Li [175] were the first to explore the effect of comparable. The MD simulations are limited in time (picoseconds,
grain size on deformation of nanocrystalline HCP magnesium by while laser shock experiments are on the order of 1-3 ns. The
using columnar grains. The turnover was found to be 22 nm for strain rates in both experiments and simulations cover similar
low temperatures and increased to approximately 30 nm for room ranges: 107–109 s 1. The grain sizes of the experimental nano-
temperature deformation. Fig. 47 shows the effect of grain size on crystals (50–150 nm) are slightly larger than the simulated ones.
the flow stress of magnesium, which has a c/a of 1.624 near to the Two metals, Ni and Ta, representative of FCC and BCC structures
theoretical value of 1.633. The inversion in the slope of the Hall- respectively, were investigated. One of the outcomes of the work
Petch slope is clear. was to establish the effect of grain size on the pressure required to
Following on this work, the IHP relationship was also shown in initiate twinning. A constitutive analysis developed by Meyers
nanocrystalline zirconium by Ruestes et al. [176] for zirconium (c/ et al. [150] was applied to the slip-twinning transition. The basic
a¼1.593). MD simulations were carried out in the nanocrystalline assumption is that slip and twinning are competing mechanisms
region with columnar grains oriented in such a manner as to avoid and that the shear stress for slip, τs equals the shear stress for
twinning, which would bring another deformation mechanism. Thus, twinning τ T at the transition:
prismatic slip was activated in the system o11-204{1-100}. The si- τs = τ T (26)
mulations were conducted at 5 108 s-1 and show an apex in strength
at 30 nm. This confirmed experimental results by Wang et al.[177] Converting into normal stresses:
which showed an inverse Hall-Petch relationship for a grain size be-
σ T ≤ σs (27)
low 30 nm. The flow stresses at 10 and 300 K are given in Fig. 48.
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 125
Fig. 40. Snapshots showing dislocation activity in nanocrystalline bcc Fe for varying potentials colored by structure as determined by CNA: yellow (bcc), blue (other, i.e. grain
boundaries, defects, and non-bcc close-packed structures). From top to bottom the potentials are: Mendelev, MEAM-p, Ackland, Voter. Compressive strains increase as
indicated from left to right [161].
Fig. 41. (a-d) Evolution of failure in nanocrystalline BCC Ta. (e,f) Twin head and crack. (f) Only defective atoms are represented, showing a twin associated with crack
opening. (g,h) Schematic of grain boundary separation and linkage of facet cracks. (i,j) Schematic of crack twin interaction. From Tang et al. [166].
There are many constitutive equations for slip, the best known strengthening due to the solid solution addition of W to Ni.
being the Zerilli-Armstrong (ZA) [178,179], which has different
forms for FCC and BCC structures. A Hall-Petch term was in- ⎛ ⎞m
σslip NiW 13% = σG + ⎜⎜ ∑ Ki1/ mCi ⎟⎟ + C2 εn exp ( − C3 T + C4 T ln ε)̇
corporated to represent the entire range of grain sizes, with a ⎝ i ⎠
prefactor ks. These equations were modified and applied by our
group. For the FCC equation, a term was added to represent the + k S d−1/2 (28)
126 E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134
Fig. 42. Maximum, 5%, 15% and 18% von Mises stress as a function of grain size d
under uniaxial compressive strain (inverse Hall–Petch behavior); note decrease in
slope as d increases (strain rate of 109 s 1). From Tang et al. [166].
Fig. 45. Peak and average flow stresses as a function of grain size for nanocrys-
talline Fe.. Dashed lines are linear regression for each stress state [160].
Table 2
Strongest Grain Size from MD simulations and experimental results.
Cu (FCC) 16 [53]
20a [13]
15 [82]
Ni (FCC) 10 [246]
Mg (HCP) 30b 22c [175]
Zr (HCP) 16b [176]
25c
Zn (HCP) 11 [247]
Ta (BCC) 30 [166]
Fe (BCC) 15 [160]
WC (hex.) 15a [248]
Fig. 43. (a) Residual displacement field for bcc Ta; (b) grain rotation; (c) grain
a
elongation [166]. experimental result.
b
T ¼ 300 K.
c
T ¼ 10 K
Fig. 46. Snapshot of nanocrystalline cobalt viewed in the (1210) plane at 9.2% and 11% strain. Atoms are colored according to structure type: hcp atoms (yellow); fcc atoms
(pink); disordered atoms (red); other 12-coordinated atoms (grey). Interactions between disordered atom segments and stacking faults are circled. Interaction between these
two defects cause a split of the stacking fault with opposite leading partial dislocations [174].
dε
= k SG P 4⋅
dt (31)
Fig. 49. Snapshots at 300 K, 10% strain, strain rate 5 108 s 1, hcp Zr for: (a) d¼ 13 nm, (b) d ¼ 53 nm, (c) d ¼ 131 nm. The critical diameter for transition to an inverse Hall-
Petch relationship is dc ¼20 nm. Grayscale atoms are indicative of atomi- displacements due to dislocation motion and reveal dislocation paths. Circular insets illustrate grain
boundary sliding through the use of purple marker atoms (originally lying in a straight line in the undeformed state) and blue grain boundary atoms. [176].
Fig. 50. Stresses for slip (ss) and twinning (sT) as a function of shock pressure for (a) Ni (d¼ 50 nm) and (b) Ni-13%W (d ¼10 nm); (c) TEM showing dislocations in grains
after 40 GPa pressure in Ni (d¼ 30 nm); TEM showing twins (circled) after 40 GPa pressure in Ni-13% W (d¼ 10 nm). (a, b from Jarmakani et al. [83]; c,d from Wang et al.
[264]).
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 129
Fig. 51. Stresses for slip and twinning as a function of pressure for (a) monocrystalline (3 mm) and (b) ultrafine grained tantalum (d ¼70 nm); (c) twins in monocrystalline Ta
shocked at P ¼ 40 GPa; (d) absence of twins in UFG Ta shocked at 40 GPa by lasers. From Lu et al. [265,266].
Fig. 53. Nanocrystalline silicon structure at deviatoric strains of (a) 0.08 and (b) 0.25. Dark atoms signify inelastic transformations, while light atoms are those that are purely
elastic. Ellipses indicate intergranular regions of localized plastic deformation. Solid lines in (b) indicate fully developed plastic flow. “C” marks a grain that undergoes
rotation during deformation. From Demkowicz et al. [187].
[202–204]. Li et al. [200] showed that Young's modulus decreased order to produce a grain size dependence of flow stress, Fu et al.
linearly with increasing hydrogen coverage. [206] considered grains as composed of two regions: grain inter-
iors and grain-boundary layers. As the grain size was varied, the
volume fractions of the two regions changed and the mechanical
5. Other computational tools
response was altered, resulting in a Hall-Petch response but with a
decreasing slope as the size decreased. This was later changed into
5.1. Finite element method
a gradient of strains, which is size dependent.
The Finite Element Method (FEM) is limited in its ability to
interrogate the response of materials at the atomic/molecular 5.2. Quasi-continuum methods
scale. Nevertheless, the effect of grain size has been investigated
by introducing an artificial length scale. A short review of com- Quasi-continuum (QC) methods are methods of mixed ato-
posite methods is presented by Mishnaevsky and Levashov [205]. mistic (namely MD) and continuum (namely FEM) approaches
Importantly, FEM predicts correctly the internal stresses due to where fully-atomistic techniques are implemented in areas of di-
compatibility requirements. As early as 1980, Meyers and Ash- rect interest and remote areas are modeled using less costly con-
worth [9] juxtaposed two grains with different orientations by tinuum mechanics. Tadmor and Miller [207] provide a thorough
transforming the stiffness matrix into equivalent elastic moduli. description of QC techniques in their recent book (in addition to
This was done for three orientations:[100], [110], and [111]. Later continuum mechanics, quantum mechanics, atomistic simulations,
analysis by Fu et al. [206] on a polycrystalline aggregate, also using and multi-scale techniques). This method is particularly appro-
FEM, produced the results shown in Fig. 55. These results are priate for the representation of polycrystalline materials where full
analogous to the ones obtained by MD and shown in Fig. 21. In atomistic resolution is too costly and continuum methods do not
Fig. 54. (a) 2D Voronoi tessellation of graphene with 30% hydrogen grain boundary coverage indicated by red atoms in (b). (c) Represents characteristic out of plane buckling
colored by out of plane displacement. (d) Effect of hydrogen content on failure strength. From Li et al. [200].
E.N. Hahn, M.A. Meyers / Materials Science & Engineering A 646 (2015) 101–134 131
Fig. 55. Stresses in polycrystalline copper loaded elastically; (a) grain configuration; three grain orientations: white [100], gray [110], and black [111]; (b) maximum principal
stress (s1). From Fu et al.[206].
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