Three-dimensional molecular dynamics simulations of void coalescence during dynamic fracture of ductile metals

0.59 MB·English

by  E. T. Seppälä

Three-dimensional molecular dynamics simulations of void coalescence during
dynamic fracture of ductile metals
E. T. Sepp¨al¨a,∗ J. Belak,† and R. E. Rudd‡
Lawrence Livermore National Laboratory, Condensed Matter Physics Division, L-045, Livermore, CA 94551, USA
(Dated: February 2, 2008)
5
Voidcoalescenceandinteractionindynamicfractureofductilemetalshavebeeninvestigatedusing
0
three-dimensional strain-controlled multi-million atom molecular dynamics simulations of copper.
0
Thecorrelatedgrowthoftwovoidsduringthecoalescenceprocessleadingtofractureisinvestigated,
2
both in terms of its onset and the ensuing dynamical interactions. Void interactions are quantified

n through the rate of reduction of the distance between the voids, through the correlated directional
a growthofthevoids,andthroughcorrelatedshapeevolutionofthevoids. Thecriticalinter-voidlig-
J
amentdistancemarkingtheonsetofcoalescenceisshowntobeapproximatelyonevoidradiusbased

6 onthequantificationmeasurementsused,independentoftheinitialseparationdistancebetweenthe
1 voidsandthestrain-rateoftheexpansionofthesystem. Theinteractionofthevoidsisnotreflected
in the volumetric asymptotic growth rate of the voids, as demonstrated here. Finally, the practice
] of using a single void and periodic boundary conditions to study coalescence is examined critically
i
c and shown to produceresults markedly different than thecoalescence of a pair of isolated voids.
s
-
l PACSnumbers: 61.72.Qq,62.20.Mk,62.20.Fe,61.72.Lk
r
t
m
I. INTRODUCTION the material, platelets of atoms, from the void and are
.
t responsiblefortheplasticdeformationsneededtoaccom-
a
m The fracture of ductile metals at high strain rates has modate significant void growth. There are also many
recent studies of fracture in ductile metals with several
- been understood at the microscopic level as a process of
d nucleation, growth and coalescence of voids.1,2 Initially holes or voids.17,18,19,20,21 While these studies model the
n voidgrowthexplicitlywithfairlysophisticatedmodelsof
voids nucleate at the weak points in the material such
o plasticity in many cases, they typically simplify the coa-
asinclusionsand/orgrainboundaryjunctions. Oncenu-
c
lescenceprocesstoinstantaneousunificationofthe voids
[ cleated, the voids grow under the tensile stress, driven
when some threshold is reached, such as once the voids
by the reduction in elastic energy. Eventually, the voids
1 growtowithinonediameterofeachother. Therearealso
growsufficientlylargethattheyinteractwitheachother,
v studies that investigate the competition between void
in some cases link through localized shear, coalesce into
9
by void growth versus multiple void interaction in crack
6 largervoids,andfinallyformthefracturesurface.3,4Con-
propagation.22 Several earlier continuum studies23,24,25
3 siderableexperimentalandtheoreticalworkhasgoneinto
andthe one atomistic study knownto us26 ofthe coales-
1 the development of our understanding of fracture. The
0 fracture process has been modeled at various levels, but cenceprocesshavebeentypicallyconductedineffectively
5 two dimensional and highly symmetric systems.
most ofthe workinvolvingthe simulationof the activity
0
of individual voids has concentrated on the void growth ThisArticlecoversindetailastudyoftheonsetofvoid
/
t anditsrelationshiptotheplasticdeformationofthesur- coalescence. Thefirstresultsofthisstudyhavebeenpre-
a
m rounding material, including effects such as the localiza- sented in a Letter.27 Here we provide a more complete
tion of this shear deformation. Relatively little work has presentation of the results of our study of void coales-
-
d gone into the explicit modeling of void coalescence, and cence. In addition to a more detailed description of the
n boththeunderstandingofthephysicsofthisprocessand resultspresentedintheLetter,wedescribedifferentmea-
o the knowledgeofhow itshouldbe implemented robustly sures of void interactions such as shape changes induced
c in continuum fracture codes remain open issues. The byaneighboringvoid,additionalanalysisofthebehavior
:
v point at which voids begin to coalesce during dynamic ofthe systemincluding stress-straincurvesandvoidvol-
i fractureisofconsiderableinterestbecausecompletefrac- umecurves,andanewanalysisofhowthecoalescenceof
X
ture of the material typically ensues rapidly thereafter. isolatedpairsofvoids differs fromthat ofvoidsinhighly
r
a As new experimental techniques have constrained the symmetric periodic arrays.
void growth models ever more stringently,5 a real need In particular the goal of this Article is to quantify the
for a well developed theory of coalescence has arisen. pointatwhichcoalescencebegins,asmeasuredbyacrit-
Computationally void growth has been studied ex- ical inter-void ligament distance (ILD), and examine the
tensively at the continuum level,6,7,8,9 also in dynamic, mechanisms involved in the transition from independent
high strain-rate, conditions.10,11 Recently we have stud- void growth to coalescence. There are several ways in
ied void growth at the atomistic level under high which two voids can interact. In the case of pure im-
strain-rate expansion, motivated by spallation exper- pingement, the voids only interact when they grow to
iments.12,13,14,15,16 The atomistic studies demonstrate the point that they intersect and join into a single void.
thatvoidsgrowby emitting dislocationsthatcarryaway In reality, the voids interact before they intersect. Their
2
range of interaction is extended due to their elastic and
plastic fields. Each void generates an elastic strain field
of the form generally associated with centers of dilata-
tion.28 The shearstressdecreaseswiththe distance from
the void like r−3. For voids sufficiently close each void’s
growth rate is altered by the stress field of the proximal B
void. The modification of the elastic field can affect the
initiationofplasticity,aswellasthesubsequentdevelop- A
mentoftheplasticzonearoundthevoids. Thevoidsmay
ILD
interactthroughtheirplasticfields,too,inwhichcasethe
fields may give rise to an increased hardening rate in a
localized region or to thermal softening and shear local-
ization. An argument due to Brown and Embury for a
transitionto shear deformation basedon simple geomet-
rical considerations suggests that the critical inter-void
ligament distance, ILD , should be equal to one diam-
c
eter of a void;29 that is, when the surfaces of a pair of
voids are separated by one void diameter, they transi-
tion from independent void growth to coalescence. It is
at this point, they argue, that the dominant void pro- FIG.1: Asketchofthesimulationconfiguration. Thesimula-
cessswitches fromthe radialplastic flowaroundisolated tionboxincludes120×120×120FCCunitcellswithperiodic
growingvoidstoalarge-scalesheardeformationallowing boundary conditions for a total of 6 912 000 atoms initially.
therapidcoalescenceofthepairofvoids. However,more From this two equal size voids are created by removing ap-
recenttwo-dimensionalstudiessuggestthatfordistances proximately 3620 atoms for each void. The thinarrow shows
between voids as large as six diameters the void growth the inter-void ligament distance (ILD), the shortest surface-
rate is enhanced.30 to-surface distance between the voids, identified as A and B.
TheinitialILDrangesbetweenone-halfandfivetimestheini-
The use of atomistic techniques permits the analysis tialvoidradius. Theboldarrowsdenotetriaxial,hydrostatic,
of the contributions of these competing mechanisms to expansion of the box. The strain-controlled expansion is ap-
the onset of void coalescence, as we describe in this Ar- plied with constant strain-rates of ε˙=108/sec and 109/sec.
ticle. We demonstrate the existence of, and compute,
the criticalinter-voidligamentdistanceILD bystarting
c
withtwovoidswellseparatedfromeachotheranddetect-
ing the point at which correlatedgrowthbegins, marked
both by the accelerated rate at which the two void sur-
TABLE I: The initial position for void B relative to void
faces approach each other and by biased growth causing
A and the center-to-center distances for various initial inter-
the voids to start to extend toward each other. These
void(surface-to-surface)ligamentdistancesILD0 usedinthis
changesgiveanindicationoftheonsetofthecoalescence
study.
process, and it tests the argument by Brown and Em-
bury.29 We also test the setup by Horstemeyer et al.30 ILD0 position of void B separation between the
relative to void A centersof voids A and B
by varying the initial distances between the voids and
0.50 [0.1334,0.0622,0.0290]L 0.150L=6.510 nm
measuring the asymptotic growth rate of the voids. The
1.00 [0.1778,0.0829,0.0387]L 0.200L=8.680 nm
initialvoid-to-voiddistancebelowwhichthe growth-rate
1.20 [0.1956,0.0912,0.0426]L 0.220L=9.548 nm
isenhancedshouldgiveanothercandidateforthecritical
1.50 [0.2223,0.1037,0.0483]L 0.250L=10.85 nm
distance and measure it in a volumetric sense. It should
1.81 [0.2500,0.1166,0.0544]L 0.281L=12.20 nm
be noted, however, that the three-dimensional void coa- 4.62 [0.5000,0.2332,0.1087]L 0.562L=24.40 nm
lescencestudiedherewithmoleculardynamics,asindeed
any 3D coalescence of roughly spherical voids, does not
admitthetwo-dimensionalshearmechanismproposedby
Brown and Embury in its simplest form, and therefore
these different analyses are not fully comparable. Also
Horstemeyer et al. used a more symmetric setup than
SectionIV: SectionIVAusestwodifferentdistancemea-
the simulations coveredin this Article.
surements to study the interaction; Section IVB intro-
The Article is organized as follows. The simulation duces a shape parameter for the purpose. Finally the
method and the performed computations are introduced volume effects of the void growth are studied in Sec-
in Section II. The basic mechanism, dislocation driven tion V: two separate voids are studied in Section VA
void growth, is demonstrated and the key reference pa- and one void and the interaction with its periodic image
rameter, mean linear void size, is introduced in Sec- are studied in Section VB. The paper ends with conclu-
tion III. The interaction between voids are studied in sions, Section VI.
3
II. METHOD AND SIMULATIONS
We have performed a series of large-scale classical
molecular dynamics (MD) simulations31 in single crys-
talface-centeredcubic(FCC)systemsusinganempirical
embedded-atom model (EAM) potential for copper.32,33
The three dimensional (3D) simulation box consists of
120×120×120 4-atom FCC cells with periodic bound-
ary conditions for a total of 6 912 000 atoms, in most
cases. In Section VB where results from smaller system
sizesimulationsarepresented,thedetailsofthe sizesare
introduced. Forthenearly7millionatomsimulationsthe
MD code was parallelized using spatial domain decom-
position and run in a massively parallel computer using
from 64 to 256 processors. FIG.2: (Gray-scale)Athreedimensionalsnapshotofthetwo
In the simulations the system is initially equilibrated voids with the prismatic dislocation loops forming from the
using a thermostat34 at room temperature, T = 300 K, voids. Only those atoms belonging to the void surfaces or to
and a constant volume L3 (with L = 43.3 nm) cho- dislocation cores, stacking faults or other defects are shown
(andasmallnumberofextraneousatomsduetothermalfluc-
sen to give ambient pressure, P ≃ 0 MPa. Once the
tuations). Thestackingfaultribbonsarethebroadplatesbe-
system has reached equilibrium, atoms are removed to
tween leadingand trailing partialdislocations. Thesnapshot
create two spherical voids in the system with radius isfromthesimulationwiththeinitialinter-voidligamentdis-
r0 = 0.05L = 2.17 nm: one centered in the box and tance ILD0=1.81 diameters, and the strain-rate ε˙=109/sec.
the other located a distance ILD0+2r0 away in the di- The strain at thesnapshot is ε=2.93%.
rection uˆ=[0.8892054,0.41464327,0.19335135]from the
first void. We refer to these as void A and void B, re-
spectively, see Fig. 1. ILD , the initial inter-void liga- including analysis of growth of a single void of the same
0
ment distance, is the closest surface-to-surface distance initial radius r in non-triaxial expansion.
0
betweenthevoids35 anditisvariedhere,buttherelative
orientation of the voids is kept fixed. For ILD we have
0
used the values: 1.00, 1.20,1.50,1.81, and in some cases III. DISLOCATIONS AND VOID GROWTH
0.50 and 4.62. The unit for the ILD is the void diame-
ter, d = 2r. The positions for the center of the void B
Letusstartreviewingthesimulationresultsbylooking
andthe distancesbetweenthe voidsarelistedinTableI.
at some figures to visualize growth of the voids. While
Initially, the voids are equal in size, with approximately
somevoidgrowthtakesplacethroughelasticstretchingin
3620 atoms removed for each. This removal of atoms
the initial phases of the box expansion, significant void
canbe interpretedas aninstantaneousdebonding oftwo
growth and void-void interaction take place only once
infinitely weakly bound inclusions.
plastic deformation has begun. The important role of
Once the voids are formed, the thermostat is turned
plasticityleadsustoconsiderinsomedetailhowdisloca-
off, and dilatationalstrain is applied uniformly at a con-
tionsaregeneratedandtheeffectofdislocationdynamics
stant strain-rate ε˙. The strain-controlled simulations12
on voidcoalescence,which are the topics of this Section.
are carried out using the scaled coordinate formula-
In Fig. 2 a three-dimensional snapshot of the voids is
tion typically employedin the constant pressure method
shown from a simulation with the initial inter-void liga-
due to Parrinello and Rahman.36 Use of scaled coordi-
ment distance ILD =1.81 diameters and the strain-rate
0
nates prevents the spurious generation of elastic waves ε˙ = 109/sec. In the plot only the atoms at crystallo-
at the box boundaries. In this method the positions of
graphic defects such as void surfaces, dislocation cores,
the atoms are stored using rescaled coordinates between
and stacking faults are shown. The decision of which
[0,1). When calculating the forces and new positions of
atomstoplotisbasedonageometricalcriterion,afinite-
the atoms their coordinates are multiplied by a diagonal
temperature generalization of the centrosymmetry devi-
scaling matrix H = {Lx,Ly,Lz}. This scaling matrix is ation.15,37 The power of this method of selecting atoms
updated each time step, when the expansion is applied,
based on the centrosymmetry deviation is that the visu-
by multiplying the initial matrix H = {L,L,L} with
0 alization can be done at finite temperature and on-the-
thesumoftheunitmatrixandthestrainmatrixE =tE˙, fly, so that the system need not to be cooled to zero-
H(t) = H0(I+tE˙). Applied strain-rates of ε˙ = 108/sec temperature and the atoms to be selected based on po-
and 109/sec have been used with perfectly triaxial, or tential energy, where quenching can influence the sys-
hydrostatic,expansion,E˙ ={ε˙x,ε˙y,ε˙z}={ε˙,ε˙,ε˙}. Thus temandpreventseeingtherealconfiguration. Thesnap-
L (t) = L (t) = L (t) = L(t) = V1/3, where V is the shot in Fig. 2 is when the voids have started to grow by
x y z
volume of the box. A time step of 6.7 fs was used. More emitting dislocations, and thus the system has already
detailsofthesimulationmethodcanbefoundinRef.16, evolvedsomewhatthroughplasticflow. Thebroadplates
4
A B
(a) (b) (c)
(d) (e) (f)
FIG. 3: Dislocation activity at six instants of time shown on one particular slice through the system in order to expose the
plasticity near the void surfaces. These snapshots are from the simulation with ε˙ = 109/sec and again only those atoms in
dislocation cores, stacking faults, void surfaces or other defectsare shown (see textfor more details andFig. 2 for afull three-
dimensional figure). The dashed loop in panel (c) is drawn around a slice of a prismatic dislocation loop. The plane shown
passes through the centers of both voids with normal [0.145, 0.145, -0.979]. The snapshots show the initial plasticity (a),(b),
interacting plastic zones (c),(d) and the final coalescence (e),(f). The frames correspond to the following values of strain ε:
1.72%, 2.42%, 3.47%, 3.89%, 4.52%, and 5.21%, respectively.
in the figure are stacking fault ribbons between leading ity around each void is essentially symmetric [Figs. 3(a)
and trailing partial dislocations. The first generation of and(b)],asexpectedforindependentvoidgrowth,butas
dislocation loops have not yet totally formed and sepa- theplasticfieldsevolvethevoid-voidinteractionisclearly
rated from the voids in this snapshot, but are about to evident both through interactions between the two plas-
do so, as can be seen as their first halves have already tic zones and bias due to the elastic fields [Fig. 3(c)].
separated from the void surface. Once the dislocation density grows sufficiently high in
Figure 3 shows a series of visualizations of the crys- the ligamentregionbetweenthe voids[Fig.3(d)], voidB
tal defects within a slice of width 4.5 ˚A about a plane begins to grow in the direction away from void A. Next
including centers of both voids at six different instants the voids coalesce [Fig. 3(e)], and continue to grow as
during coalescence. These snapshots are from the same one until ultimately the void coalesces with the its peri-
simulation as in Fig. 2, and the same criterion for show- odic images [subsequent to Fig. 3(f)] so that the cavity
ing the atomsis used. Now onlyaslice is showninorder percolates through the periodic system. The details of
to revealthe dislocationactivitynearthe voidsurfaceas this final stage depend strongly on the periodic bound-
the dislocation density increases. Figure 2 is a snapshot ary conditions and will not be of interest here.
from a state of the system between panels Fig. 3(b) and The dislocationformationis closely relatedto the vol-
(c). Fromthe snapshots it is apparentthat the deforma- ume evolution of the voids. The voids grow by emit-
tion mechanism involves the nucleation and propagation ting dislocation loops, driven by the reduction in the
ofdislocations,accommodatingthevoidgrowth,andthe elastic energy as the increase in void volume allows the
interaction of the dislocations.38 For example, the pris- strained matrix material to relax. This relaxation can
matic dislocationloops punched out by the voidsappear be detected from the saturation of the increasing stress
as roughly parallel line traces (due to the stacking fault in the system. All of these phenomena happen simul-
ribbons) in the slice Fig. 3(c), as verified in the full 3D taneously, see Ref. 16. In Fig. 4(a) the mean (hy-
configuration (cf. Fig. 2). Initially the dislocation activ- drostatic) stress σ is plotted with respect to strain ε
m
5
10.0
TABLEII: Criticalstrainε andcriticallinearmeanvoidsize
Pa ] 89..00 (a) fd¯ci1s/t3anfrcoemcrFosigse.s5,thceallciunleatIeLdcDa=s0w.5hde:nf¯tc1h/3e =intfe¯r1-/v3o(IidLDli=ga0m.5edn)t,
G 7.0 εc =ε(ILD=0.5d).
[
) m 6.0 ILD0 ε˙ (sec−1) f¯c1/3 εc (%)
σ( 5.0 1.81 108 0.153 2.46
ress 4.0 11..8510 110099 00..115302 33..4282
n st 3.0 1.20 109 0.121 3.06
ea 2.0 ε.=109/sec 1.00 109 0.106 2.74
m ε.=108/sec
1.0
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
strain (ε) [ % ]
) (b)
3
1/
−f
( the boxattime t. The technique forcalculatingthe void
e
iz volume Vvoid is described in Ref. 16. The void growth,
oid s10−1 GPa ] 10.0 sattressigsnsiafitcuarnattliyonsmanadlleevresntrtahiensvofiodrcsolaowleescresntcreaitna-kreatpelsa,c1e6
n v ) [ m 9.0 as can be seen from the figures, too. Therefore we con-
near mea σmean stress ( 678...0000.08 0.10 0.12 0.14 0.16 rcsvtilevurraedsdiuens-qtrhluaianattneeatasiritnmyva)outiluidasrtapsiolirznweefs,aeiyrfnr1te/tod3h.petloPosatplomlqtouettiatnfiinnggtguivtrieveeressirsusfrustoosfmpt1h/ldo3eitff(steathrredaenmient-
li linear mean void size (f−1/3) because it reduces strain-rate effects. Thus by choosing
1.75 2.25 2.75 3.25 3.75 4.25 f1/3 from Fig. 4(b) as the reference quantity, the start
strain (ε) [ % ] of the deviation from the elastic behavior can be syn-
chronized for the different strain-rates, see an example
forthe meanstressfromtheinsetofFig.4(b), wherethe
FIG. 4: (a) Mean stress σ versus strain ε from the simula-
m meanstressstartstosaturateforbothofthestrain-rates
tainodnsthweithstrtahien-inraittieaslεi˙nt=er-1v0o8i/dselicga(mdaesnhteddisltinaen)ceanILdD100=9/1s.8e1c at f¯1/3 ≃ 0.09. In the two-void case we have chosen to
(solid line). (b)Linear mean void size, f¯1/3, (seethetextfor use the mean void fraction f¯, which is calculated as the
average void fraction of the two voids.
its definition) versus thestrain from thesame simulations as
the data in (a). The void sizes are calculated until the co-
alescence of the voids. The inset shows the stress from (a)
versus linear mean void size from (b). Note the strain scales
are different. Throughout the remainder of the Article we will typi-
cally use the linear mean void size f¯1/3 as the reference
quantity. Theinitiallinearmeanvoidfractionatambient
pressure in these simulations is f¯1/3 ≃(6.0×10−4)1/3 =
(the control parameter in these simulations) for strain- 0
rates ε˙ = 108/sec and 109/sec with ILD =1.81. The 0.084. Theinitialmeanvoidsizef¯=6.0×10−4 issome-
0 what larger than the value one gets from 4π(r /L)3 =
stress is calculated with the virial expression σm = 3 0
− 1 |p~ |2/m + ~r ·f~ , where for atoms 5.2×10−4. This potential source of confusion arises for
3V (cid:16)Pi i i Pi,j|j>i ij ij(cid:17) two reasons: first, atoms that have their centers within
i and j the~r is the relative position, f~ is the force, p~ the radiusr fromthe voidcenterare removedforcreat-
ij ij i 0
is the the momentum and m is the mass (for a recent ing the void. On the other hand, when the void volume
i
discussion of atomistic stress calculations see Ref. 39). is calculated the surface of the void is defined based on
Comparing Fig. 4(a) with Fig. 4(b) it is apparent that the centers of the remaining surface atoms. Second, the
the start of the accelerated void growth due to plastic- void surface relaxes somewhat after the void is formed.
ity is accompanied by, and indeed causes, the stress to For more discussion of the void volume calculation, see
plateau (at the same strain value) as the elastic dilata- Ref.16. Forreference,themeanlinearvoidsizeinFig.2
tionisrelieved. ThevoidgrowthisshowninFig.4(b)by is f¯1/3 = 0.111, and in the first four snapshots of Fig. 3
plotting linear void size f1/3, where the single void frac- f¯1/3 =0.089,0.094,0.149,and0.195,respectively. After
tionisf =V /V andV istheinstantaneousvolumeof coalescence f¯is not measured.
void
6
8.0 the void surfaces (the ILD) serves as something akin to
a reaction coordinate for the coalescence: the voids coa-
7.0
lesce when it goes to zero. Other indications include the
]
m 6.0 displacement of the center of a void as it grows prefer-
n
) [ 5.0 ethnetiavlolyidtogwroawrdththraeteneaisghtbhoerivnogidvsoiindtearnadctt.heInchthanisgeseicn-
D
L 4.0 tion we now quantify two of these effects, the evolution
I
( of the ILD and the void center movement, in order to
ce 3.0 analyze the coalescence. In Section IVB we study the
n
a voidshapeevolution,andthevoidgrowthrateisstudied
t 2.0
s
i in Section V.
d
(a)
1.0
InFigs.5(a)and(b)thedynamicevolutionoftheILD
hasbeenplottedforstrain-ratesε˙=108/secand109/sec
0.0
1.70 2.20 2.70 3.20 3.70 4.20 andforvariousinitialclosestsurface-to-surfacedistances
strain (ε) [ % ] between the voids ILD . (In Figs. 2-4 the data from the
0
case ILD ≃ 1.81 were shown.) The dynamic ILD has
0
1.40 two spherical voids been derived as the separation distance of the two sur-
one void, ε.=109/sec face atoms from the two voids that are closest to the
1.20 ILD=1.8, ε.=109/sec line connecting the original centers of the voids. The
d] ILD0=1.8, ε.=108/sec raw data of the ILD have been plotted in Fig. 5(a) and
D/ 1.00 ILD0=1.5, ε.=109/sec in Fig. 5(b) scaled data are shown. For the horizontal
L 0
[I 0.80 ILD0=1.2, ε.=109/sec axisof Fig.5(b), the linear meanvoidsize f¯1/3 has been
ce ILD0=1.0, ε.=109/sec used in order to collapse data from different strain-rates
an 0.60 ε˙ = 108/sec and 109/sec in the same figure. For the
t
s vertical axis the inter-void ligament distance has been
di 0.40
divided by the current diameter of a void, d (note, that
0.20 (b) it is not the initial value d0 = 2r0). The diameter d
is calculated from the void volume assuming a spheri-
0.00 cal shape (the formula is given below). This scaling is
0.084 0.124 0.164 0.204 0.244
motivated by the ansatz that the void diameter sets the
−1/3
linear mean void size (f ) length scale for the system, and hence the relevant dis-
tance between the voids is not the pure distance, but
FIG. 5: Evolution of the ILD and the critical ILD. (a) Dy- its ratio with the diameters of the voids. The void di-
namical ILD, the distance between the surfaces of the voids ameter affects both the elastic and plastic fields around
alongthelineconnectingtheoriginalcenterpositionsforvari- the void. Initially the void separationdistance decreases
ousinitialILD0=1.81,1.50,1.20,1.00,plottedversusstrain. essentially smoothly until plasticity begins, eventually
For ILD0=1.81 thethick solid line and thick medium dashed reaching zero. A transition occurs when the ILD starts
line denote ε˙ = 109/sec and 108/sec, respectively. The thin
to decrease noticeably faster than the free impingement
solidlinesshowthehypotheticalILDforsphericalvoidswith
line (the thin solid lines calculated from two indepen-
thesameILD0 impingingfreelyoneachother(seetext). The
dent spherical voids) indicating void interactions at the
short dashed line shows the hypothetical ILD computed by
duplicating a single void (in the same box size as the two onset of coalescence. The thin solid lines are derived
void simulations) at fixed centers (here the duplication is to by calculating from the same simulations the average
the position with ILD0=1.81). (b) The same as in (a), but sizes of the two voids V¯void and deriving the diameter
now plotted versus the linear average void size, f¯1/3 and the as d=2 3 V¯ 1/3. Then the free impingement curve
distances are given in the units of the current average diam- (cid:0)4π void(cid:1)
is given by ILD = (ILD +d )(1+tε˙)−d. The ac-
free 0 0
eter d of the voids, calculated from their volumes assuming
celeratedILDdecreaseassociatedwithtransitionto coa-
that they are spherical (see text). The horizontal line is at
lescencetakesplacewhentheILDreachesapproximately
ILD=0.5diameters,thevalueweidentifyastheILD . (Panel
c one half: ILD =0.5±0.1 diameter or one radius, inde-
(b) is after Ref. 27.) c
pendently of ILD or the strain-rate. Also note that a
0
curve derived from a single void growth is provided to
estimate the contributionofuncorrelatedfaceting effects
IV. THE INFLUENCE OF THE NEIGHBORING
(the “one void” curve at ILD =1.81), and these effects
VOID 0
are seen to be relatively small. They are most notice-
able near the start of void growth where there is a rela-
A. Distance Measurements of the Voids
tively large upward fluctuation in the “one void” curve.
The critical ILD of one radius is much lower than the
Figure 3 offered severalvisual indications of the inter- Brown-Embury estimate, and it corresponds to a strain
action between voids. Clearly, the separation between of 3.48% (f¯1/3 ≃ 0.15) for ILD =1.81 at ε˙ = 109/sec,
0
7
0.100 Another measure of void interactions is whether the
0.075 (a) ε.=109/sec voids grow preferentially towardtheir neighbor. This ef-
fect is quantified in Fig. 6, which shows the movement
0.050 of the center of mass of the void surface for the voids
]
/d0 0.025 shown in Fig. 3 (ILD0=1.81). We use here the center of
v massofthe voidsurfaceasthe definitionforcalculations
dc
[ 0.000 of the void center. The void surface has been derived by
e
c Voronoi triangulation based on the center points of the
n−0.025
a surface atoms (the same is done when the volume of the
t
s
di−0.050 void A voidis calculated). See Ref.16forthe detailsofthe void
void B surface derivation. The distance
−0.075 one void
−0.100 dcv =h(cid:0)x′−x0cv(cid:1)2+(cid:0)y′−yc0v(cid:1)2+(cid:0)z′−zc0v(cid:1)2i1/2 (1)
0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25
linear mean void size (f−1/3) is calculated between the original center of mass of the
voidsurface [x0 ,y0 ,z0 ]and the projection[x′,y′,z′]of
0.200 cv cv cv
thecurrentcenterofvoid[x ,y ,z ],wheretheprojec-
cv cv cv
0.150 (b) ε.=108/sec tion is onto the line connecting the originalvoid centers.
The sign is positive if the void center has moved toward
0.100
the other void and negative in the opposite case.
]
/d0 0.050 Letuslookfirstatthecasewithε˙=109/sec,Fig.6(a).
v
dc Afterthevoidgrowthstarts,thecenterofvoidAinitially
e [ 0.000 movesonlyslightly,butataboutf¯1/3 =0.15(ILD=0.5d
anc−0.050 inFig.5(b)andf¯c1/3 inTableII),itstartstomoveinthe
st direction of the other void as the void growth becomes
di−0.100 void A biased toward its neighbor. Just before coalescence the
void B
centerofvoidAbeginstomoveawayfromvoidB,asthe
−0.150 one void
growth is biased in the opposite direction. During this
−0.200 sequence, void B initially grows away from void A, then
0.09 0.11 0.13 0.15 0.17 0.19 0.21
linear mean void size (f−1/3) roughlyinunisonwithvoidA(f¯1/3 =f¯c1/3 =0.15)itbe-
gins to grow towardits neighbor, and before coalescence
it too switches to growth away from the proximal void.
FIG. 6: Void center-of-mass displacement. The distance d
cv Thisretrogradegrowthhappensatthe samepoint(after
fromtheoriginalcenterofvoidtotheinstantaneousvoidcen- f¯1/3 = 0.19) as the decrease of the ILD begins to slow
ter, projected onto the line connecting the original void cen-
downinFig.5(b)[seealsothesnapshotinFig.3(d)]. The
ters, is plotted versus the average void size to the point of
coalescence (ILD ≃ 0). The sign of the displacement d is samephenomenon–firstslowmovementorrepulsionfrom
cv
the void; then growth toward the nearby void at about
positive for movement toward the other void. Solid and long
dashed lines are for voidsA and B, respectively. Strain-rates f¯1/3 = 0.15 (f¯1/3)–holds in the ε˙ = 108/sec case, too,
c
are ε˙=109/sec in (a) and ε˙=108/sec in (b). The thin solid Fig. 6(b). However, the retrograde growth phenomenon
line is for a single void in the same size of the box and with is less pronounced in the ε˙ = 108/sec case, as is the
the same radius and strain-rate projected to the same line. slowing down of the decrease of the ILD in Fig. 5(b).
Herethedistanced isgivenintheunitsoftheoriginalvoid
cv Indeed, only the growth toward the neighboring void is
diameter d0. ILD0=1.81 in both (a) and (b). well above the noise in the ε˙=108/sec case. As a refer-
encethemovementofthecenterofasinglevoid(insame
box size) projected to the same line is plotted for both
close to frame (c) of Fig. 3. The values for criticalstrain strain-rates, too. Comparing the single void case with
and linear mean void size when ILD=0.5d, derived from the interactingvoidswiththe samestrain-rates,onesees
Fig. 5(b), are tabulated for the simulated systems in Ta- that the maximum distance the centers of the interact-
ble II. In the very final stages the ligament is drawn ing voids have moved is 2.5 to five times larger than the
under biaxial stress, and the flow switches from radial nanoscale random walk of the single void center, except
materialtransportto tangentialtransportas the mecha- for the void B in Fig. 6(b). The noise is these curves
nismswitchesfromlooppunchingtodrawing. Thistran- appears to be dependent on the history, reflecting the
sition is visible in Figs. 5(a) and (b) as slowing down of nature of the plastic deformation processes involved. It
thereductionofILD.Atthispoint,thematerialishighly is difficult to quantify the level of noise in such a non-
defective butitremainsductile. Thereisnoabruptfrac- Markovprocess,butitshouldbeclearthatthemovement
ture, as might be expected at larger length scales. Here of the void center, especially the movement toward the
the final coalescence involves an extended drawing and neighboring void, is statistically significant and not just
thinning of the ligament until rupture. due to fluctuations at the void surface.
8
B. Shape Evolution of the Voids 0.15
0.10 (a) void A
)
The presence of a nearby void not only affects the po- m 0.05
2
Q
sitionofthevoidbutitsshape,aswell. Theshapecanbe 0.00
(
quantifiedbycalculating multipole momentsofspherical nt −0.05
harmonics Qlm ≡ r¯12 RYlm(θ,φ)r2(θ,φ)dΩ, see Refs. 16 me −0.10 m=0
and 40. Under fully triaxial expansion the void tends to o −0.15
m m=1, Re
an octahedral shape because of the index of the active
e 0.10 m=1, Im
glide planes in FCC crystals, the anisotropic elastic con- ol 0.05 m=2, Re
stants, and anisotropic surface energies. See also Ref. 41 p m=2, Im
u 0.00
for astudy ofvoidshapes inFCC crystals. Onthe other dr
hand,underuniaxialexpansionthevoidsareofpredomi- ua −0.05
q −0.10
nantlyellipsoidalshapesalignedalongthepreferredaxis,
and they make a transition from a prolate to an oblate −0.15
0.084 0.104 0.124 0.144 0.164
shape.16 References 42,43,44 have also considered oblate linear mean void size (f−1/3)
void shapes under uniaxial loading through continuum
0.15
modeling. One may enquire whether the presence of a
nearby void causes evolution to an ellipsoidal shape, see 0.10 (b) void B
)
e.g. void A in Fig. 3(c). Ellipsoidal shapes can be quan- m 0.05
2
Q
tified with quadrupole moments: 0.00
(
nt −0.05
Q20 = 14qπ5r¯12 R3z2−r2dΩ, me −0.10 m=0
Re Q21 = −21q21π5r¯12 RxzdΩ, e mo −00..1150 mm==11,, RIme
Im Q21 = −12q21π5r¯12 RyzdΩ, (2) pol 0.05 mm==22,, RIme
Re Q22 = 41q21π5r¯12 Rx2−y2dΩ, adru −00..0005
Im Q = 1 15 1 xydΩ, u
22 2q2πr¯2 R q −0.10
−0.15
where r¯2 = 1 r2(θ,φ)dΩ is averaged over the surface 0.084 0.104 0.124 0.144 0.164
4π R linear mean void size (f−1/3)
of the void. In calculating the quadrupole moments the
origin of the coordinates is taken to be the center of the 0.15
void. In Eq. (2) the quadrupole moments are calculated
0.10 (c) one void
with zˆas the preferred axis, whereas the physically pre- )
m 0.05
ferred axis is the line connecting the void centers. Thus, Q2
0.00
we transform the moments to the more natural coordi- (
nates using D-matrices45: nt −0.05
e
m −0.10 m=0
DQmllm′m((θα′,,φβ′)) == eP−lmim′=′α−dllmD′mml (′mβ)(,α,β)Qlm′(θ,φ), (3) le mo −00..1150
o 0.05
where the Eulerangle αdefines the rotationbetweenco- p
u 0.00
ordinatesaxesin(xy)plane(correspondingangleφ);and dr m=1, Re
the Euler angle β describes the rotation in z-axis (corre- ua −0.05 m=1, Im
sponding angle θ). dl (β) can be found from tables.46 q −0.10 m=2, Re
mm′ m=2, Im
ThequadrupolemomentsareplottedforthevoidsAand −0.150.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25
B from the simulation with ILD0=1.00 in Figs. 7(a) and linear void size (f1/3)
(b), respectively. One sees from Fig. 7(a) that void A
becomes markedly elliptical in the direction of the other
FIG. 7: (a) Quadrupole moments (2) of the surface of void
void, as represented by Q20, when f¯1/3 ≃ f¯c1/3 ≃ 0.106 A from the start of the simulation until coalescence. The
(from Table II). The quadrupole data from an iden- coordinate axes have been rotated using Eq. (3) so that the
tical simulation but with only one void are plotted in z-axis is aligned with the center of void B. The initial inter-
Fig. 7(c) as a control, and a smaller variation in the void ligament distance is ILD0=1.00 and the strain-rate is
quadrupole moment is observed simply due to fluctua- ε˙ = 109/sec. (b) Quadrupole moments for the void B, lo-
cated at [0.1778,0.0829,0.0387]L from void A. Now Eq. (3)
tions in the atomistic growth. Other cases with larger
has been used so that the positive z-axis is toward the void
ILD lookaboutthe sameasFigs.7(a)and(b)although
0 A. (c) Quadrupole moments (not rotated) for a single void
the trend and especially the transition point may not
in a simulation with same box size as in (a) and (b) and the
be as clear. This variation may be due to the elastic strain-rate ε˙=109/sec.
9
and plastic interactions causing random shape evolution 10−2
longer before the critical ILDc (ILD=0.5d) is reached in one void
cases when the voids are initially separatedfurther from (a) ILD=4.6
0
eachother(largerILD ). SeeFig.7(c)forthesinglevoid ILD=1.8
0 0
cf¯a1s/e3a=sa0n.0e9xaismwplheeonftthheerdainsldoocmatisohnapderievveonluvtoioidn.gTrohwetrhe n (f) IILLDD00==11..52
o ILD=1.0
starts, see inset of Fig. 4(b). Thus, in larger ILD cases i 0
0 ct ILD0=0.5
the random shape evolution suppresses the transition to a exp(200ε)
r
the shape evolution due to the other void. d f
i
o
v10−3
V. VOID VOLUME EVOLUTION UNDER THE
INFLUENCE OF THE SECOND VOID
1.50 2.00 2.50 3.00 3.50 4.00
In Ref. 27, we briefly discussed how void growth after
the onsetofinteractionbut priorto coalescenceis differ- strain (ε) [ % ]
ent from void growth for an isolated void. We address
]
this further here. How does the correlated growth differ )
ε
from the exponential growth of an isolated void?12,16 00 (b)
2
p(10−5
x
e
A. A Pair of Voids /
f
[
n
o
We first examine the volume evolution for the same i
t
c
two-voidsimulationsdescribedabove. Figure8(a) shows a
r
thevoidfraction,f =Vvoid/V,forthevoidA(alsointhe d f
case of a single void in the same box size) with respect i
o
tothestrainbeforecoalescence. As canbe seenfromthe v
d
figure the voidgrowsas exp(200ǫ),at leastforthe larger e
l
ILD0’s and for the single void in the box. In Fig. 8(b) ca10−6
we have factored out the asymptotic growth rate from f s 0.09 0.11 0.13 0.15 0.17 0.19
in order to emphasize the differences between the curves 1/3
linear void size [f ]
andplottedversuslinearvoidsizef1/3. Thevoidgrowth
dataforILD =4.62and1.81coincidewiththesinglevoid
0
FIG.8: (a)Growth of thevoidAuntilcoalescence presented
curve. The void growth rate with smaller ILD ’s reach
their asymptotic growth rate earlier. In the fi0gure we by void fraction f = Vvoid/V versus strain for ILD0=0.50,
1.00,1.20,1.50, 1.81, and 4.62 diameters as well as in a single
have drawn as circles the void size values, where the dy- void case (in the same box size) at ε˙=109/sec. The asymp-
namic ILD’s cross the line ILD=0.5d in Fig. 5(b). As toticbehavior(beforefinitesizeeffects)isexponentialgrowth
can be seen from figure, there is no significant change in with exp(200ǫ) as seen from the line drawn as a guide to the
the void volume behavior when the voids start to inter- eye. (b)Thevoidsizef presentedin(a)hasbeenscaledwith
act. Therefore we conclude that the void growth rate is theexponentialand plotted versuslinear void size f1/3. The
not affected by the interaction between the voids: thus circles point where the dynamical ILD’s cross the horizontal
the interaction cannot be detected through the growth line ILD=0.5d in Fig. 5(b).
rate. The key factor for the void growth rate is the rate
at which the dislocations separate from the void, and it
appears to remain unchanged in the vicinity of the sec- boxsizesV =L3 inordertofindthecoalescenceprocess
0
ondvoid. However,the locationatthe voidsurfacefrom ofthevoidwithits(six)periodicimage(s),similartothe
wherethedislocationloopsseparateisaffectedbythein- manner in which some continuum calculations of coales-
teraction with another void, as seen in Section IV in the cence havebeen done (cf. Refs.9 and43). The details of
accelerated reduction of ILD, in the movement of void thedifferentboxsizesarereportedinTableIII. Figure9
center and in the shapes of the voids. shows the data from this series of simulations. There we
havescaledthevoidfractionf =V /V withthe expo-
void
nential,exp(200ǫ)(asinFig.8)aswellas(r /L)3inorder
0
B. A Single Void Interacting with its Periodic to take into account different initial volumes V = L3 of
0
Images the box. Also (r /L)scalingis applied to the linear void
0
f1/3(horizontalaxis). Fromthefigureoneconcludesthat
Wehavealsoperformedaseriesofsimulationsofasin- thebehaviorisoppositetotheresultsfromthecaseofan
gle void with fixed initial radius size r in various initial isolated pair of voids, Section VA, in an important way.
0
10
a single void forces interstitial loops awayfrom the void.
TABLE III: Sizes for simulated systems with various r0/L This field decreases with the distance from the void as
(r0= 2.17 nmisfixed)asthenumberofFCC cells, theequi- 1/r3. At the near side of a neighboring void, this field
librium side length of the cube L at ambient pressure and
would tend to drive interstitial loops into that void; in
room temperature,andthenumberofatomsintheboxafter
combination, it reduces that void’s own stress field. At
the void is formed. The shortest initial inter-void ligament
distanceILD0 ofthevoidwithitsperiodicimageintheunits thefarside,itaddstotheothervoid’sfield,buttheeffect
of the void diameter d is reported in the last column. is smaller due to the greater distance. For a symmetric
array,theeffectistoreducethemaximumresolvedshear
r0/L FCC cells L atoms ILD0 stress across the surface and delay dislocation emission.
1/3 18×18×18 6.50 nm 23328 0.50
2/9 27×27×27 9.75 nm 78732 1.25
1/6 36×36×36 13.0 nm 186624 2.00
1/8 48×48×48 17.3 nm 442368 3.00
1/10 60×60×60 21.7 nm 860396 4.50
1/20 120×120×120 43.3 nm 6908379 9.00 VI. CONCLUSIONS
] Tosummarize,interactionandcoalescenceoftwovoids
}
)
ε in copper under tension have been simulated in multi-
0
(20 100 rr0//LL==12//39 million-atom MD simulations. The effects of interac-
p 0 tionsbetweenvoidshavebeenquantifiedbytheincreased
x r/L=1/6
3L)e rr00//LL==11//810 rthedeiurctcieonnt-errast,eaonfdthbeyirthseeiprarsahtaipone,ebvyoluthtieonms.ovTemheenvtoiodf
{(r/010−1 r00/L=1/20 interactionhas also been visualizedby detecting the dis-
f/ locationsmovinginthesystemusingthegeneralizedcen-
[
n trosymmetryparameter. Thecriticalinter-voidligament
o
cti distancehasbeenfoundtobeclosetoonevoidradius,in-
ra10−2 dependent ofthe strain-rateorthe initialseparationdis-
f
oid tanceILD0. Theonsetofcoalescenceoccursatthe point
v that the plastic zones surrounding the voids first inter-
d
e act strongly. Signatures of coalescence have been found
al10−3
sc 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 in the dynamic ILD curves and the void center move-
scaled linear void size [f1/3/(r /L)] ments, as explained in detail here, including reference to
0 the stress-strain and void volume curves. A weaker sig-
nature of the onset of coalescence has also been found
FIG. 9: Growth of a single void with varying initial box
in the void shape curves giving the quadrupole moment
size. The initial radius of the void is kept constant r0 =
2.17nm, while the initial side length of the cube is varied as evolution. Ithasbeendemonstratedthatthe interaction
L =6.50nm, 9.75nm, 13.0nm, 17.3nm, 21.7nm, and 43.3nm. ofthe voidsis notreflectedin the volumetric asymptotic
The void fraction f has been divided by (r0/L)3 in order to growthrateofthevoids. Finally,thecoalescenceprocess
takeintoaccountdifferentinitialvolumesV0 =L3 ofthebox. ofanisolatedpairofvoidshasbeenshowntobemarkedly
The vertical axis has been divided also with the exponential differentthanthecoalescenceofasinglevoidwithitspe-
of the strain in order to show the asymptotic behavior as in riodic images, so the latter would not provide a reliable
Fig. 8 for the largest box size. The strain-rate is for all the
description of coalescence in typical low-symmetry con-
simulations ε˙=109/sec.
figurations.
Inthefuture itwouldbeinterestingtostudythecases
with uniaxial expansion (and the various orientations of
Thesmallertheboxsize,andhencethesmallertheILD ,
0 thevoidswithrespecttotheexpansiondirection),differ-
the later the void starts to grow. This is easy to under-
entsizes ofthe voidsrelative to eachother, othercrystal
stand since measured from afar the two separate voids
structures as body-center cubic and hexagonal lattices,
act as one big void, and thus grow faster than a smaller
and systems including larger collections of voids.
void. Inasinglevoidwithitsperiodicimagethatpicture
doesnothold. Forexample,thereisnodistanceatwhich
the stress field approaches a single void stress field. The
Acknowledgments
periodicimageonlyrestrictsthegrowthofthevoid. Ata
moremechanisticlevel,theneteffectofthearrayofvoids
is to reduce the resolved shear stress driving dislocation This work was performed under the auspices of the
emission from the void surface, whereas a single nearby U.S. Dept. of Energy by the University of California,
void enhances this resolved shear stress on some regions LawrenceLivermoreNationalLaboratory,undercontract
of the void surface. An intuitive way to understand this no. W-7405-Eng-48. We would like to thank R. Becker
phenomenon is to consider that the shear stress field of for useful discussions.