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Philosophical Magazine
ISSN: 1478-6435 (Print) 1478-6443 (Online) Journal homepage: http://www.tandfonline.com/loi/tphm20
Multiscale dislocation dynamics simulations of
shock-induced plasticity in small volumes
Mutasem A. Shehadeh
To cite this article: Mutasem A. Shehadeh (2012) Multiscale dislocation dynamics simulations
of shock-induced plasticity in small volumes, Philosophical Magazine, 92:10, 1173-1197, DOI:
10.1080/14786435.2011.637988
To link to this article: http://dx.doi.org/10.1080/14786435.2011.637988
Published online: 04 Jan 2012.
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Philosophical Magazine
Vol. 92, No. 10, 1 April 2012, 1173–1197
Multiscale dislocation dynamics simulations of shock-induced
plasticity in small volumes
Mutasem A. Shehadeh*
Department of Mechanical Engineering,
American University of Beirut, Beirut, Lebanon
(Received 28 June 2011; final version received 26 October 2011)
Multiscale dislocation dynamics plasticity (MDDP) was used to investigate
shock-induced deformation in monocrystalline copper. In order to enhance
the numerical simulations, a periodic boundary condition was implemented
in the continuum finite element (FE) scale so that the uniaxial compression
of shocks could be attained. Additionally, lattice rotation was accounted
for by modifying the dislocation dynamics (DD) code to update the
dislocations’ slip systems. The dislocation microstructures were examined
in detail and a mechanism of microband formation is proposed for single-
and multiple-slip deformation. The simulation results show that lattice
rotation enhances microband formation in single slip by locally reorienting
the slip plane. It is also illustrated that both confined and periodic
boundary conditions can be used to achieve uniaxial compression;
however, a periodic boundary condition yields a disturbed wave profile
due to edge effects. Moreover, the boundary conditions and the loading rise
time show no significant effects on shock–dislocations interaction and the
resulting microstructures. MDDP results of high strain rate calculations are
also compared with the predictions of the Armstrong–Zerilli model of
dislocation generation and movement. This work confirms that the effect of
resident dislocations on the strain rate can be neglected when a homoge-
neous nucleation mechanism is included.
Keywords: dislocation dynamics; shock wave; homogeneous nucleation;
lattice rotation; boundary condition; multiscale simulation; dislocation
structure; slip band
1. Introduction
Most of the work that has been carried out in the area of shock compression of solids
is experimental in nature. This can be attributed to the transient nature of shocks,
complexities of geometries, nonlinearities of the material [1] and the multiscale
nature of the deformation process. Nonetheless, there has been considerable progress
in simulating the dynamic response of solids. In the case of shock-induced
deformation, the vast majority of the simulations have been carried out at a single
length scale, namely the finite element (FE) continuum scale [2–6] or molecular
dynamics (MD) atomistic scale [7–10]. Motivated by the limitations of the
*Email: ms144@aub.edu.lb
ISSN 1478–6435 print/ISSN 1478–6443 online
ß2012 Taylor & Francis
http://dx.doi.org/10.1080/14786435.2011.637988
http://www.tandfonline.com
continuum constitutive relations and the restrictions of the time and length scales in
atomistic simulations, dislocation dynamics (DD) and multiscale modelling have
attracted much attention in the past two decades.
Several DD models have been developed to simulate plastic deformation in single
crystals [11–15]. DD has also been coupled with continuum FE to correct for the
actual boundary conditions. The principle of superposition was used for the two-
dimensional (2D) coupling [16] and was extended later for the three-dimensional
(3D) coupling [17–20]. Although DD/FE models have been utilised to investigate
different deformation problems, such as hardening mechanisms, size effects and
dislocation–particle interactions, limited research on multiscale modelling of shock
compression has been reported. Smirnova et al. [21] used a combined MD and FE
model to study the propagation of laser-induced pressure in a solid. Li and Sheng
[22] developed a multiscale MD framework to investigate among other things the 2D
shock wave propagation and interaction with a single dislocation. Barton et al.[23]
used a dislocation density approach to develop a multiscale model of material
strength under shock loading. Shehadeh et al. used multiscale dislocation dynamics
plasticity (MDDP) to investigate shock-induced plasticity due to the collective
behaviour of dislocations in copper and aluminium single crystals [24–29].
The planar shock condition in solids is attained over a rise time period that
ranges between hundreds to fractions of nanoseconds. Consequently, a uniaxial
strain condition with no edge effects from the boundaries is produced. In modelling
the uniaxial compression, different boundary conditions are used depending on the
length scale. For example, in FE simulations, uniaxial strain is achieved either by
applying a confined boundary condition (CBC), where the sides of the simulation
domain are prevented from moving in the lateral direction [2,24–29], or by using
infinite elastic elements, which act as non-reflective boundaries [3]. At the atomistic
scale, on the other hand, a periodic boundary condition (PBC) is commonly used to
simulate the repeated nature of the crystal lattice. Additionally, it has been shown
that a PBC is suitable for shock simulations as the edge effects are minimised and the
uniaxial strain condition is achieved [8].
In contrast to single length scale modelling, multiscale simulations require the
implementation of relevant and consistent boundary conditions at different length
scales. In their previous works on multiscale shock deformation, Shehadeh et al.
[24–26] and Cheng and Shehadeh [27,28] used different boundary conditions at
different length scales whereby a CBC is used at the macroscale to generate the shock
condition, while a PBC is used in the microscale to mimic the periodicity of the
lattice. This raises the issue of the inconsistency in boundary conditions in multiscale
simulations.
It is well known that under finite deformation, crystalline materials accommodate
the applied strain not only by dislocation glide but also by plastic spin [30,31]. In
shocked samples, the observed and simulated dislocation microstructures consist
mainly of heavy deformation bands coincident with the slip planes [24,25,32–34].
These deformation bands are locally inhomogeneous leading not only to significant
lattice strain but also to lattice rotation [35–38]. In spite of the possible significance
of lattice rotation on the resulting dislocation microstructure, this effect has been
neglected in the published simulations of shock deformation.
1174 M.A. Shehadeh
The aim of this paper is to enhance the multiscale simulations of shock-induced
plasticity in single crystal copper by investigating the following issues: (i) the
evolution of the dislocation microstructure due to shock interaction with pre-existing
dislocation sources; (ii) the effect of boundary conditions and the shock rise time on
wave profile characteristics and dislocation–shock interaction; (iii) the role of lattice
rotation on strain localisation and microband formation in a single slip deformation.
Finally, MDDP results will be compared with Armstrong–Zerilli (AZ) model of
strain-rate-dependent dislocation movement and generation.
2. Simulation model
2.1. MDDP
MDDP [39] is a multiscale model that links dislocation dynamics computations with
continuum FE analysis. In this framework, plastic deformation of single crystals is
obtained in a dislocation dynamics (DD) code by the explicit evaluation of
dislocation evolution history. Dislocation lines and curves are represented by discrete
straight segments that interact with each other over short and long ranges. At each
time step, the Peach–Koehler (PK) force is calculated at each dislocation segment by
accounting for the dislocation interactions among each other, the dislocation
interactions with other defects, the image force from the boundary, the Peierls force
and the externally applied load. It is assumed that the dynamics of the dislocations is
of a Newtonian type, which consists of inertia, viscous and loading terms such that:
ms_
vþ1
MsðT,pÞv¼Fs:ð1Þ
In the above equation the subscript sstands for the dislocation segment, msis
defined as the effective dislocation segment mass density, Msis the dislocation
mobility and Fsis the PK force. Hirth et al. [40] derived the following expressions for
the effective mass per unit dislocation length for screw, (ms)
screw
, and edge, (ms)
edge
,
dislocations when moving at a high speed:
ðmsÞscrew ¼W0
v2ð1þ3Þ
ðmsÞedge ¼W0C2
v4ð16l401
lþ83
lþ14þ501223þ65Þ
ð2Þ
where l¼ð1v2=C2
lÞ1
2,¼ð1v2=C2Þ1
2,Clis the longitudinal sound velocity, Cis
the transverse sound velocity, is the dislocation velocity, W0¼Gb2
4ln R=r0
ðÞis the
rest energy for the screw per unit length and Gis the shear modulus. The value of Ris
typically equal to the size of the dislocation cell (about 1000 b, with bbeing the
magnitude of the Burgers vector). The solution of the Newtonian equations yields
the dislocation nodal velocities, which are then used to calculate plastic strain rate
and plastic spin:
_
"p¼X
N
i¼1
livgi
2Vnibiþbini
ðÞ,ð3Þ
Philosophical Magazine 1175
Wp¼X
N
i¼1
livgi
2Vnibibini
ðÞ,ð4Þ
where _
"pis the plastic strain rate, Wpis the plastic spin, l
i
is the dislocation segment
length, v
gi
is the dislocation glide velocity, n
i
is a unit normal to the slip plane, bis the
slip direction, Vis the volume of the representative volume element (RVE) and Nis
the total number of dislocations segments within a given finite element. At the
macroscale, it is assumed that the material obeys the basic laws of continuum
mechanics, i.e. the linear momentum balance and the conservation of energy equation:
div S¼_
vð5Þ
Cv
_
T¼Kr2TþS_
"p,ð6Þ
where Sis the stress, Tis the temperature, vpis the particle velocity and ,C
v
and K
are mass density, specific heat and thermal conductivity, respectively. For elasto-
viscoplastic behaviour, the strain rate tensor _
"is decomposed into an elastic part _
"e
and plastic part _
"p:
_
"¼_
"eþ_
"pð7Þ
For most metals, the elastic response is linear and can be expressed using the
incremental form of Hooke’s law for large deformation such that:
S
¼Ce
½
_
"_
"p
½,ð8Þ
where S
is the co-rotational stress rate and Ceis the elastic stiffness tensor.
Under high pressure loading, the elastic properties of metals become pressure
dependent. The experimental results of Hayes et al. [41] for the shear modulus (G)
and Poisson’s ratio () of isotropic copper are utilised in the constitutive equation to
account for the dependence of elastic properties on pressure such that:
G¼G0þ0:89P05P560
G0þ53:4þ0:40P60 5P5100
,ð9Þ
¼
0þ1:70 P1012,ð10Þ
where Pis in GPa and G
0
and
0are the shear modulus and Poisson’s ratio under
normal static loading conditions.
The momentum equation is then solved using dynamic FE such that:
M½
€
U
þC½_
U
þK½Ufg¼ f
,ð11Þ
where M½is the mass matrix, C½is the damping matrix, K½is the stiffness matrix,
Ufgis the nodal displacement and f
is the force vector. The implementation of the
above nonlinear elastic model in the FE framework requires the adjustment of the
stiffness matrix at every time step to account for the pressure-dependant elastic
properties. In the dynamic FE model, a forward explicit integration scheme is used to
solve for the displacement vector Ufgsuch that:
Ukþ1
i¼Uk1
iþ2Uk
it2Ki
Mi
Uk
iþfit2
Mi
þDfact
ð1þDfactÞUk1
i,ð12Þ
1176 M.A. Shehadeh
where, Dfact ¼Cit=2Mi. In the following two sections, modifications that have
been introduced to implement periodic boundary conditions in the FE model and
slip rotation in DD are presented.
2.2. Implementation of FE periodic boundary condition
The implementation of the periodic boundary condition is based on the idea of
master and slave nodes where the displacement of a slave node on one side is linearly
dependent on the displacement of the corresponding master node on the opposite
side of the simulation domain. Using the periodicity assumption for the nodal
displacement yields:
U
S
i¼U
M
i,ð13Þ
where Mdenotes a master node, Sdenotes a slave node and iis the nodal degree of
freedom. In addition to the displacement condition, the traction distribution on the
slave and the master nodes must be the same, i.e.
f
S
i¼f
M
i:ð14Þ
It can be proven that the traction condition is automatically satisfied by imposing
the displacement condition [42].
By combining Equations (13)–(14) with Equation (12), the nodal displacements
for the master and the slave nodes can be expressed as:
U
Mkþ1
i¼
U
Mk1
iþ2U
Mk
i1
2t2K
M
i
M
M
i
þK
S
i
M
S
i
0
@1
AU
Mk
iþ1
2f
M
i
t21
M
M
i
þ1
M
S
i
0
@1
A
þ1
2
Dfact
M
ð1þDfact
M
Þ
þDfact
S
ð1þDfact
S
Þ
0
@1
AU
Mk1
i:ð15Þ
When using a computer program, it is necessary to specify the slave and the
master nodes [43]. The selection of the master and slave nodes on each side of the
computational cell is randomly generated. Hence, the number of master and slave
nodes on each side of the cell may not be identical.
2.3. Computation of slip rotation
Crystal lattice responds to the induced plastic spin given by Equation (4) by
reorienting its slip systems via the relations:
_
n¼!n,ð16Þ
_
b¼!b,ð17Þ
Philosophical Magazine 1177
where _
nis the unit normal rotation, _
bis the Burgers vector rotation and !is the spin
of the microstructure and is given as the difference between the material spin Wand
the plastic spin Wp:
!¼WWp:ð18Þ
In DD analysis, dislocation lines are sorted in each FE (subcell) to account for
the elemental plastic spin, which is then used to compute lattice rotation. In fcc
materials, there are 12 possible slip systems as combinations of four slip planes and
three slip directions. By looping over all the dislocation segments in each subcell, the
glide planes and directions are identified and then used to update both nand b
such that:
nnþ1¼nnþ!nnt,ð19Þ
bnþ1¼bnþ!bnt:ð20Þ
3. Simulation setup
MDDP simulations were performed to investigate shock deformation in copper
single crystals. The material properties used in this work are summarised in Table 1.
As illustrated in Figure 1, the simulation domain is a prismatic bar with a square
cross-section 2.5 mm2.5 mm and height 5 mm. In the FE code, a ramp shock wave is
generated by applying a velocity-controlled boundary condition on the upper surface
such that the velocity increases linearly to its maximum value over a finite rise time
period. The velocity is then held constant for a period equal to the shock holding
time. The total pulse duration in the current simulation was nearly 2.0 ns,
corresponding to a strain rate of approximately 10
8
s
1
. To achieve the uniaxial
strain condition of shock waves, the four sides of the domain are either confined such
that the surfaces can only move in the loading direction and fixed in the transverse
directions, or are periodic, as discussed in Section 2.2. The simulation domain is
oriented in the [0 0 1] direction and a few dislocation sources are randomly placed on
the slip planes (Figure 1). A periodic or symmetric boundary condition is applied in
DD to ensure the conservation of dislocation flux and the continuity of dislocation
lines across the boundaries.
Table 1. Material properties for copper.
Shear modulus (Pa) 48.3 10
9
Poisson’s ratio 0.32
Mass density (kg/m
3
) 8900
Heat capacitance (J/kg K) 385
Heat conductivity (W/m K) 398
Dislocation mobility (1/Pa s) 1000
Burgers vector (m) 2.6 10
10
1178 M.A. Shehadeh
4. Results
4.1. Dislocation source–shock interaction
Figure 2 shows the dislocation microstructure developed in copper single crystal,
during the passage of an 8 GPa shock wave. Clearly, the morphology of the
microstructure consists of dense and intersecting microbands confined on the slip
planes. The separation distances between these dense bands are of the order of
hundreds of nanometres, and the spaces between them are full of dislocation slip
bands. This simulated microstructure is similar to what has been observed
experimentally using TEM and SEM of shocked copper monocrystals [44].
Moreover, it is interesting to see that during the first two nanoseconds of shock–
dislocations interaction, the thickness of the primary dense bands grow to
approximately 30 nm.
The microstructure presented in Figure 2 was investigated in more detail by
taking snapshots of its evolution, as depicted in Figure 3. The dislocation sources are
impacted and activated at a very high rate leading to the accumulation of the
dislocation slip on the primary slip planes (Figure 3a). As the dislocation glide
proceeds, both cross slip and double cross slip mechanisms become possible
(Figure 3b), giving rise to microband formation (Figure 3c).
Figure 1. Simulation setup showing the boundary conditions and the initial dislocation
distribution.
Philosophical Magazine 1179
Figure 3. The evolution of the microstructure during shock–dislocation interaction: (a) initial
configuration; (b) dislocation activation and accumulation on the slip plane; (c) activation of
cross slip on the conjugate plane; (d) double cross slip where the dislocations resume their glide
on the primary slip plane.
Figure 2. Dislocation microstructure formed in the first two nanoseconds of an 8 GPa shock
propagation in copper. Microbands are formed on all available slip planes.
1180 M.A. Shehadeh
It is worth pointing out that due to the extremity and the transient
nature of the loading, thousands of dislocation loops are formed in fractions
of a nanosecond. Therefore, the simulations can only track shock-induced
plasticity in the first few nanoseconds during the transmission of the wave.
In spite of the time limitation, MDDP simulations are very powerful in the
sense that in situ analysis of the dislocation microstructure is possible. In the
future, these results can hopefully be compared with in situ microstructure
characterisation instead of the post-deformation analysis that is available
nowadays.
4.2. The effect of FE boundary conditions
In order to investigate the effect of FE boundary conditions on shock-induced
plasticity, the simulation carried out in Section 3 was repeated with the exception
that a PBC is used in both FE and DD. Figure 4 shows the deformed shape of the
upper surface when using a PBC. It is obvious that the deformed sides are of uneven
distribution of slave and master nodes, and thus discrepancies with the confined case
are inevitable.
Figure 4. Deformed FE mesh for PBC.
Philosophical Magazine 1181
Figure 5 compares the stress components of the confined and periodic cases. The
following can be deduced from the plots:
(1) PBC can capture the longitudinal component of the shock wave to a good
level of accuracy. As manifested by Figure 5a, the longitudinal component
(
33
) of the PBC is slightly lower but in a very good agreement with that of
the CBC.
(2) PBC results in relaxed peaks and disturbed wave plateau of the lateral
stresses (
11
,
22
).
(3) While confined compression yields no shear components of stress, consid-
erable components of the shear stresses (1 GPa) are generated in the
periodic case, as illustrated in Figure 5b.
The discrepancies in the PBC components can be attributed mainly to the edge
effects of the boundaries resulting from the uneven distribution of slave and master
nodes. Since the selection of the slave and master nodes is random, it is expected that
the difference between the number of slave and master nodes diminishes as the FE
mesh density increases.
Figure 6 compares the overall influence of the boundaries on the effective von
Mises stress. Although the shear components developed in the PBC notably alter the
wave plateau and generate higher peak stress, the dislocation multiplication rate
decreases, as can be seen in Figure 7. This contradictory effect is mainly attributed to
Figure 5. Shock wave profile generated using confined and periodic boundary conditions. (a)
Normal stress components for periodic and confined boundary conditions. (b) Shear stress
components developed in the periodic and confined boundary conditions.
1182 M.A. Shehadeh
the lower value of the resolved shear stress in PBC due to the fluctuations in the signs
of shear components (Figure 5b). In the effective stress calculations, the absolute
values of the shear stresses are accounted for with no consideration of their sense
which undoubtedly affects the resolved shear stress on the dislocations.
4.3. The effect of the DD boundary conditions
The characterisation of post-deformation microstructure is very important to
develop a better understanding of strain localisation. Strain localisation is inherently
related to dislocation pattern configuration, which is affected by the highly
correlated dislocation motion, imposed strain rate and dislocation boundary
condition. In this section, the effect of DD boundary conditions on the dislocations
interaction and slip activation are investigated assuming CPC in FE. In order to
simulate a repeated unit cell, either periodic or symmetric (reflective) boundary
Figure 6. Effective von Mises wave profile.
Figure 7. Dislocation density histories of the confined and the periodic boundaries.
Philosophical Magazine 1183
condition is used. Symmetric boundary condition ensures the continuity of the
dislocation curves across the boundary while periodic boundary condition ensures
both the conservation of the dislocation flux across the boundary as well as
continuity.
Figure 8 depicts the results of slip contributions to the overall dislocation density
using periodic and symmetric DD boundary conditions. It can be seen that
essentially the same slip planes are activated in both cases, and their relative
contributions to the overall dislocation density has not changed much. However,
periodic boundary condition results in significantly higher dislocation density due to
the conservation of the dislocation flux.
4.4. The effect of lattice rotation
Experimental examination of the dislocation microstructure of shocked samples
shows that in addition to the formation of localised bands, the material also
accommodates the applied strain by lattice rotation [36–38,45]. The purpose of this
section is to quantify the effect of lattice rotation on shocked samples by carrying out
two identical simulations with and without the consideration of lattice rotation.
In doing so, a ramp shock wave of 14 GPa peak pressure and 0.05 ns rise time is
launched in the simulation domain, which consists of a prismatic bar of
1.0 mm1.0 mm2.5 mm dimensions. In order to simplify the interpretation of the
results under such extreme condition, the crystal is oriented for a single slip, and only
a single dislocation source is placed in the middle of the simulation domain.
Figure 9 depicts the dislocation microstructure with and without incorporating
lattice rotation. Dislocation walls are formed in both cases due to the accumulation
of slip bands; however, lattice rotation yields much thicker walls that are locally
misoriented with respect to the original (111) plane. It is interesting to see that
although the crystal is oriented for a single slip with no possibility of dislocation
cross slip, the dislocation walls thicken from 0.8 nm to around 40 nm in less than
Figure 8. The effect of DD boundary conditions on dislocation densities. Slip planes 1, 2, 3
and 4 are ð
1
11Þ,ð111Þ,ð
111Þand ð111Þ, respectively.
1184 M.A. Shehadeh
1.0 ns. Since cross slip is prohibited, the underlying mechanism of wall thickening is
likely due to local lattice rotation and can be examined by taking snapshots of the
evolution of the microstructure.
Figure 10 shows that lattice rotation activates a kink mechanism that allows
neighbouring dislocations to locally change their slip planes. Thus, the originally
straight slip planes become curved due to their local reorientation, leading to the
observed thickening of the walls. A closer examination of the substructure within the
wall illustrates that ‘immobile junction-like’ segments are formed especially near
the sample boundaries where the dislocation density is very high (Figure 11). One can
envision the formation mechanism of these immobile segments as a consequence of
the short-range interactions of neighbouring dislocations that are gliding on the
newly reoriented and intersecting slip planes. In some ways, this mechanism is similar
to the coplanar junction formation whereby a sessile junction is formed as a result of
the short-range interaction between two dislocations having the same Burgers vector
and moving on conjugate slip planes. To the best of the author’s knowledge, such a
Figure 10. Snapshots showing: (a) initially the dislocations are confined on the original slip
plane; (b) as the dislocation density increases the slip plane becomes disoriented and a kink-
like mechanism is activated and the slip band becomes thicker; (c) the slip band becomes a
microband.
Figure 9. Dislocation microstructure (a) without lattice rotation, (b) with lattice rotation.
Philosophical Magazine 1185
mechanism has not been reported elsewhere probably due to the common belief that
such an interaction cannot occur in single slip deformation.
In accordance with recent experimental observations [38], the local distributions
of the dislocation densities within the deformation bands are not homogeneous
(Figure 12). Moreover, the inclusion of lattice rotation makes the localised bands
more heterogeneous. This manifested by the increase in the local dislocation density
from 1.5 10
17
m
2
without lattice rotation to 2.5 10
17
m
2
with lattice rotation.
The increase in the dislocation density can be further investigated by quantifying the
Figure 12. Contour plots of the dislocation density distribution (a) without lattice rotation,
(b) with lattice rotation.
Figure 11. Dislocation substructure across the slip plane showing a few (red coloured)
‘immobile’ dislocation segments formed near the boundary.
1186 M.A. Shehadeh
effect of lattice rotation on the resolved shear stress (res ) which is related to the
applied stress () via Schmid equation:
res ¼cos cos ,ð21Þ
where is the angle between the axis of loading and the normal to the slip plane, and
is the angle between the axis of loading and the slip direction. The time evoluation
of Schmid factor (cos cos ) is tracked for all dislocations located in the central
region of the computational cell. As can be seen in Figure 13, the Schmid factor
gradually increases from its initial value of 0.408 to around 0.47, leading to higher
rates of dislocation multiplication.
The emergence of lattice rotation is a direct consequence of plastic distortion in
the crystal. The FE results for lattice rotation (Figure 14) show that distinct regions
of local 3D in- and out-of-plane lattice rotation are formed. Moreover the
deformation within the microbands is highly heterogeneous and sub-boundaries
Figure 14. Contour plot of the effective plastic strain (a) without lattice rotation and (b) with
lattice rotation.
Figure 13. Time evolution of the Schmid factor in one FE subcell.
Philosophical Magazine 1187
are formed within the deformation band. The misorientation angles within the bands
may exceed 0.30 rad (20) which is of the same order of magnitude as what is
observed in electron backscattering results [37,39] and X-ray microdiffraction of
polycrystalline copper [46]. Therefore, the plane deformation treatment of lattice
rotation need to be revisited [47]. Figure 15 compares the effective plastic strain with
and without the incorporation of lattice rotation. Obviously, the magnitude of the
maximum effective plastic strain has increased considerably from less than 0.20
(point 1) to more than 0.28 (point 1*). This increase in the effective plastic strain is
accompanied by the increase in the deformation bandwidth.
4.5. Rise time and mesh sensitivity
MDDP was previously used to investigate the effect of rise time on the homogeneous
nucleation of dislocations [26]. In this section, the effect of rise time on shock wave
profile characteristics and the resulting dislocation sources activation is examined.
Figure 16 shows the normalised shock wave profiles produced at different rise times
ranging between 0.17 ns to 0.70 ns. Clearly, the wave plateau fluctuates about steady
Figure 16. Effect of rise time on the wave profile.
Figure 15. Contour plot of the plastic spin components (a) w1, (b) w2, (c) w3.
1188 M.A. Shehadeh
state pressure that increases with rise time, while the period required for plateau
relaxation (number of fluctuations) decreases with rise time. For any FE signal, the
particle velocity (Up), the bandwidth (), the frequency ($) and the FE size (Dx) are
related by:
Up¼$,ð22Þ
¼Dx
NFE
,ð23Þ
where N
FE
is the number of simulation steps within each FE. Determination of the
time step under extreme dynamic loading conditions is critical. In DD, the time step
is dictated by the shortest flight distance for short-range interaction between
dislocations and the time step used in dynamic FE modelling. In this analysis, the
critical time in FE (t
c
) and the time step (t) for DD which yield a stable solution are
given by tc¼l
Cland t¼tc
10, where lis the characteristic length, which is the shortest
dimension in the FE mesh, and C
l
is the longitudinal wave speed. Therefore N
FE
is
taken to be equal to 10.
In ramp waves, the particle velocity and thus the frequency would increase
incrementally from zero to their maximum magnitudes. For short rise time the
maximum velocity is attained almost instantaneously; therefore the material is
accelerated at high rates leading to the pronounced fluctuations in the wave profile
due to inertia effect. For long rise time on the other hand, the loading rate is
considerably smaller, thus; more time is given to the material to accommodate the
increments of loading. Consequently, smaller fluctuations are observed and the
plateau converges faster.
Figure 17 shows a mesh sensitivity analysis of the wave profile at constant rise
time (same rate of loading). Obviously, the frequency of wave fluctuations increases
with mesh density, which can be explained by referring to Equations (16) and (17). It
can be seen that the bandwidth decreases with decreasing mesh size, and hence more
fluctuations are detected. It can also be seen that as the mesh size increases, the wave
profile converges and the wave velocity approaches its theoretical value, similar to
what was reported by Shehadeh et al. [24].
Figure 17. Mesh sensitivity of a wave profile with a rise time of 0.17 ns.
Philosophical Magazine 1189
The effect of rise time on dislocation density evolution is shown in Figure 18.
Two simulations were carried out where a copper crystal is shocked to a peak
pressure of 7.5 GPa at rise times of 0.2 ns and 0.5 ns. Since the width of the wave
front and the increment of pressure are inversely proportional to the shock rise time,
it is found that in the short rise time simulation, the onset of plasticity takes place
first, and that the dislocation multiplication occurs at a higher rate. Contrary to what
was seen in the homogeneous nucleation simulations [26], the rise time in the current
source activation simulations has little effect on the microstructure the dislocation
density.
4.6. Comparison with the Armstrong–Zerilli (AZ) model
Armstrong and Zerilli [48,49] have recently proposed a model for strain rate
dependent dislocation generation in shocked metals. In their model, contributions of
the resident dislocation velocity and the shock induced dislocation generation and
movement are taken into account such that:
d"
dt¼1
mNbvNþdG
dtbDxdþGbvG
,ð24Þ
where Nis the resident dislocation density, vNis the resident dislocation velocity, dG
dt
is the rate of dislocation generation (nucleated at the wave front), Dx
d
is the self-
forming distance of the generated dislocation, Gis the generated dislocation density
with its own average velocity vG,bis the Burgers vector in copper (0.26 nm) and mis
the Taylor factor taken as 2.45. As conventionally done in the low strain rate
regime, the contribution of the rate of multiplication of resident dislocations
dN
dtbDxN
on the strain rate has been neglected in Equation (24). In order to check
the validity of this treatment in the context of MDDP simulations, this term is added
such that:
d"
dt¼1
m
dN
dtbDxNþNbvNþdG
dtbDxdþGbvG
,ð25Þ
Figure 18. The evolution of the dislocation density at different rise times.
1190 M.A. Shehadeh
where Dx
N
is the average dislocation displacement for resident dislocations and
considered here to be equal to the forming distance of the dislocation, 6b. In this
section, the AZ model is compared with MDDP calculations for two cases:
(1) A weak shock where no homogeneous nucleation at the wave from is
considered and only the contributions from resident dislocations movement
and rate of multiplication are computed.
(2) A strong shock where contributions from homogeneously nucleated dislo-
cations as well as resident dislocations are accounted for (all the four terms of
Equation (25)).
4.6.1. Weak shock simulation
A shock wave is launched in a copper single crystal oriented in the [0 01] orientation
at two rise times of 0.2 and 0.5 ns, and a particle velocity U
p
¼65 m s
1
. The
simulation domain consists of a prismatic bar of 1 mm1mm5mm dimensions and
few dislocation sources are randomly distributed on the slip planes with an initial
density of 2 10
11
m
2
. The peak pressure for this loading is about 2.8 GPa and the
average strain rate of the entire domain can be estimated via: d"
dt
ave¼Up
L¼
65
5106¼1:3107s1.
Since the peak pressure is below the threshold for homogeneous nucleation of
loops at the wave front (see Section 4.6.2) the last two terms in Equation (25) can be
dropped.
It is worth mentioning that depending on the FE mesh density, the local values of
the strain rate can reach much a higher value than the average value of 1:3107s1.
During the passage of the shock wave, the overall dislocation density and the average
velocity of the entire dislocation segments are tracked at each time step, as shown in
Figures 18 and 19. By taking the time derivative of the dislocation density evolution,
one can estimate the rate of resident dislocations multiplication, dN
dt.
Figure 20 depicts the evolution of the strain rate resulting from substituting
MDDP results in the AZ model for the two rise times. The instantaneous strain rate
Figure 19. The average velocity of dislocations impacted to an average strain rate of
1.3 10
7
s
1
.
Philosophical Magazine 1191
keeps increasing until it saturates at 2.0 10
8
s
1
, which is an order of magnitude
higher than the average strain rate. The time evolution behaviour of the strain rate
can be explained by referring to the average dislocation velocity (Figure 18). Due to
the relativistic effect, the dislocation velocity cannot exceed the shear wave velocity
(3000 m s
1
in copper). At the instant the shock front impacts the dislocation sources,
the dislocations are accelerated to the limiting shear wave velocity, however, at that
same instant the dislocation density and the rate of multiplication are not high
enough to attain the maximum strain rate. As the dislocation–shock interaction
continues the rate of multiplication and the dislocation density increase while the
average dislocation velocity decreases until the collective behaviour of all the
dislocations yields the maximum value of strain rate.
It is worth pointing out that if the last two terms in Equation (24) are deleted,
then the classical Orowan relation will be reproduced, such as:
d"
dt¼1
mGbvðÞ:ð26Þ
This shows that in the original derivation of the Orowan equation, the
contribution of the rate of multiplication of resident dislocations has been neglected.
In order to examine the contribution of the dislocation rate of multiplication on
the overall strain rate, the results obtained by the AZ model are compared with those
of the Orowan relation, as shown in Figure 21. It can be clearly noticed that the
difference between the AZ model and the Orowan relation, which is due to the rate of
multiplication, is very small. This is attributed to the fact that there is no
homogeneous nucleation of dislocation loops at the wave front. In other words, when
no homogeneous nucleation is present, the contribution of the rate of multiplication
of resident dislocation to the strain rate is negligible when compared to the
dislocations’ movement.
In MDDP, the local values of plastic strain components are computed in all FE
subcells. By taking the time derivative of the effective plastic strain, one can obtain
the von Mises plastic strain rate, which is compared with the Orowan relation. As
expected, Figure 22 reveals that the local value of plastic strain rate is in a very good
agreement with Orowan predictions.
Figure 20. Strain rate evolution as predicted by combining MDDP simulations with the AZ
model at two rise times.
1192 M.A. Shehadeh
4.6.2. Strong shock simulation with homogeneous nucleation
Based on numerous MD shock simulations [50], two conditions must be satisfied for
loop nucleation at the wave front in copper: (i) the critical resolved shear stress is
greater than 2 GPa; (ii) the longitudinal stress along the wave propagation is greater
than a threshold value of 30 GPa. In fact the second condition automatically satisfies
the first condition for uniaxial compression in the [0 01] orientation. In this section, a
MDDP simulation which accounts for homogeneous nucleation of loops at the wave
front is described. The simulation domain consists of a prismatic bar of
0.4 mm0.4 mm1mm dimensions and few dislocation sources are distributed on
the slip planes with an initial density of 6 10
13
m
2
. A shock wave is launched in a
copper single crystal oriented in the [0 0 1] direction at a particle velocity
U
p
¼1000 m s
1
. The peak pressure of this loading is 32 GPa at an average strain
Figure 22. Comparison between the strain rate evolutions obtained using the Orowan relation
and MDDP computations of the effective plastic strain rate.
Figure 21. Comparison between the strain rate evolutions obtained using the AZ equation
and the Orowan relation.
Philosophical Magazine 1193
rate of 1:0109s1. Dislocation loops are nucleated in all available slip planes with
a probability that is proportional to their Schmid factor.
Figure 23 shows snapshots of the dislocation evolution during the travel of the
shock wave in the material. Clearly, the resident dislocation sources are activated
once the resolved shear stress reaches the critical value of source activation
(Figure 23a). As the peak pressure builds up and the threshold value of
homogeneous nucleation is attained, dislocation loops are nucleated and grow
quickly leading to very high dislocation production rate (Figure 23b and c).
Figure 24 shows the dislocation density histories of the pre-existing dislocation
sources, the nucleated loops and the total density. Since the source activation stress is
much lower than the homogeneous nucleation stress, one can see that the total
dislocation density in the initial stages of shock propagation is entirely due to the
Figure 23. MDDP simulation of a 32 GPa shock in copper. (a) The sample initially contains
few pre-existing sources; (b) as the shock wave travels, the sources are activated followed by
homogeneous nucleation of loops at the wave front; (c) the process of loop nucleation
continues while the already nucleated loops grow and move at high rates.
Figure 24. MDDP calculations of the average dislocation density evolution for resident pre-
exiting dislocations, homogeneously nucleated loops and total density.
1194 M.A. Shehadeh
source activation. However, once the nucleation criterion is satisfied, the homoge-
neous nucleation mechanism overwhelms any dislocation production from the pre-
existing sources.
A similar trend is also seen when substituting MDDP results in Equation (25).
Figure 25 reveals that the contribution of the nucleated loops generation and
movement dominates the deformation process. In fact, the strain rate due to
homogeneous nucleation is two orders of magnitude higher than that due to source
activation.
As discussed in Section 4.6, strain rate has contributions from four terms, namely
resident dislocations movement, resident dislocations rate of generation, homoge-
neous loops movement and growth, and homogeneous loops rate of generation.
Detailed analysis of the deformation process is presented here by quantifying the
percentage contribution of the four terms on the induced plasticity. Figure 26
illustrates that resident dislocations movement and rate of generation have negligible
effect on the strain rate. Additionally, the effect of nucleated loops movement is
much higher (480%) than that of loops generation rate.
Figure 26. Percentage contributions of the movement and generation of resident and
nucleated dislocations.
Figure 25. Strain rate evolution due to contributions from the resident dislocations and the
homogeneously nucleated loops. The total strain rate is the sum of the two contributions.
Philosophical Magazine 1195
5. Conclusions
Multiscale dislocation dynamics plasticity (MDDP) simulations were carried out to
address several issues of shock-induced plastic deformation in small volumes. The
effects of boundary conditions, shock rise time and lattice rotation on the wave
profile characteristics and the evolution of the microstructure were studied in detail.
The current results illustrate that:
(1) While uniaxial strain is achieved with high accuracy using confined boundary
conditions, periodic boundary conditions yields a disturbed wave profile due
the edge effect.
(2) For single slip deformation, the incorporation of lattice rotation yields higher
dislocation density and more localised plastic strain.
(3) When heterogeneous nucleation of dislocation sources is considered, shock
rise time has little effect on the induced plasticity and the resulting dislocation
microstructure.
Furthermore, MDDP results are compared with the AZ model of high strain rate
deformation for pre-existing resident dislocations and homogeneous nucleation of
dislocation loops at the wave front. MDDP results are in very good agreement with
the predictions of the AZ model, and confirm that, at high rates of deformation, the
contributions of resident dislocations can be neglected.
Acknowledgments
This research was supported by the Research Board of the American University of Beirut. The
author would like to thank Dr. Hussein Zbib from Washington State University for the useful
discussions during the course of this work.
References
[1] J.A. Zukas and D.R. Scheffler, Int. J. Impact Eng. 24 (2000) p.925.
[2] D.J. Benson, V.F. Nesterenko, F. Jonsdottir and M.A. Meyers, J. Mech. Phys. Solid 45
(1997) p.955.
[3] W. Braisted and R. Brockman, Int. J. Fatig. 91 (1999) p.719.
[4] X.M. Qui, J. Zhang and T.X. Yu, Int. J. Impact Eng. 36 (2009) p.1231.
[5] M. Stoffel, R. Schmidt and D. Weichert, Int. J. Solid Struct. 38 (2001) p.7659.
[6] P.J. Tan, S.R. Reid, J.J. Harrigan, Z. Zou and S. Li, J. Mech. Phys. Solid 53 (2005)
p.2206.
[7] E.M. Bringa, B.D. Wirth, M.J. Caturla, J. Stolken and D. Kalantar, Nucl. Instrum.
Meth. Phys. B 202 (2003) p.56.
[8] E.M. Bringa, J.U. Cazamias, P. Erhart, J. Sto
¨lken, N. Tanushev, B.D. Wirth, R.E. Rudd
and M.J. Caturla, J. Appl. Phys. 96 (2004) p.3793.
[9] L. Holian and P.S. Lomdahl, Science 280 (1998) p.2085.
[10] K. Kadau, T.C. Germann, P.S. Lomdhal and B. Holian, Science 296 (2002) p.1681.
[11] N.M. Ghoneim and L.Z. Sun, Phys. Rev. B 60 (1999) p.128.
[12] L.P. Kubin and G. Canova, Scripta Metall. Mater. 27 (1992) p.957.
[13] K.W. Schwarz and F.K. LeGoues, Phys. Rev. Lett. 79 (1997) p.1877.
[14] H.Y. Wang and R. LeSar, Phil. Mag. A 71 (1995) p.149.
[15] H.M. Zbib, M. Rhee and J.P. Hirth, Int. J. Mech. Sci. 40 (1998) p.113.
[16] E. Van der Giessen and A. Needleman, Model. Simulat. Mater. Sci. Eng. 3 (1995) p.689.
1196 M.A. Shehadeh
[17] M.C. Fivel, C.F. Roberston, G. Canova and L. Bonlanger, Acta Mater.46 (1998) p.6183.
[18] B. Devincre, L.P. Kubin, C. Lemarchand and R. Madec, Mater. Sci. Eng. A 309–310
(2001) p.211.
[19] C. Lemarchand, B. Devincre and L.P. Kubin, J. Mech. Phys. Solid 49 (2011) p.1969.
[20] H. Yasin, H.M. Zbib and M.A. Khaleel, Mater. Sci. Eng. A 309–310 (2001) p.29.
[21] J.A. Smirnova, L.V. Zhigilei and B.J. Garrison, Comput. Sci. Comm. 118 (1999) p.11.
[22] S. Li and N. Sheng, Int. J. Numer. Meth. Eng. 83 (2010) p.998.
[23] N.N. Barton, J.V. Bernire, R. Becker, A. Arsenlis, R. Cavallo, J. Marian, M. Rhee,
H.S. Park, B.A. Remington and R.T. Olson, J. Appl. Phys. 109 (2011) p.073501.
[24] M.A. Shehadeh, H.M. Zbib and T.D. de la Rubia, Phil. Mag. 85 (2005) p.1667.
[25] M.A. Shehadeh, H.M. Zbib and T. de la Rubia, Int. J. Plast. 21 (2005) p.2369.
[26] M.A. Shehadeh, E.M. Bringa, H.M. Zbib, J.M. McNaney and B.A. Remington, App.
Phys. Lett. 89 (2006) p.171918.
[27] G.J. Cheng and M.A. Shehadeh, Int. J. Plast. 22 (2006) p.2171.
[28] G.J. Cheng and M.A. Shehadeh, Scripta Mater. 53 (2005) p.1013.
[29] H.M. Zbib, M.A. Shehadeh, S.M. Khan and G. Karami, Int. J. Multiscale Comput. Eng.
1 (2003) p.73.
[30] V.S. Deshpande, A. Needleman and E. Van der Giessen, J. Mech. Phys. Solid 51 (2003)
p.2057.
[31] V.S. Deshpande, A. Needleman and E. Van der Giessen, Mater. Sci. Eng. A 400 (2004) p.154.
[32] M.A. Mogilevskii and L.S. Bushnev, Combustion, Explosion Shock Wave 26 (1990)
p.215.
[33] J.M. Rivas, S.A. Quinones and L.E. Murr, Scripta Metall. Mater. 33 (1995) p.101.
[34] X. Liu, J. Scaropas and J. Blaauwendraad, Int. J. Solid Struct. 42 (2005) p.1883.
[35] P. Klimanek, V. Klemm, M. Motylenko and A. Romanove, Adv. Eng. Mater. 6
(2004) p. 861.
[36] M.A. Meyers, Y.B. Xu, Q. Xue, M.T. Perez-Prado and T.R. McNelley, Acta Mater. 5
(2003) p.1307.
[37] H. Paul, J.H. Driver, C. Maurice and A. Piatkowski, Acta Mater. 55 (2007) p.575.
[38] M.T. Perez-Prado and G. Doncel, Scripta Mater. 54 (2006) p.915.
[39] H.M. Zbib and T.D. de la Rubia, Int. J. Plast. 18 (2002) p.1133.
[40] J.P. Hirth, H.M. Zbib and J. Lothe, Model. Simulat. Mater. Sci. Eng.6 (1998) p.165.
[41] D. Hayes, R.S. Hixson and R.G. McQueen, High pressure elastic properties, solid-liquid
phase boundary and liquid equation of state from release wave measurements in shock-loaded
copper,inShock Compression of Condensed Matter, M.D. Furnish, L.C. Chhbildas and
R.S. Hixson, eds., American Institute of Physics, Melville, NY, 1999, p.483.
[42] Z. Xia, C. Zhou, Q. Yong and X. Wang, Int. J. Solid Struct. 43 (2006) p.226.
[43] H.L. Soriano and C.C. Nunes, Comput. Struct. 57 (1995) p.439.
[44] B. Gao, D. Lassila, C. Huang, Y. Xu and M.A. Meyers, Mater. Sci. Eng. A 527 (2010) p.424.
[45] R.I. Barabash, Y.F. Gao, G.E. Ice, O.M. Barabash, J.-S. Chung, W. Liu, R. Kro
¨ger,
H. Lohmeyer, K. Sebald, J. Gutowski, T. Bo
¨ttcher and D. Hommel, Mater. Sci. Eng. A
528 (2010) p.52.
[46] H. D. Joo, K. H. Kim, C. W. Bark, Y. M. Koo and N. Tamura, in Proceedings of the
Eight International Conference on Synchrotron Radiation Instrumentation, T. Warwick,
J. Arthur, H.A. Padmore and J. Sto
¨hr, eds., 2004, p.1094.
[47] H. Chen, J.W. Kysar and Y.L. Yao, J. Appl. Mech. 71 (2007) p.714.
[48] R.W. Armstrong and F.J. Zerilli, in Proceedings of the 16th APS Topical Conference on
Shock Compression of Condensed Matter, Nashville, TN, AIP Conference Proceedings,
2009, p.1195.
[49] R.W. Armstrong and F.J. Zerilli, J. Phys. D: Appl. Phys. 43 (2010) p.492002.
[50] J. Li, K.J. Van Vliet, T. Zhu, S. Yip and S. Suresh, Nature 418 (2002) p.307.
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