Biomedical Engineering Reference
In-Depth Information
motion are used to relax the pressure on structure in all three directions.
During stretching, the MD computational cell is stretched in the loading
direction using a modified version of the NPT equations of motion [83].
NPT equations ensure that the structure has lateral pressure relaxed to
atmospheric values during deformation [32, 39]. In this algorithm, the rate
of change of a simulation cell volume, V(t), is specified using a barostat
friction coefficient parameter
η
such that:
d t Þ
dt ¼
1
Nk B T ext t
V ð t Þð P P ext Þ
½ 5
:
3
2
P
and
dV ð t Þ
dt ¼ 3 t Þ V ð t Þ
½ 5
:
4
where P is the instantaneous pressure, P ext is the externally applied pressure,
N is total number of atoms in the system, k B is the Boltzmann constant, T ext
is the external temperature, and
τ P is a specified time constant for pressure
fluctuations. For a given cross-sectional area, the specification of
in
equation 5.4 is equivalent to specifying strain rate for the change in
simulation cell length. Further, for a given
η
η
, equation 5.3 can be modified
as:
d t Þ
dt ¼
1
Nk B T ext t
V ð t Þð P P ext ÞgZ
½ 5
:
5
2
P
In equation 5.5, the term
acts as a damping coefficient for reducing
fluctuations in pressure during the stretching of the simulation cell. During
the simulations, the system is initially equilibrated at T ext =300 K. After
equilibration, the computational cell is stretched in the loading direction
using
γη
=0.1 psec 1 . The values of
η
γ
=0.5 and P ext =1 atmospheric
￿ ￿ ￿ ￿ ￿ ￿
pressure are used.
The values for
are calculated in trial calculations that focused on
achieving the best balance between simulation time (low
η
and
γ
results in long
simulation times and vice versa) and pressure fluctuations (high
η
γ
results in
excessive pressure damping with increase in residual stresses along periodic
boundaries). In the analyses reported in this chapter, the MD equilibration
time in between the periods of stretching is chosen as 2.0 psec.
5.6
Dynamic fracture analyses
As pointed out earlier, simulations focus on comparing the CFEM analyses
of the effect of the second-phase SiC particles with the MD analyses. MD
results are analyzed using visuals as well as virial stress-strain relations.
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