SUPPLEMENTARY INFORMATION
The Effect of Contact Conditions on the Onset of Shear Instability
in Cold-Spray
Fanchao Meng, Huseyin Aydin, Stephen Yue, and Jun Song*
Department of Mining and Materials Engineering,
McGill University, Montréal, Québec H3A 0C5, Canada
S1. Deformation Control Techniques
To cope with the convergence challenges in cold-spray finite element simulations, many
different deformation control techniques have been explored. For example, adaptive meshing has
been largely adopted to confine the excessive mesh distortion
[1-4]
. However, very frequent
remeshing in adaptive meshing can cause unrealistic deformation shapes for both the particle and
substrate and non-conserving energy variations [1, 3-5], and thus is not adopted in our simulations.
Figure S1 illustrates a typical deformation response in the particle/substrate contact using
adaptive meshing frequency of 30 and mesh sweeps per increment of 1, showing a much
smoothed “jetting” periphery compared with the one without adaptive meshing (see Fig.4 and
Fig. 8 in the main text). Hourglass control and element distortion control have also been studied
by Li et al.
[5]
as alternative deformation control techniques. As reported in Ref. [5], hourglass
control with stiffness technique is particularly useful in problems using reduced integration
elements, whose physical response will not be constrained under this control
[5, 6]
. The element
distortion control can be employed to generate more realistic deformation shapes with good
convergence if a small element distortion control is used (e.g., using a distortion length ratio <
*
Corresponding author: Jun Song. Email: jun.song2@mcgill.ca Tel.: +1 514-398-4592 Fax : +1 514-398-4492
Other Emails: fanchao.meng@mail.mcgill.ca (Fanchao Meng), huseyin.aydin@mcgill.ca (Huseyin Aydin),
steve.yue@mcgill.ca (Stephen Yue).
S1
0.1)
[5, 6]
. In addition, hourglass control and element distortion control can be used together to
further improve the simulation convergence.
In the present study, the effects of hourglass control with stiffness technique and element
distortion control on the onset of shear instability were examined. Figure S2 plotted the
temperature evolution of the corresponding most critical particle element under different
deformation control techniques for a representative full-particle model with friction coefficient 
= 0.5. As shown in Fig. S2, the shear instability initiates around 32 ns when no control technique
is used, being very close to onset time under the element distortion control (with distortion length
ratio 0.1). However, under hourglass control with stiffness or hourglass control together with
element distortion control, the shear instability is completely refrained. Consequently, element
distortion control with distortion length ratio 0.1 is chosen for our simulations in this study.
Fig. S1. Deformed particle/substrate configurations and temperature contours with adaptive meshing. The
simulation is conducted using the half-particle model with the particle velocity being 500 m/s, friction
coefficient being 0.5, and surface separation allowed.
S2
Fig. S2. The simulated temperature evolution curves for the most critical particle element under different
distortion control techniques (DC: distortion control; EDC: element distortion control; HGC: hourglass
control). The simulation is conducted using the full-particle model with the particle velocity being 500
m/s, friction coefficient being 0.5, and surface separation allowed.
S2. Frictional Dissipation
In the work by Li et al.
[5]
, it was demonstrated that in cold spray, the thermal energy from
frictional dissipation is negligible compared to the energy from plastic dissipation for low
friction coefficients (i.e.,  < 0.4). In our study, we further examined the energy contribution
from the frictional dissipation for intermediate and high values of friction coefficient, and found
that the contribution remains negligible. Figure S3 plotted a representative energy evolution
curve under friction coefficient μ = 4, showing the thermal energy generated by friction is order
of magnitude smaller than the one coming from plastic deformation. Therefore we believe that
friction dissipation does not modify the simulation results. The insignificant frictional dissipation
likely is because a) the contacting surface is only limited between the particle outer surface and
the substrate, and b) the relative sliding between particle/substrate is small.
S3
Fig. S3. A representative energy evolution curve extracted from the overall particle/substrate model. The
simulation is conducted using the full-particle model with the particle velocity being 500 m/s, friction
coefficient being 4, and surface separation allowed.
S3. Perturbation Induced Axi-asymmetric Deformation
When the full-particular model is used in the simulation, axi-asymmetric deformation may
happen (see Fig. 5). This axi-asymmetry is in essence induced by the numerical randomness
which serves as a perturbation. The perturbation would affect the exact details of the temperature
evolution, e.g., onset time of shear instability, just as those details can be easily influenced by
contact conditions. This is exactly the reason why the onset of shear instability cannot be used as
a precise indication of bonding. However, the perturbation (due to numerical randomness or
other sources) is not expected to modify the physics of the problem. As discussed in the
manuscript, the trends shown in Figs. 9 and 10 reflect the actual deformation characteristics (i.e.,
physics), which remain unchanged regardless of the presence of perturbation.
S4
To further illustrate this point, we performed a series of simulations with manually
introduced perturbations. Below we demonstrated two case studies, shown in Figs. S4 and S5,
corresponding to simulations with friction coefficient being 0.25 and 8 respectively (both with
surface separation allowed). In both cases the system was perturbed by removing a rectangular
region of a thickness of only one element (0.0006 mm) on the substrate (see Fig. S4). From the
simulation results we can see that though the general features remain similar, the exact details of
the temperature evolution are altered. In particular for the case with  = 0.25, the introduction of
the perturbation is seen to delay the onset of shear instability on the left side of the particle, while
for the case with  = 8 the perturbation is shown to sway the (side of) occurrence for axisymmetry as illustrated in Fig. S5c. Incorporating the data from those simulations into Figs. 9-10
(see Figs. S6 and S7), we note that they nicely collapse into the previously observed trends,
confirming that the underlying physics remains intact.
Fig. S4. The schematic particle/substrate model constructed for the FEA simulation with exaggerated
particle size. Red rectangular region with height of 0.0006 mm (one element size in this study) is deleted
in order to apply a small geometrical perturbation.
S5
Fig. S5. (a) Deformed particle/substrate configurations and temperature contours at the onset of shear
instability in models without (top row) and with perturbation (bottom row). Temperature evolution curves
for the most critical particle element and its corresponding symmetric element at friction coefficient equal
to (b) 0.25 and (c) 8 for the full-particle model with and without perturbation. The acronyms L and R
represent the two symmetric elements from the left and right sides of the particle with one being the most
critical particle element and the other being its symmetric element.
S6
Fig. S6. (a) Onset time and (b) von Mises stress versus the ̅̅̅̅̅̅̅̅
𝑃𝐸𝐸𝑄 at the onset of shear instability for
models with different contact conditions with (open symbols) and without (solid symbols) normal
constraint. The dashed line in (a) is drawn to guide the eyes while the line in (b) represents the linear
fitting of data. The red triangles are data calculated from simulations with added perturbation.
̃ 𝑃 (see Eq. 3) versus the onset time at the onset of shear instability for
Fig. S7. The energy measure, 𝑊
models with different contact conditions with (open symbols) and without (solid symbols) normal
constraint. The dashed line represents the linear fitting of data. The red triangles are data calculated from
simulations with added perturbation.
S7
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