Modeling Impacts for Cold-Gas Dynamic Spray 

by William S. de Rosset 

ARL-TR-3890 September 2006 

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Army Research Laboratory 

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ARL-TR-3890 September 2006 

Modeling Impacts for Cold-Gas Dynamic Spray 

William S. de Rosset 

Weapons and Materials Research Directorate, ARL 

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Modeling Impacts for Cold-Gas Dynamic Spray 

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William S. de Rosset 

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14. ABSTRACT 

A one-dimensional penetration model has been developed to analyze the cold-gas dynamic spray process in hopes of 

establishing a constant minimum normalized penetration depth for particle adhesion. The simple model was in disagreement 

with results extrapolated from published copper-on-steel impact data. Possible reasons for the disagreement are discussed. A 

possible bonding mechanism is proposed that is dependent on the flow properties of both the particle and substrate. 

15. SUBJECT TERMS

cold-gas dynamic spray, particle penetration mechanics, particle adhesion 

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William S. de Rosset 

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iii

Contents 

List of Figures iv

Acknowledgments v

1. Introduction 1

2. Data Analysis 2

3. Model Development 4

4. Discussion 9

5. General Comments 12

6. Summary 14

7. References 15

Appendix. Calculation Procedure 17 

Distribution List 18 
iv

List of Figures 

Figure 1. Crater volume vs. particle volume for copper particles impacting and sticking to 

stainless steel at an average velocity of 521 m/s (Dykhuizen et al. [1])....................................3 

Figure 2. Surface of 304 stainless steel showing grain structure....................................................6 

Figure 3. Calculated penetration vs. time for 22-μm copper particles impacting 304 stainless 

at 500-m/s steel for two values of β..........................................................................................8 

Figure 4. Autodyne calculation of normalized penetration depth (X/R) vs. time; 10-mm 

diameter copper sphere impacting stainless steel at 500 m/s...................................................10 

Figure 5. Normalized penetration vs. impact velocity (model calculation)..................................12 

Figure 6. Exaggerated picture of incipient bonding mechanism showing cross section of 

crater lip and deformed particle. ..............................................................................................13 

Figure 7. Incipient bonding mechanism (frictional/mechanical) for nondeformed particle.........14 
v

Acknowledgments 

The author would like to acknowledge the help of Constantine Fountzoulas, who ran several 

Autodyne calculations, and Marc Pepi, who polished and etched the sample of stainless steel. 

Mr. Pepi also made the hardness measurements on the sample. Special thanks go also to 

Dr. Steven Segletes for pointing out the inherent strain rate effects in the cold spray process and 

for his helpful comments on the modeling effort. 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

Ever since Supersonic Particle Deposition (SPD), otherwise known as cold-gas dynamic spray, 

showed the promise of being able to coat substrates with a variety of materials, there has been 

speculation as to how the bond is formed between the impacting particle and the substrate. Most 

of the successful spray depositions have occurred for metal particles impacting metal surfaces, 

which has led some authors to suppose that an oxide surface layer must be broken before 

bonding will occur (1). The fact that there is a minimum particle impact velocity required for 

particle adhesion is consistent with this hypothesis. Others have seen a correlation between 

impact velocities and calculated values of the onset of adiabatic shear instability in the particle 

(2, 3). The bonding mechanism invoked in this case is that the adiabatic shear occurs at and 

welds the contacting surfaces. In some instances, the penetration of the particle is deep enough 

to provide a mechanical rivet of the particle to the substrate (4). Another supposition is that a jet 

is formed between the spherical particle and the substrate, similar to that formed in explosive 

bonding (1–3). The jet acts to clean the substrate surface of any oxide layer. 

Clearly, a particle will adhere to a metal substrate if the impact velocity is high enough to supply 

a mechanical bond produced by deep penetration of the target. (Of course, the velocity cannot be 

so high as to vaporize the particle.) What is more interesting is the nature of the bonding process 

at the lowest velocity at which a particle will stick to the substrate. A minimum particle impact 

velocity for adhesion indicates that there is a minimum value of the penetration depth for any 

particle-substrate material combination. From the data in the literature a normalized penetration 

depth, defined here as the penetration depth divided by the particle radius, can be determined. If 

the value of the normalized penetration depth were the same for all material combinations, then 

some insight might be gained on the nature of the bond between the particle and substrate. 

Unfortunately, there is very little information available on penetration depths at the onset of 

adhesion. The approach taken here was to develop a simple predictive model and use available 

data to test its accuracy. The model would then be used to predict the penetration depths at 

reported minimum impact velocities for other material combinations. Section 2 describes a 

particular experiment that contains an extensive amount of impact data for one particular 

material combination, copper and stainless steel. With several reasonable assumptions, the 

normalized penetration depth is determined. Section 3 contains the development of a onedimensional impact model used to predict depth of penetration. The model is found to deviate 

from the observed results. In section 4, possible reasons for the discrepancy are discussed and 

sensitivity analyses are conducted to determine the impact of certain input parameter changes. 

Section 5 offers some general observations on the incipient bonding process and suggests that 

mechanical bonding is possible even for low-velocity impacts. Section 6 provides a summary of 

the report.
2

2. Data Analysis 

In order to interpret SPD experimental data correctly, one must understand the statistics behind 

the process. First, the particles that are used in the process are not a single size but come in a 

distribution with a mean size and a variance. In most cases, the distribution is normal 

(Gaussian), but this may not always be the case. The distribution of particle sizes leads to a 

distribution of particle velocities for any particular set of operating parameters (e.g., gas 

pressure, gas type, nozzle design, temperature, and nozzle-to-target distance) used with the SPD 

equipment. Aerodynamic considerations and certain approximations lead to the simple 

relationship between a particle’s mass and its velocity: 

 Vp = V(mo/m)1/2, (1) 

where Vp is the particle velocity, V is the average particle velocity, m is the particle mass, and 

mo is the average particle mass (5). In general, the smaller particles have a higher velocity than 

the mean, and the larger particles have a smaller velocity than the mean. It has been found that 

there is not good quantitative agreement between this simple formula and experimental data for 

copper and aluminum particles (6), although the overall trend is correct. In addition, the velocity 

of a given size particle will decrease the farther it is from the center of the gas stream. 

Dykhuizen et al. (1) conducted a series of experiments in which they sprayed copper particles on 

a 304 stainless steel substrate and measured both the crater volume and the particle volume of 

those particles that stuck. The copper powder had a mean particle diameter of 22 μm, with a 

5.6-μm standard deviation. Four average particle velocities were chosen, and the actual 

velocities achieved were 403, 521, 606, and 704 m/s. 

Figure 1 shows the data for the 521 m/s data set. A straight line has been added to the data. All 

but one of the data points fall above the line. The particle sizes fall between 10 and 30 μm in 

diameter, with more of the data points clustered at the lower end of the range. (The volume 

corresponding to a 22-μm-sized particle is 5575 μm3

.) The main conclusion to be drawn from 

these data points is that regardless of the particle size, there is a minimum crater-volume-toparticle-volume (CV/PV) ratio. The particular line drawn in figure 1 gives CV/PV as 0.058. 

(There is one outlier below the line at a particle volume of about 7500 μm3 which is neglected in 

this analysis.) Similar data for an impact velocity of 606 m/s is also presented in Dykhuizen 

et al. (1). All but one of the 68 data points in this set have CV/PV greater than 0.06. 

Both the experiments conducted at 521 and 606 m/s contained particles that had the necessary 

minimum velocity to stick to the target. For those particles with a CV/PV ratio greater than 0.06, 

it is assumed that they impact the target at a velocity that is higher than the minimum needed to 

stick. However, the same authors found that no copper particles stuck to the stainless steel at a 

403 m/s impact velocity. Some of the smaller particles have an impact velocity in excess of this
3

Particle Volume, cubic microns

0 2000 4000 6000 8000 10000 12000 14000 16000

Crater Volume, cubic microns

0

200

400

600

800

1000

μm3

μm3

Figure 1. Crater volume vs. particle volume for copper particles impacting and sticking to stainless steel at an 

average velocity of 521 m/s (Dykhuizen et al. [1]). 

value, so that the minimum impact velocity needed for sticking would certainly be higher than 

403 m/s, but less than 521 m/s. 

Some simple geometric considerations provide an estimate of the penetration depth to particle 

radius ratio for the particles that stuck to the target. First, we assume that the crater wall stays in 

contact with the particle over the complete extent of the crater and that the particle radius does 

not change significantly until the particle ceases its axial motion. This assumption appears to be 

reasonable based on the cross-section measurement of a particle that is sticking to the substrate 

shown in figure 8 of Dykhuizen et al. (1). (The model development in section 3 will allow the 

particle to deform, but most of the deformation occurs during the latter stages of the penetration 

process.) In this case, we have for 0 < = X < = R 

 CV = 1/3 π X2 (3R – X), (2) 

where X is the final penetration depth and R is the particle radius. The volume of a spherical 

particle is given by 

 PV = 4/3 π R3. (3) 
4

Setting CV/PV to the experimentally observed value (see figure 1) of 0.06 and solving for the 

value of X/R, we get 

 X/R = 0.298. (4) 

For a particle with a 22-μm diameter, the penetration depth would be 3.3 μm. This is a 

surprisingly shallow penetration depth, but it is consistent with figure 8 of Dykhuizen et al. (1), 

which shows a crater profile whose depth is a little over 2 μm. 

3. Model Development 

There exist many computer codes of various types (Eulerian, Lagrangian, smooth-particle 

hydrodynamic, etc.) that can easily be used to model the impact of a metal sphere onto a metal 

target. These codes employ very sophisticated constitutive relations and come with a fixed 

library of constants that are selected based on the type of constitutive relation chosen for the 

calculation. They are capable of solving complicated problems in three dimensions, if required, 

although the use of symmetry is used to reduce run time and memory requirements. The full 

dynamic process can be handled with stress wave propagation calculations and material failure 

evolution models. This high-powered computing comes with a cost, however. In many cases, it 

is difficult to determine which parameters are the most important in determining the outcome. It 

takes a skilled operator versed in the inner working of the code to differentiate computational 

artifacts (created, say, by the mesh size) from real effects. The output of data from a code run 

has become so large that we depend a great deal on scientific visualization to make sense of 

them. This leads to additional work to adjust the output presentation. 

There also exist simple analytical approaches to impact problems. They have the advantage that 

anyone with a reasonable mathematics and scientific background can not only understand how 

the model works, including assumptions and limitations, but also be able to exercise the model. 

The approach taken with this work was to use a simple one-dimensional penetration model, 

adapted to a ball-on-plate geometry. The modeling was used as an aid to understanding lowimpact-velocity phenomena, with the ultimate goal of understanding how particles first begin to 

adhere to surfaces. 

The basis for the model is the classic Bernoulli equation, modified by Tate (7, 8) and 

Alekseevskii (9): 

Σ = 1⁄2 ρp(v – u)2

 + Yp = 1⁄2 ρt u

2

 + Rt . (5) 

Here, ρp and ρt are the particle and target densities, respectively, v is the speed of the rear of the 

particle, u is the penetration speed, Yp is the strength of the particle, and Rt is the target 

resistance. Σ is the axial stress along the centerline of the particle. It will be assumed that the 
5

particle is a sphere and that it penetrates as a rigid body. This is a reasonable assumption since 

the particle does not erode but maintains its mass throughout the penetration process. However, 

the assumption is inconsistent with the fact that the particle deforms. At some point, possibly 

late in the penetration process, the penetration will cease but there will still be a center-of-mass 

velocity. While this assumption will undoubtedly lead to some disagreement between the model 

calculations and observed results, it is made in order to move the analysis forward. The 

assumption dictates that v = u, and equation 1 reduces to 

Σ = 1⁄2 ρt v

2

 + Rt . (6) 

Since the penetration process occurs in three dimensions, some reasonable assumptions must be 

made to allow the one-dimensional model to agree with experimental results. The first 

assumption is that the retarding force on the particle is given by the axial stress times the 

projected contact area of the particle. The projected contact area is calculated from the circle 

formed by the intersection of the particle projected onto the original surface. The second 

assumption is that the target resistance is proportional to the target flow stress. That is, 

 Rt = β Yt , (7) 

where β is the constant of proportionality and Yt is the target flow stress. 

The main problem then becomes one of finding a reasonable value for the proportionality 

constant. A one-dimensional model that incorporates plasticity theory has been developed by 

Goodier (10). He gives the retarding force (target resistance) as 

 Fr = q (2a)n , (8) 

where n is generally taken as 2 and q = 2.77 Yt. Given that 

 Fr = π a

2

 Rt , (9) 

we get β = 3.5. Walker and Anderson (11) estimate the value of β from a penetration mechanics 

analysis involving the integration of momentum along the centerline of a penetrator and making 

certain reasonable assumptions about the velocity profiles in the target and penetrator materials: 

β = 7/3 ln(α) , (10) 

where α is the extent of the plastic zone in the target material. These authors calculate the value 

of α for a rigid body penetrator to be 

α = (2Et/3Yt)

1/3 , (11) 

where Et is the elastic modulus of the target material. Using a yield strength of 215 MPa and a 

Young’s modulus of 200 GPa for stainless steel, we get α = 8.5 and β = 5.0. Of course, the 

particular example we are considering for the cold spray application is not rigid in the usual 

sense.
6

For static indentations, a rule of thumb is that the Brinnell hardness number in kg/mm2 is three 

times the ultimate tensile strength of many steel alloys (12). If the Brinnell hardness can be 

considered a “target resistance,” then β = 3. 

These considerations point to the likelihood that the value of β lies in the range of 3 to 5. Rather 

than pick a value of β that fits the data, this range will be used to see how close the calculations 

come to the observed values. 

Of even more concern is the value to be used as the target flow stress. Micrographs taken of a 

304 stainless steel sample confirmed the supposition that the grain size diameter of this material 

can be much larger than the diameter of an individual particle. A representative micrograph is 

shown in figure 2. 

Figure 2. Surface of 304 stainless steel showing grain structure. 

Grain size will depend, to a large extent, on the processing history of the material. The 

processing history of the material used by Dykhuizen et al. (1) is not known. Neither is the 

processing history of the material shown in figure 2 known. However, it is interesting to note 

that in the latter case, the craters made by several 20-μm particles could be totally enclosed in a 

single grain. (The length scale shown in the figure is 20 μm.) Since the experiments suggest 

that a particle penetrated only a third of its radius, the impact area would have a diameter less 

than 20 μm. This suggests that individual grain properties should be used in the calculations, 

rather than the bulk properties. 
7

Figure 2 shows grains with two different appearances. One has a feathery appearance, and the 

other is more clear. Knoop microhardness measurements were made on both types of grains 

using a 50 gram-force load. The hardnesses were converted to tensile strength values and the 

results were 110 and 98 ksi for the strengths of the feathery and clear grains, respectively. In 

order to examine a single grain at a time, standard procedures were not followed. Consequently, 

the results must be considered approximate at best (13). Nevertheless, they indicate a strength 

level several times that of the bulk material (~30 ksi). For these calculations, the flow stress will 

be taken to be 700 MPa. 

While the yield strength of the copper does not enter the calculation directly, it must be assumed 

that the strength of the copper is greater than the target resistance for there to be rigid body 

penetration. The yield strength of annealed copper is below 100 MPa (the room temperature 

yield strength used in the Johnson-Cook constitutive model is 90 MPa [12]). However, the 

copper particles are so small that they are most likely single crystals. Therefore, the strength of 

an individual particle may be much higher than that of the bulk material. In addition, the strain 

rates at which cold spray takes place are extremely high and can be estimated by dividing the 

impact velocity (500 m/s) by a typical dimension (for instance, the particle diameter, 20 μm). 

Thus, strain rates on the order of 107 s

–1 or greater are produced during impact. These rates are 

far higher than even split-Hopkinson bar experiments can reach. The yield strength of the copper 

particles can be expected to increase with increasing strain rate, but the amount of increase is 

unknown. Finally, spherical-nosed (or simply spherical) penetrators are constrained by the crater 

they make in the target so that the penetrator material appears to be stronger than it actually is 

(14, 15). 

From geometric considerations, the projected contact area of the particle and the target, πa

2

, is 

given by 

πa

2 

= π(Dx – x2), (12) 

where D = 2R and x(t) is penetration depth as a function of time. When x = R, then the projected 

contact area is simply πR2

. The equation of motion then becomes 

2 2

t t

2 2

t t

mdv/dt = –(1⁄2 v + R ) (Dx – x ), 0 < x < R

 = –(1⁄2 v + R ) R , x > R . 

ρ π

ρ π

 (13) 

The initial conditions are x(0) = 0 and v(0) = impact velocity. This equation is solved for x as a 

function of time through an iterative method using an Excel spreadsheet.* The appendix contains 

the details of these calculations. A time step of 0.2 ns was used; the results did not change 

significantly when a time step of 0.5 ns was used. Figure 3 shows the calculated penetration depth 

as a function of time for two values of β. The final penetration depths are 10.0 and 7.67 μm for 

 *

Excel is a registered trademark of the Microsoft Corporation (Redmond, WA). 
8

Time, s

0 1e-8 2e-8 3e-8 4e-8

Penetration Depth, m

0.0

2.0e-6

4.0e-6

6.0e-6

8.0e-6

1.0e-5

1.2e-5

β = 3

β = 5

Figure 3. Calculated penetration vs. time for 22-μm copper particles impacting 304 stainless at 

500-m/s steel for two values of β. 

values of β equal to 3 and 5, respectively. This equates to a normalized penetration depth, X/R, 

of 0.91 and 0.70 for values of β equal to 3 and 5, respectively. From the experiments, X/R = 0.3, 

so that there is a large difference between the model results and the observations. This disparity 

is most likely attributable to the rigid body assumption. 

Several attempts were made to include particle deformation effects in a more complicated model 

that did not use the rigid body assumption. These attempts did no better in producing a model 

that could be used to estimate the value of X/R accurately. 

Frank and Zook (16) have modified equation 5 for the noneroding rod penetrator case (u = v) and 

added a term that accounts for unsteady motion (8, 10): 

 Yp = kt ρt v

2

 + Rt + k1 ρt D dv/dt . (14) 

Here, kt and k1 are fitting parameters, and D is the penetrator diameter. (Note that kt takes the 

place of 1/2 in equation 5.) Define the rate of penetration as dx/dt = v. The penetrator is 

decelerated according to equation 15: 

 dv/dt = – Yp/Lρp, (15)
9

where L is the penetrator length. Equations 14 and 15 can be solved to obtain 

 X/R = ((L/D ρp/ρt +k1)/2kt)ln(1+ktρtV2

/Rt). (16) 

In this equation, V is the impact velocity. Frank and Zook (16) fit high-density, low-L/D-ratio 

penetrator data to equation 16 to obtain the following values of the fitting parameters and Rt: 

1

t

9

t

k = 0.557,

k = 1.046,

R = 9.13 10 Pa . × (17)

Here, Rt is the target resistance value for rolled homogeneous (steel) armor (RHA). The authors 

recognized that the value of the target resistance was about twice that ordinarily used for eroding 

long rod penetrators. They suggested that the energy lost in deforming the penetrator (but not 

eroding it) was responsible for this increase in apparent target resistance. 

The Frank-Zook formulation can be used to make a first estimate of copper sphere impact depths 

into stainless steel. First, assume that a sphere is the mass equivalent of a cylinder with L/D = 1. 

Therefore, the L/D ratio for the sphere equivalent is 2/3. Next, substitute the appropriate density 

values in equation 16 and use the parameters listed in equation 17. This produces a value of 

0.278 for X/R. This value is slightly less than that obtained with the analysis in section 2. The 

data that Frank and Zook used to fit their equation involved impacts into steel RHA. Therefore, 

the strength difference between RHA and stainless steel might account for some of the 

difference. If the value of 7.6 × 109 Pa is used for Rt, then equation 16 gives 0.297 for X/R. 

The Frank-Zook formulation suffers from the same drawback as other approaches. That is, the 

target resistance cannot be determined a priori from first principles. Consequently, it cannot 

make independent predictions for low-velocity impacts of a variety of material combinations. It 

is interesting to note that this approach comes closer to the result determined from the data 

analysis than the other approaches tested. This may be fortuitous, considering the material 

properties and small size of the impacting particles. 

4. Discussion 

A one-dimensional penetration model has been adapted to the case of a copper particle impacting 

stainless steel at 500 m/s. A rigid body assumption was used to enable the calculations to be 

carried out. It is not surprising, therefore, that the calculated penetration depth is not in good 

agreement with experimental observations. The normalized penetration depths derived from the 

model are over two times too high (0.9 or 0.7 vs. 0.3).
10

As a point of comparison, a standard penetration mechanics code (Autodyne*) was used for the 

same calculation. The values of the copper and steel yield stress that were used for the 

calculation were 120 and 300 MPa, respectively. For reasons already mentioned, these values 

may not be appropriate for this calculation. In addition, the version of Autodyne that was used 

for these calculations cannot formulate its cells with a length scale smaller than 1 mm. 

Therefore, it was assumed that in the absence of any rate-dependent material properties scaling 

would be valid. The calculations were carried out for a spherical copper particle 10 mm in 

diameter. A plot of the penetration depth, normalized by the original radius, is shown in figure 4. 

The final normalized penetration depth (X/R) fluctuated around 0.52 for times greater than 

40 μs. 

Time, microseconds

0 10 20 30 40

Normalized Penetration

0.0

0.1

0.2

0.3

0.4

0.5

0.6

μs 

Figure 4. Autodyne calculation of normalized penetration depth (X/R) vs. time; 10-mm diameter copper 

sphere impacting stainless steel at 500 m/s. 

In addition, measurements taken from the figures generated by the CTH code in Dykhuizen et al. 

(1) showed a normalized penetration depth of 0.41 for a 25-μm copper particle impact on stainless 

steel for an impact velocity of 500 m/s. The CTH code takes into account such effects as the 

pressure dependence of the yield strength and strain hardening. It predicts impact pressures up to 

5 GPa for this calculation. Thus, predictions from more sophisticated codes also over-estimate 

the depth of penetration.

 *

Autodyne is a registered trademark of the Ansys Corporation. 
11

Several possible causes for the over-estimation can be offered. First, all the approaches assume 

that the copper particles are perfectly spherical at impact. If they are not perfectly spherical and 

hit the target with a flat surface, then the penetration depth will decrease. This is because they 

have a relatively larger contact surface from the beginning of the calculation. 

One other source of error is the neglect of any elastic rebound. An effect called “shallowing” has 

been observed in many cases of static indentation (17). When an indenter (e.g., Brinnell 

hardness ball) is lifted from the surface of a sample after having made an indentation, it is 

observed that the indentation has a larger radius of curvature than the indenting ball. That is, the 

indentation depth achieved by the ball is not retained but relaxes to a lower value. Because of 

the shallowing effect, depth measurements are unreliable and are not used in the hardness 

determination. Such an elastic relaxation may have occurred in the work reported by Dykhuizen 

et al. (1) as a result of removing the copper particles from the stainless steel. 

The model has been pursued assuming that averages of experimental parameters are appropriate 

for comparing with model predictions. In particular, we have used an average particle size 

traveling at an average velocity for the calculations. It may be that for any given spraying 

operation, the smaller particles travel faster than the larger ones and produce deeper craters in 

relation to their diameters (see figure 1). As previously mentioned, there is a range of particle 

velocities in any given particle stream. (The particle velocity distribution approximates a 

Gaussian distribution with about a 50-m/s mean deviation.) If the average velocity of the stream 

is low, then only the smaller, faster particles will adhere. As the average particle velocity 

increases, larger particles will have sufficient velocity to adhere to the target and deposition 

efficiency will increase. It is the slower particles in the stream that will produce the incipient 

conditions for particle sticking for elevated average particle velocities. Model calculations 

showed that the value of X/R for different size particles was the same. 

However, the model does predict that effect of particle velocity on X/R is more pronounced. The 

velocity was varied from 400 to 500 m/s, and the results are presented in figure 5. A particle 

radius of 11 μm and a β = 5 were used for the calculations. Dykhuizen et al. (1) did not observe 

any particle sticking at an average particle velocity of 400 m/s. If there were any particles in this 

stream with a velocity of 450 m/s, they did not adhere or were too few in number to be observed. 

Particle adhesion was observed for an average particle velocity of 521 m/s. However, the 

deposition efficiency was low at this velocity, suggesting that only faster-than-average particles 

were adhering. Therefore, velocity distribution effects cannot explain the difference between the 

model predictions and observed penetration depths. 

One final point can be made regarding the target resistance. If the target experiences a strain rate 

similar to that experienced by the particle, then it would be reasonable to assume that the target 

resistance would also increase. This would lead to a smaller depth of penetration.
12

Impact Velocity, m/s

380 400 420 440 460 480 500 520

Normalized Penetration, X/R

0.54

0.56

0.58

0.60

0.62

0.64

0.66

0.68

0.70

0.72

Figure 5. Normalized penetration vs. impact velocity (model calculation). 

5. General Comments 

At the outset of this work, it was anticipated that a model could be developed that would be able 

to predict accurately the value of X/R for a range of material combinations. However, the goal 

of developing an accurate model was not achieved. This is due in large part of the unknown 

material properties of both the target and particle material in the scale of sizes and strain rates 

pertinent to cold spray. In addition, the possibility that there exists a single minimum value of 

X/R for all material combinations that would assure particle adherence is unlikely, given the 

wide range of material properties. 

One of the most intriguing points brought out by the analysis of the cold-spray impact data is that 

particle sticking can occur for low penetration depths. Analysis of data involving copper 

particles sprayed onto a 304 stainless steel target indicate that the normalized penetration depth, 

X/R, is ~0.3. That is, a copper particle needs only to be buried less than a third of its radius 

(one-sixth of the diameter) for adhesion to take place on stainless steel. Certainly, at a very high 
13

velocity that produces deep penetration of the target, there is a strong frictional and/or 

mechanical bond that welds the particle to the substrate. How and to what extent incipient 

bonding occurs is not so clear. 

One possibility for a frictional or mechanical bond is presented in figure 6, which shows a 

stylized and somewhat exaggerated cross-section of the edge of a crater and the shape of the 

deformed particle. Here, a shallow crater has been formed, and material has been ejected from 

the crater to form a lip. At some later time, the particle has ceased to penetrate the target. 

However, it is assumed that the rear of the particle continues to deform and be decelerated until it 

comes to rest, encompassing, to some extent, the crater lip. This drawing is not unlike the actual 

crater and particle formation shown in figure 8 of Dykhuizen et al. (1). However, the crosssection shown in Dykhuizen, et al. appears to lack any mechanical interference that would be the 

source of bonding. This lack of detail is not surprising, considering the small scale of the crosssection. 

Crater Lip 

Deformed Particle

Substrate 

Figure 6. Exaggerated picture of incipient bonding mechanism showing cross section 

of crater lip and deformed particle. 

The bonding mechanism pictured in figure 6 may be unique to the copper-stainless steel 

combination of materials, or other material combinations with densities and material strengths 

that are similar. Consider next a very hard material impacting a soft substrate, such as tungsten 

particles impacting annealed aluminum. In this case, the bonding may occur as a result of 

embedment of the particle in the substrate, giving a high value of the normalized penetration 

depth. The bond would depend to some extent on friction and elastic recompression of the 

substrate material. A possible representation of this bonding mechanism is shown in figure 7. 

The value of X/R for this figure is a little over one; this value might be different for different 

material combinations. Note that the impact velocity for actually building up a thick layer of 

tungsten may be substantially different from the value needed for initial adhesion. Finally, there 
14

Substrate

Nondeformed 

Particle 

Figure 7. Incipient bonding mechanism (frictional/mechanical) for nondeformed particle. 

is the case where the particle strength is much less than that of the substrate. Unless the density 

of the particle is high enough to make a crater (i.e., the ρu

2

 term is large), then a mechanical 

bond might not be possible. 

6. Summary 

It is widely accepted that for any given combination of particle and substrate used in the cold-gas 

dynamic spray process, there is a minimum impact velocity required to achieve particle 

adhesion. A first hypothesis was that there might exist a normalized penetration depth, X/R, that 

would provide adhesion for all material combinations. If this were the case, then some insight 

might be gained as to the nature of the bond between the particle and substrate. To test this 

hypothesis, a one-dimensional model was developed to describe the penetration process. In 

addition, published data on the impact of penetration of copper particles onto a 304 stainless steel 

substrate have been analyzed to estimate the value of X/R when adhesion first begins. From 

these data, the critical normalized penetration depth was determined to be ~0.3 for this particular 

particle-substrate combination. The one-dimensional model developed here was found not to 

agree well with the observed results. Computer code results from CTH and Autodyne were also 

found to over-estimate the observed penetration depths. Several possible explanations for the 

discrepancy were discussed. The primary problem centers around the unknown material 

properties at the strain rates and scale sizes pertinent to cold spray. Another plausible 

explanation is that there was some elastic relaxation of the material in the crater after the copper 

particle was removed, leading to a decrease in the observed penetration depth. Mechanical 

bonding was suggested as the primary mechanism for particle adhesion. The exact nature of this 

process would depend on the relative flow properties and densities of the particle and substrate 

materials. 
15

7. References 

1. Dykhuizen, R. C.; Smith, M. F.; Gilmore, D. L.; Neiser, R. A.; Jiang, X.; Sampath, S. 

Impact of High Velocity Cold Spray Particles. Journal of Spray Technology 1999, 8 (4). 

2. Grujicic, M.; Zhao, C. L.; de Rosset, W. S.; Helfritch, D. Adiabatic Shear Instability Based 

Mechanism for Particle/Substrate Bonding in the Cold-Gas Dynamic-Spray Process. 

Materials and Design 2004, 25 (8). 

3. Hammad, A.; Gartner, F.; Stoltenhoff, T.; Kreye, H. Bonding Mechanism in Cold Gas 

Spraying. Acta Materialia 2003, 51. 

4. Grujicic, M.; Saylor, J. R.; Beasley, D. E.; de Rosset, W. S.; Helfritch, D. Computational 

Analysis of the Interfacial Bonding Between Feed Particles and the Substrate in the ColdGas Dynamic Spray Process. Applied Surface Science 2003, 219, 211–237. 

5. Dykhuizen, R. C.; Smith, M. F. Gas Dynamic Principles of Cold Spray. Journal of Thermal 

Spray Technology 1998, 7 (2). 

6. Smith, M. F.; Brockman, J. E.; Dykhuizen, R. C.; Gilmore, D. L.; Neiser, R. A.; Roemer, 

T. J. Cold Spray Direct Fabrication – High Rate, Solid State, Material Consolidation. 

Proceedings of the Fall 1998 Meeting of the Materials Research Society, Boston, MA, 

30 November–4 December 1998. 

7. Tate, A. A Theory for the Deceleration of Long Rods After Impact. J. Mech. Phys. Solids

1967, 15, 387–399. 

8. Tate, A. Further Results in the Theory of Long Rod Penetration. J. Mech. Phys. Solids

1969, 17, 141–150. 

9. Alekseevskii, V. P. Penetration of a Rod Into a Target at High Velocity, in Combustion, 

Explosion and Shock Waves, Faraday Press: New York (translated from Russian), 1966; 

Vol. 2, pp 63–66. 

10. Goodier, J. N. On the Mechanics of Indentation and Cratering in Solid Targets of StrainHardening Metal by Impact of Hard and Soft Spheres. Proceedings of the Seventh 

Hypervelocity Impact Symposium, Vol. III, Martin Company, Tampa, FL, 1965. 

11. Walker, J. D.; Anderson, C. E., Jr. A Time-Dependent Model for Long Rod Penetration. 

Int. J. Impact Engng. 1995, 16, 19–48. 

12. Florence, A. L.; Gefken, P. R.; Kirkpatrick, S. W. Dynamic Plastic Buckling of Copper 

Cylindrical Shells. Int. J. Solids Structures 1991, 27 (1), 89–103.
16

13. Pepi, M. U.S. Army Research Laboratory, Aberdeen Proving Ground, MD. Private 

communication, 21 February 2006. 

14. Segletes, S. B. Analysis of the Noneroding Penetration of Tungsten Alloy Long Rods Into 

Aluminum Targets; ARL-TR-3075; U.S. Army Research Laboratory: Aberdeen Proving 

Ground, MD, September 2003. 

15. Segletes, S. B. Analysis of Erosion Transition in Tungsten-Alloy Long Rods Into Aluminum 

Targets; ARL-TR-3153; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, 

March 2004. 

16. Frank, K.; Zook, J. Chunky Metal Penetrators Act Like Constant Mass Penetrators. In 

Proceedings of the 12th International Symposium on Ballistics, San Antonio, TX, 

30 October–1 November 1990; Vol. 1, pp 441–449. 

17. Tabor, D. The Hardness of Metals; Oxford at the Clarendon Press: London, E.C.4, 1951. 
17

Appendix. Calculation Procedure 

The force law is assumed to be given by 

Σ (π a2) = (1⁄2 ρt u

2

 + Rt) π a2 , (A-1) 

where a is the radius of the contact area, ρt is the target density, u is the penetration velocity, and 

Rt is the target resistance. The target resistance is assumed to take the form 

 Rt = β Yt , (A-2) 

where Yt is the target yield strength and β is a constant of proportionality. 

The contact area, πa

2

, is calculated from 

πa

2

 = π(Dx – x2), (A-3) 

where D is the particle diameter and x(t) is the coordinate of the center of mass. The initial 

conditions are that the center-of-mass velocity is the impact velocity, the center of mass os 

located at the origin. The calculation is done in 0.2-ns intervals. For the first time step, x is 

approximated by multiplying the center-of-mass velocity by 0.1 ns. This gives the mid-point 

value of x for the time interval of 0.2 ns. The value of the acceleration is then calculated. This 

mid-point approximation of the acceleration is then used to calculate the velocity at the end of 

the time interval. This velocity is the initial velocity for the next time interval. The same 

procedure is repeated for the next time step, except that the time interval now is the entire 0.2 ns. 

The calculation is continued until the center-of-mass velocity is less than 1 m/s. The first few 

time steps are shown below. 

Table A-1. Rigid body penetration. 

R = 1.10E-05 

Density = 8.90E+03 

M = 4.96E-11 

D = 2.20E-05 

Initial vel = 500 

Rt = 3.50E+09 

Time dx/dt initial x d2x/dt2 dx/dt final 

 0 500 0 0 

 2.00E-10 500 5.00E–08 3.11E+08 5.00E+02 

 4.00E-10 5.00E+02 1.50E–07 9.30E+08 5.00E+02 

 6E-10 5.00E+02 2.50E–07 1.54E+09 4.99E+02 

 8.00E-10 4.99E+02 3.50E–07 2.15E+09 4.99E+02 
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