ABSTRACT The modeling of high velocity impact is an important topic in impact engineering. Due to variousconstraints, experimental data are extremely limited. Therefore, detailed numerical simulation can be considered as adesirable alternative. However, the physical processes involved in the impact are very sophisticated; hence a practical andcomplete reproduction of the phenomena involves complicated numerical models. In this paper, we present a smoothedparticle hydrodynamics (SPH) method to model two-dimensional impact of metal sphere on thin metallic plate. Thesimulations are applied to different materials (Aluminum, Lead and Steel); however the target and projectile are formed ofsimilar metals. A wide range of velocities (300, 1000, 2000, and 3100 m/s) are considered in this study. The goal is tostudy the most sensitive input parameters (impact velocity and plate thickness) on the longitudinal extension of theprojectile, penetration depth and damage crater.
KEYWORDS smoothed particle hydrodynamics, high velocity impact, sensitivity analysis
1 Introduction
Impact in mechanical engineering problems can be definedas a high force or shock which is applied over a short timeperiod. Modeling and simulating the collision of two ormore bodies can help predict the consequences of theimpact and protect the surrounding area against theseeffects.Since experimental studies of the behavior of solid
material or fluid are conducted with very expensivedevices and equipment, computational modeling andnumerical simulation have become the proper and acceptedways to investigate behavior of materials. Meshfreemethods [1–3] are a powerful alternative to finite elementmethods to model large deformations, dynamic fractureand fragmentation.SPH method is a fully Lagrangian, non-grid based
computational technique that is typically used for con-tinuum dynamics simulations. SPH was mainly used tosimulate the motion of compressible flow. SPH hasrecently been extended to simulate incompressible fluidmotions. In SPH modeling, particles of the fluid can be
modeled as spheres, and the interacting forces betweenthem can be calculated by applying dynamics and fluiddynamics formulas like gravity effect, inertia effect,viscosity effect, and pressure effect. Smoothed ParticleHydrodynamics method (SPH) was first introduced in the1970s to model astrophysical applications [4,5], butrecently, SPH can be found in a wide range of researchareas. SPH method plays a key role in modeling ofengineering problems including heat conduction [6], multi-phase flows [7], chemical explosions [8], as well asdeformation and impact problems [9]. The SPH methodcan overcome the difficulties in large deformations, largeinhomogeneities, and large discontinuities as well as inmaterial interface tracking and therefore it is a goodalternative to traditional grid-based numerical methods insimulating impact problems.In the following, first a brief summary of the SPH
method will be presented. The difficulty in modeling theprimary steps, in particular the number of particles used forinitial particles placement, is then described. The effects ofinitial velocity variations for three types of materials(aluminum, lead and steel) are then described. For thevalidation of our SPH code, our research is compared withpreviously published results. Finally a sensitivity analysisArticle history: Received Mar. 19, 2012; Accepted Apr. 12, 2012
of the output variables is conducted with respect to theinitial projectile velocity and plate thickness.
2 SPH methodology and numerical aspects
2.1 Governing equations
In SPH method, the fluid is represented by particles. In thistechnique the interpolation function A is defined as
AkðrÞ ¼ !υ
Aðr#Þwðr – r#,hÞdr#, (1)
where the integral is over the domain v and w(r – r′, h)signifies the smoothing kernel function. r and r′ are theposition vectors at different points and h, the smoothinglength, is the effective width of smoothing.When using interpolation for a model, it is necessary to
divide it into a set of small elements which are representedas mass elements. For instance, the properties of element jwill be expressed in terms of mj (mass of element j), ρj(density of element j), and rj (position of element j). Thevalue of parameter A at particle j is denoted by Aj.Moreover, in numerical techniques the integral of
interpolation function is generally approximated with aweight interpolating function. The integral can beestimated by summing up the mass elements. Thesummation interpolant can be written as
As rð Þ ¼Xj
mjAj
�jwðr – rj,hÞ: (2)
One of the most well-known kernels used in SPH isbased on the cubic spline function.
w r – r#,hð Þ ¼ αhυ
�1 –
3
2s2 þ 3
4s3, 0£s<1;
1
4ð2 – sÞ3, 1£s<2;
0, 2£s:
8>>><>>>:
(3)
where s ¼ r – r#j jh
, α is a normalized constant and v is the
dimension of the problem. α is equal to2
3,10
7π, 1=π for one,
two, and three dimension respectively.The second derivative of the above kernel is continuous,
and the dominant error term in the integral of interpolationfunction is o(h2). The compact support length of this kernelis 2h which means that there is no interaction between theparticles in distance beyond 2h. Figure 1 shows how akernel function acts on its compact support.The two-dimensional SPH expressions for equations of
continuum mechanics, conservation of mass, momentumand energy, are given by
d�idt
¼ �Xj
mj
�jðVi –VjÞ⋅rwij, (4)
dV t
dt¼Xj
mj�i
p2iþ �j
p2j–Πij
!:rwij, (5)
deidt
¼ –Xj
mjðVi –VjÞ�i
ρ2iþ 1
2Πij
� �:rwij: (6)
In above equations �i, mi, Vi, �i, and ei are the density,mass, velocity, stress and specific internal energy of the ithparticle respectively. Πij denotes the artificial viscositywhich is introduced to capture the shock wave. For morecomprehensive details about effects caused by variation ofartificial viscosity, one may refer to [10].In order to reduce the numerical noise (numerical error)
in simulation, some spatial filtering practices are oftenemployed which improve the stability and convergence ofthe model. Generally there are various correction methodsin the SPH method and XSPH is one of them. Particularlyin the simulation of high speed flow or impact thistechnique must be considered. For instance, numericalnoise in velocity regardless of direction may even causeunwanted penetration between particles resulting in thefailure of the simulation. To clarify, consider closelyspaced particles which have similar but not equal velocity.The particle with higher speed has priority and passes overthe other particles; this situation results in particlepenetration which is not allowed for the continuummaterial. The main purpose of using the XSPH correctiontechnique is to correct and modify velocity to the propervalue which is related to the velocities of the neighborparticles. In this case, particles can move smoothly [11].The equations below are used to calculate the velocity.
dυidt
¼ υi þ εΔυi, (7)
Fig. 1 Spline kernel acting on its compact support
102 Front. Struct. Civ. Eng. 2012, 6(2): 101–110
where,
Δυi ¼Xj
mj
�jðυj – υiÞW ðri – rj,hÞ: (8)
In this study, the Von-Mises yield criterion is used. Thefollowing steps show the procedure of applying governingequations:1) First, reduced tensions must be compared with the
yield condition. If S2x þ S2y þ 2S2xy£2
3ð�Þ2, then the value
of stress corresponds to the yield stress and particles show
elastic behavior. But if S2x þ S2y þ 2S2xy >2
3ð�Þ2, then the
material yields and it is in the plastic region.2) If the material yields in the first stage, in this step all
stress elements are multiplied by the following factor toreturn to the yield level
3) Stresses can be calculated from the equation below
�ij ¼ –Pδij þ Sij, (10)
where �ij is the stress tensor, P is pressure and Sij is thedeviatoric stress tensor. In Eq. (10) δij is the Kroneckerdelta. It is noted that the mean stress σ0 is given by σ0 = –P.Assuming Hooke’s law with shear modulus, the evolutionequation for the deviatoric stress S [12] is:
dSij
dt¼ 2� є_
– ij –1
3δijє_ – ij
� �þ ΩjkSik þ ΩikSjk , (11)
where,
є_– ij ¼ 1
2
∂υi
∂xjþ ∂υj
∂xi
� �, Ωij ¼ 1
2
∂υi
∂xjþ ∂υj
∂xi
� �: (12)
3 Modeling and results
In this section we discuss about the modeling of our casestudy. The codes used in this study are written inFORTRAN V.90 and the results are simulated inTECPLOT V.360 software package. The simulations aredone for different velocities and materials, however, foreach case the projectile and the target plate are made of thesame metals. As it is shown in Fig. 2, particles are initiallylocated in a regular grid with the initial inter-particledistance of 0.01 cm. This involves a spherical projectilewith a 5 mm radius impacting on 2 mm thick target plate.The simulation is started at the moment of impact;therefore particles of the projectile are assigned theprojectile velocity which are 300, 1000, 2000 and3100 m/s.
The total number of particles used in this study is 17985and the ratio of h/d is equal to 1.5 (where h is thesmoothing length and d is the initial particles distance).This simulation is done in 2D planar geometry carried outfor the total duration of 8 µs after impact.At the beginning of modeling, due to the insufficient
number of particles (less than 10000 particles), no jet ofparticles is observed coming out of the wall where theprojectile is impacting. A rupture also occurred in the wall.These issues are clearly seen in Fig. 3.
Three quantities are tracked during simulation. Resultsfor crater diameter, final longitudinal diameter of projectile
Fig. 2 Particles initial placement
Fig. 3 Modeling failure
Hossein ASADI KALAMEH et al. High velocity impact of metal sphere on thin metallic plate using SPH method 103
and projectile penetration length are presented in Tables 1,2, and 3 relating to the AL-AL, Lead-Lead, and Steel-Steelimpact respectively. Note that the steel alloy used in thisproject is Steel-SAE-1030.According to the results obtained from the above
models, it can be seen that the projectile penetrationincreases by increasing the velocity. The rate of change ofthe crater diameter, for all three types of materials,decreases with each step of velocity changes. Accordingto the above tables, for all materials with velocity of3100 m/s, the targets are entirely penetrated and a debriscloud is formed to the back of the target entailing materialof both projectile and target. Comparing the above tables,the obtained results from (AL-AL, Lead-Lead and Steel-Steel) simulations are quite similar to each other.Some of the above cases are simulated using the
TECPLT 360 software package. Figures 4–7 depict impactof aluminum projectile hitting aluminum target plate atvelocity of 3100 m/s. It can be observed that the particles’continuity is maintained over the time.Figure 8 displays a general view of the above figures.In the following, Figs. 9–12 illustrate velocity variation
in X direction related to the AL-AL impact with the initialvelocity of 3100 m/s.The upper half of the velocity variation in X direction for
Lead-Lead impact (at 8 µs after impact) for the velocitiesof 300, 1000, 2000, 3100 m/s are shown in Figs. 13–16respectively.
4 Validating data with published literature
In this section, our results for a specific case (AL-ALimpact at the velocity of 3100 m/s) are validated bycomparison with a previous paper published by Mehra andChaturvedi [13]. They have studied high velocity impact ofaluminum sphere on thin aluminum plate as a smoothparticle hydrodynamics study. They used four differentshock capturing patterns which are used in SPH, and thenapplied the equations in the high velocity impact and hypervelocity impact of projectile on thin aluminum plate whichare around 3 and 6 km/s respectively. These two simula-tions are applied for the smooth geometrically plane. Anelastic––perfectly plastic constitutive prototype has beenused. Hence, it is possible to compare the simulation withsimulation done by Howell and Ball [14]. They comparedthe results from the simulation with the data which hadbeen previously examined and published. These patternsare different in the manner of treating of artificial viscosity(AV). These schemes are included regular SPH artificialviscosity, Monaghan and Morris’s artificial viscosity(MON), Balsara’s artificial viscosity (BAL), Riemannbased contact algorithm (CON) of Parshikov. Mehra andChaturvedi [13] found that in the impact problems withmoderately high velocity, CON achieved best resultoverall. Figure 6 shows the arrangement of particles usedin SAV, BAL, MON, and CON methods.The main differences between their modeling and what
Table 1 Results for AL-AL impact, 8 µs after impact
initial velocity/(m$s–1) crater diameter/mm final longitudinal diameter of projectile/mm projectile penetration/mm
300 10.414 9.249 1.582
1000 15.554 8.173 5.767
2000 18.842 7.006 11.987
3100 20.993 6.154 18.992
Table 2 Results for Lead-Lead impact, 8 µs after impact
initial velocity/(m$s–1) crater diameter/mm final longitudinal diameter of projectile/mm projectile penetration/mm
300 10.643 9.264 1.597
1000 15.557 8.169 5.776
2000 18.820 6.996 11.983
3100 21.361 6.050 18.981
Table 3 Results for Steel-Steel impact, 8 µs after impact
initial velocity/(m$s–1) crater diameter/mm final longitudinal diameter of projectile/mm projectile penetration/mm
300 10.543 9.256 1.588
1000 15.574 8.181 5.772
2000 18.889 7.018 11.980
3100 20.881 6.177 18.941
104 Front. Struct. Civ. Eng. 2012, 6(2): 101–110
Fig. 4 Spatial variation in X direction for AL-AL impact at 2 µsafter impact
Fig. 5 Spatial variation in X direction for AL-AL impact at 4 µsafter impact
Fig. 6 Spatial variation in X direction for AL-AL impact at 6 µsafter impact
Fig. 7 Spatial variation in X direction for AL-AL impact at 8 µsafter impact
Fig. 8 Spatial variation in X direction for AL-AL impact at 2, 4,6 and 8 µs after impact
Fig. 9 Velocity variation in X direction for AL-AL impact for2 µs after impact
Hossein ASADI KALAMEH et al. High velocity impact of metal sphere on thin metallic plate using SPH method 105
is presented here are number of particles, h/d and howparticles are initially distributed. As shown in Fig. 17,particles are initially located on a regular mesh. The meshis rectangular except that the circularity of the projectile issurrounded by two rings of equidistant particles along thecircumference [13]. In Mehra and Chaturvedi, the initialdistance between particles is 0.01 cm. Wall thickness andradius of the projectile are the same as in the modeldiscussed in this paper. They have a total of 17,850particles in the simulation including 7820 particles in theprojectile. The results are given for h/d = 1.4 for all AVs,with the exception of CON, where h/d = 1.7 is used.Upper half of the configuration obtained by AL-AL
impact at a velocity of 3100 m/s, at the time, 8 µs afterimpact, are shown in Figs. 18 and 19 for different AVs andthe method used in this paper respectively.Comparing the position of particles in different methods,
CON method and the SPH method used in this study havethe least trouble with numerical fracture and the tendencyto form clumps at moderately high velocity impact.Results of three mentioned parameters produced by AL-
AL impact at the velocity of 3100 m/s, for this specificcase, are shown in Table 4.As shown in Table 4, the SPH technique used in this
project generally produces slightly larger crater diametersthan those reported by Mehra and Chaturvedi [13] butslightly smaller final longitudinal diameter of projectile.All SPH simulations are in agreement among themselveson geometrical measurement of penetration length, craterdiameter and final longitudinal diameter of projectile.In summary, the SPH code used in this paper is validated
for a specific case which is impact of AL-AL at the velocityof 3100 m/s by the work done by Mehra and Chaturvedi[13]. The results demonstrate that there is a positivecorrelation between the particles cohesion and the
Fig. 10 Velocity variation in X direction for AL-AL impact for4 µs after impact
Fig. 11 Velocity variation in X direction for AL-AL impact for6 µs after impact
Fig. 12 Velocity variation in X direction for AL-AL impact for8 µs after impact
Fig. 13 Velocity of 300 m/s
106 Front. Struct. Civ. Eng. 2012, 6(2): 101–110
accuracy of the methods which means that CON methodand the SPH method used in this study can produce thebest results.
5 Sensitivity analysis
In this section, we will examine the sensitivity of the modelwith respect to small changes in the input variables.Sensitivity analysis (SA) is becoming an importantcomponent of model validation and is part of the moregeneral subject of uncertainty quantification. All input dataare subject to uncertainty of varying degrees, either due to
Fig. 14 Velocity of 1000 m/s
Fig. 15 Velocity of 2000 m/s
Fig. 16 Velocity of 3100 m/s
Fig. 17 Geometry and the initial particle placement for Al-Alimpact
Fig. 18 Upper half of the configuration obtained by AL-ALimpact, at a time 8 µs after impact
Hossein ASADI KALAMEH et al. High velocity impact of metal sphere on thin metallic plate using SPH method 107
assumptions made or due to measurement errors. To assessthe validity of a model, it is important to quantify the effectof these uncertainties on the output variables. As a sidebenefit, SA requires several test runs of the model whichcan be used for validation and for discovering softwaredefects.There are several types of methods that can be employed
for SA such as local methods, variance based methods,metamodeling or Monte Carlo filtering. Many of thesemethods require a large number (500–1000) of model runswhich is prohibitively expensive for this problem sinceeach run requires more than 10 hours of CPU time on astandard desktop computer. In the following, we will limitourselves to estimating the local sensitivity for two inputvariables (initial velocity and plate thickness) and three
Fig. 19 Upper half of the configuration obtained by AL-ALimpact, at a time 8 µs after impact using SPH technique used in thispaper
Table 4 Results obtained using different methods
simulation model crater diameter/cm final longitudinal diameter of projectile/cm projectile penetration/cm
SAV1 2.0 0.7 1.8
SAV2 1.9 0.7 1.8
BAL 2.0 0.7 1.8
MON 2.0 0.8 2.0
CON 2.1 0.7 1.9
SPH technique used in this paper 2.1 0.6 1.9
Table 5 Elasticity indices for the input variables V = initial velocity and W = wall thickness and output variables L = longitudinal extension, P =
penetration, and C = crater diameter
elasticityindex
impact velocity
300 1000 2000 3100
EIVL 0.061065 0.165715 0.285776 0.371782
EIWL 0.040613 0.110824 0.177229 0.226165
EIVP 1.083322 1.056758 1.050761 1.048952
EIWP 0.262322 0.214559 0.188727 0.177789
EIVC 0.240607 0.437752 0.310203 0.277102
EIWC 0.087269 0.318836 0.278214 0.176652
Table 6 Percentage contribution to the variation in the outputs L = longitudinal extension, P = penetration, and C = crater diameter for the input
variables V = initial velocity and W = wall thickness
percentagecontribution
impact velocity
300 1000 2000 3100
PC[V, L] 69.33% 69.10% 72.22% 72.99%
PC[W, L] 30.67% 30.90% 27.78% 27.01%
PC[V, P] 94.46% 96.04% 96.87% 97.21%
PC[W, P] 5.54% 3.96% 3.13% 2.79%
PC[V, C] 88.37% 65.34% 55.42% 71.10%
PC[W, C] 11.63% 34.66% 44.58% 28.90%
108 Front. Struct. Civ. Eng. 2012, 6(2): 101–110
outputs (crater diameter, final longitudinal diameter of theprojectile and projectile penetration).First we calculate the sensitivity index [15], which, for
an input parameter P, and output parameter Q is defined bythe equation
SIPQ ¼ Q0 –QL
P0 –PL
��������þ
Q0 –QH
Q0 –QH
��������
� �=2, (13)
where index 0 represents the base value, and the index Land H represent an increase and decrease from the basevalue of the parameter P, or the corresponding outputs Q.In all of the calculations, we considered a 5% increase anddecrease from the base value of the two input parameters(initial velocity and plate thickness). Note that SIPQ is anapproximation of the partial derivative ∂Q=∂P at P0, whichmeasures the absolute change in the output and isdependent on the units used. A measure of sensitivitythat is dimensionless and measures relative change is theelasticity index, defined as
EIPQ ¼ SIPQ⋅ðP0=Q0Þ: (14)
This index can be interpreted as an estimate of theproportional change in the output Q caused by a 1%change in the input P at P0. We evaluate EIPQ at the fourbase initial velocity values and a base plate thickness of 2mm, and show the results in Table 5. We observe that thethree outputs are more sensitive to the initial velocity thanthe plate thickness, especially in the case of projectilepenetration. The elasticity index is generally less than 1,except for the elasticity index of the velocity with respectto penetration which is slightly greater than 1.Next, we will consider a first-order sensitivity analysis
that takes into account the variance of the input parameters.In particular, given a probability distribution of the inputparameters, we would like to determine the variance of theoutput and how much each input contributes to the outputvariance. The percentage contribution of each inputparameter Pi can be written as [16]
PC½Pi, Q� ¼ð∂Q=∂PiÞ2Var½Pi�Xjð∂Q=∂PiÞ2=Var½Pi�
⋅100: (15)
Here the partial derivatives ∂Q=∂Pi can be approximatedfrom the sensitivity indices SIPQ. In the following, we willassume that the range between the high and low values ofthe input parameters correspond to a 5–95 percentile rangefor a normally distributed variable (i.e., a 90% probabilityinterval or 3.3 standard deviations). Then the variances ofthe input parameters can be calculated from the equation
Var½P� ¼ PH –PL
3:3
� �2
: (16)
In Table 6, we summarize the percentage contribution ofthe two input variables for the four pairs of base values,
corresponding to the four different initial velocities and thewall thickness of 2 mm.We note again that more than 50% of the contribution
comes from the variance in the initial velocity for all threeinput variables, and the contribution is particularlysignificant for the variance in projectile penetration(close to 95%). Therefore we can conclude that outputsare more sensitive to changes in initial velocity than wallthickness.
6 Conclusions
SPH is a particle based method for modeling fluid flow andsolid deformation and it has been used successfully tomodel impact problems. For these impact problems, SPHhas the advantages of being able to follow very highdeformations (further than what is possible with finiteelement method and finite volume methods) and to keeptrack of the specific history of each part of the metal andpotentially direct prediction of many types defectsoccurring during impact.In this paper, an implementation of the SPH method to
deal with two-dimensional impact has been presented. Thiswork is mainly focused on recording three parameterswhich are crater diameter, final longitudinal diameter ofprojectile and projectile penetration; then this method iscompared with four different types of SPH techniques.Elastic-perfectly plastic constitutive equation is used inorder to compare with free Lagrangian simulation of [13].The numerical simulation showed a good illustration of
the basic effects such as penetration of projectile andformation of the crater.A sensitivity analysis of the result showed that the
elasticity index for the input/output pairs examined atvarious data points is generally less than or close to one,which indicates the outputs are not overly sensitive withrespect to the inputs. It was also shown that most of thevariance in the outputs is due to the variance in the initialvelocity rather than the plate thickness.
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