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1.2. Select projectile pull down menu and the relevant projectile parameters will then be filled in automatically If the projectile is not in the list the user must input the required projectile parameters himself These are listed in Table 2 1 and are illustrated in Figure 2 2 No explanation should be necessary about the projectile mass diameter and impact velocity but the two last parameters might deserve a few words Projectile diameter Projectile diameter diameter Projectile curvature radius Table 2 1 Projectile parameters q Flat part diameter Target thickness Target diameter Cavity diameter Figure 2 2 Definition of projectile and target parameters Parameter Dimension Required Projectile mass kg Yes Projectile diameter cm Yes Projectile curvature radius cm Yes Flat part diameter cm No default 0 Impact velocity m s Yes 10 The parameter projectile curvature defines the curvature radius of the ogival projectile nose The final parameter flat part diameter makes it possible to define a projectile with a truncated nose The input value is then the diameter of the front part of the nose If the nose is not truncated this field can be left blank or set to zero 2 2 Target geometry parameters The target geometry can be defined in the right part of the main menu of the GUI The necessary input parameters are listed in Table 2 2 Table 2 2 Target
3. equal to the highest yield stress Again the cavity stress p has to be calculated numerically as no analytical solution is possible This model is best suited for concrete and geological materials 3 BOUNDARY EFFECTS Boundary effects can be accounted for in various different ways This is selected in the box Boundary option on the right hand side of the menu The following values are possible 13 No boundary effects The target is thus considered semi infinite Boundary effects from the edges but not from the rear end of the target Boundary effects both from the edges and the rear end calculated separately Same as Option 2 except that distance to boundary calculated in a different way Boundary effects from the edges and the rear end calculated together PL NS If the box is left blank the default value is 0 thus no boundary effects are included and the target is considered semi infinite The boundary effects are implemented by reducing the stress on the projectile nose at various points by a decay factor a Thus p 0 gt a p There are several ways to define the decay factor In our calculations an extension of an expression derived by Littlefield 5 is used En In 1 4 x d 14In V3G Y where x is the distance to the boundary which may depend on the position on the projectile nose If is larger than one it is put equal to one The distance x is not uniquely defined but different possibilities
4. 5 Discussion It is clear that there are advantages and disadvantages with all the various options For a very sharp projectile the perpendicular distance to the rear end will for example be very large and thus boundary effects from the rear end will be very small using this method At the moment we provide no clear recommendation about which option is most correct so the user is encouraged to experiment with the various settings to obtain different estimates In many cases there will be little difference in results for the various options 4 PENETRATION PHASES A penetration process can be divided into various phases In the calculations these are dealt with in different ways During the tunneling phase when the projectile nose is completely inside the target expressions from cavity expansion are used to determine the stresses on the projectile nose These stresses are then integrated over the whole nose to obtain the total force on the projectile In the cratering phase when the projectile nose is not yet completely embedded in the target only the part of the nose that is inside the target is integrated over The force thus grows larger as the projectile enters the target If the target has a finite thickness there is a third phase when part of the projectile nose has exited the target Also in this case only the nose that is still inside the target is integrated over This exit phase model is probably not very realistic for brittle mat
5. Penetration of a rigid projectile into a target material is often calculated using the theory of cavity expansion In this approach the stresses on a projectile during penetration is related to the radial stresses necessary for expanding a cavity at a given velocity Analytical models of this kind do not yield a physical description comparable to full numerical simulations using Autodyn or a similar code but they are nonetheless very useful It is much easier and quicker to obtain results from an analytical model than from a hydrocode as the latter often requires a great deal of computation and modelling time Analytical models are therefore useful for obtaining quick estimates or performing sensitivity studies varying various parameters Furthermore lack of accurate or reliable material data can render hydrocode simulations useless There are three steps involved in using the cavity expansion theory CET to calculate penetration First we calculate the radial stress p on a cavity boundary as a result of a forced expansion at some velocity v Secondly we use this expression to estimate the stress p 0 over a projectile surface during a penetration process and integrate the stress to obtain the total force F on the projectile as a function of velocity Finally Newton s 2 law is applied to calculate the projectile motion Of these steps the first one is by far the most difficult as it involves solving a complicated set of differential
6. can be selected by the user The various options are shown in Figure 3 1 and are described in detail below Figure 3 1 Options 1 4 left to right for defining the distance to the boundary and the related decay factors 3 1 Option 1 Using this option x is set equal to the target diameter Thus a is independent of position on the projectile nose and only boundary effects from the target edges are considered 3 2 Option 2 In this case boundary effects from the rear end of the target are considered as well In addition to the factor a from Option 1 the force is also multiplied by a factor z where x is 14 the distance from the projectile tip to the rear boundary so that the total decay factor is a a a As a consequence boundary effects are larger when the projectile is close to perforating 3 3 Option 3 Again the decay factor is the product of two separate factors and z The first factor is the same as in Option 1 The second factor is found by defining x as the perpendicular distance from a given point on the projectile nose to a line parallel to the rear end of the target Since this depends on which point on the nose is considered we then average over the complete nose to obtain a 3 4 Option 4 The boundary effects from the edge and the rear end of the targets are taken care of with one expression The parameter x is given by the smallest perpendicular distance either to the rear end or the target edge 3
7. equations Depending on whether the target material model is simple enough this can however be done analytically For instance this is the case for a Mises plasticity model However for a more complicated plasticity model like a piecewise linear pressure dependent yield surface an analytical solution is not possible In this case it becomes necessary to obtain a numerical solution for the cavity expansion force before proceeding to steps 2 and 3 In order to find a numerical solution easily a numerical Matlab tool has been developed at FFI It was first formulated in 1999 and documented in Norwegian 1 but has later been extended with several new options including a graphical user interface This report provides a brief documentation in English and also serves as a user manual for the new program For details about the methods for numerical solution of the CET equations the reader is referred to 1 This plasticity model has incorrectly been referred to as Mohr Coulomb in Autodyn terminology However in Autodyn 5 it will be called Drucker Prager which is somewhat more accurate although still misleading For the Norwegian Defence Weapon Effects Handbook 2 a similar tool has been established independently The difference between these tools is that the one described here is more aimed at the scientist engineer who wants to experiment to see what happens when changes are made to material models parameter sensitivity s
8. geometry parameters Parameter Dimension Required Target diameter cm No default semi infinite Target thickness cm No default semi infinite Cavity diameter cm No default 0 Figure No 2 File Edit View Insert Tools Window Help uimenu FFI Advanced Penetration Tool Analytical Numerical Cavity Expansion Analysis Define common parameters E modulus GPa Poisson ratio Density Select plasticity model below 2 Concrete empirical Compressive strength Yield strength MPa 2 Mises Yield strength MPa gt Mohr Coulomb Yield strength for p 0 Slope 2 Piecewise linear Pressure 1 Yield strength 1 Pressure 2 Yield strength 2 Pressure 3 Yield strength 3 Pressure 4 Yield strength 4 Load data Save data Generate new cavity expansion data Figure 2 3 The graphical menu for defining material models 11 The cavity diameter is used when there is an initial cylindrical channel in the target This situation could typically arise for a tandem warhead where the first stage shaped charge has created a cavity before penetration of the second stage projectile 2 3 Target material data On pushing the button Create material a new window is opened This is shown in Figure 2 2 It contains various fields for the user to input data describing the target material The input parameters common to all materials are shown in Table 2 3 Table 2 3 T
9. FFI RAPPORT MULTIFUNCTIONAL NUMERICAL TOOL FOR PENETRATION ANALYSIS TELAND Jan Arild FFI RAPPORT 2002 04647 FFIBM 766 130 MULTIFUNCTIONAL NUMERICAL TOOL FOR PENETRATION ANALYSIS TELAND Jan Arild FFI RAPPORT 2002 04647 FORSVARETS FORSKNINGSINSTITUTT Norwegian Defence Research Establishment P O Box 25 NO 2027 Kjeller Norway Approved Kjeller 27 November 2002 Bjarne Haugstad Director of Research FORSVARETS FORSKNINGSINSTITUTT FFI UNCLASSIFIED Norwegian Defence Research Establishment P O BOX 25 SECURITY CLASSIFICATION OF THIS PAGE N0 2027 KJELLER NORWAY when data entered REPORT DOCUMENTATION PAGE PUBL REPORT NUMBER SECURITY CLASSIFICATION 3 NUMBER OF FFI RAPPORT 2002 04647 UNCLASSIFIED PAGES PROJECT REFERENCE DECLASSIFICATION DOWNGRADING SCHEDULE 18 FFIBM 766 130 TITLE MULTIFUNCTIONAL NUMERICAL TOOL FOR PENETRATION ANALYSIS NAMES OF AUTHOR S IN FULL surname first TELAND Jan Arild DISTRIBUTION STATEMENT Approved for public release Distribution unlimited Offentlig tilgjengelig INDEXING TERMS IN ENGLISH IN NORWEGIAN a Cavity expansion a Hulromsekspansjon b Numerical tool b Numerisk verkt y c Matlab c Matlab d Penetration d Penetrasjon e e THESAURUS REFERENCE 8 ABSTRACT An advanced analytical numerical tool for calculating penetration of rigid projectiles into various materials is described The program is based on cavity expansion t
10. arget material parameters Parameter Dimension Required E modulus GPa Yes Poisson ratio Yes Density kg m Yes Further the plastic behaviour of the target material must be specified The user can choose which model to apply by pushing the appropriate button next to the model There are four different yield models available e Empirical concrete model e Mises model e Mohr Coulomb model e Piecewise linear model These are described in the next sections 2 3 1 Empirical concrete model This is an empirical concrete model that was developed by Forrestal 3 using curve fitting to penetration experiments into concrete It was later modified by FFI 4 to hold for a larger range of concretes Thus the cavity radial stress is found from this formula p So pv S 49 50 where is the concrete compressive strength in MPa This model is only applicable for concrete targets The user input is the concrete compressive strength and for a finite size target the plastic yield limit For concrete the plastic yield limit depends on pressure and a value corresponding to the high pressures encountered during a penetration process should be entered Note that the yield limit is only used in the calculation of boundary effects for finite We use the term plastic yield for concrete even if yielding is related to internal microcracking and damage 12 targets If the problem involves a semi inf
11. erials where scabbing will make sure that material is released from the target rear face long before the projectile actually 15 reaches it However at the moment no good scabbing model is available but if one is developed this feature can easily be included in future versions of the program 5 USING THE PROGRAM At FFI Version 1 0 of the program is stored in the p766 penetration directory The program should be compatible with all newer versions of Matlab During this work Matlab 6 2 on a Unix platform has been used The user first has to start Matlab and then run the penetrate m file The menus described in the previous chapters are then displayed The target material data can either be loaded from a previous session or a new material can be created and even saved for later use To load an old material just select Load material and a list of available material models will be described The appropriate material is then selected by double clicking To generate a new material the user must choose Generate material after which a new window appears The various inputs have been described in Chapter 2 After entering the material data it is necessary to choose Generate new cavity expansion data A waitbar will then be displayed while the computer uses cavity expansion theory to determine a relationship between stress and velocity After the CET data has been generated the user may either save it for later use by se
12. heory CET and is a further development of an earlier FFI code to include numerous projectile geometries and target configurations The new program also contains a graphical user interface making it very simple to define materials and set up penetration simulations Moreover it is possible to run multiple simulations to determine the penetration depth as a function of velocity and the program can also determine ballistic limit and required target thickness to stop a projectile This report serves as documentation and user manual for the program 9 DATE AUTHORIZED BY POSITION This page only 27 November 2002 Bjarne Haugstad Director of Research ISBN 82 464 0693 0 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE when data entered CONTENTS 2 1 2 2 23 2 3 1 2 3 2 233 2 3 4 3 1 3 2 3 3 3 4 3 5 6 1 6 2 6 3 6 4 INTRODUCTION GRAPHICAL USER INTERFACE Projectile parameters Target geometry parameters Target material data Empirical concrete model Mises model Mohr Coulomb model Piecewise linear model BOUNDARY EFFECTS Option 1 Option 2 Option 3 Option 4 Discussion PENETRATION PHASES USING THE PROGRAM RUNNING SIMULATIONS Single simulation Multiple simulations Ballistic limit Required thickness SUMMARY Distribution list Page 10 11 11 12 12 12 12 13 13 14 14 14 14 15 15 15 16 16 16 17 18 MULTIFUNCTIONAL NUMERICAL TOOL FOR PENETRATION ANALYSIS 1 INTRODUCTION
13. inite target the yield limit needs not be specified The same goes for the elastic modulus and Poisson ratio 2 3 2 Mises model In the Mises model the yield limit is constant independent of pressure This is typical for the behaviour of metals such as steel and aluminium when no strain hardening and temperature effects are included The cavity stress is calculated according to the following analytical formula from spherical CET ms 1 In 4e 3 1 v 2 3 3 Mohr Coulomb model In the Mohr Coulomb model the yield limit depends linearly on pressure in the following way Y Y fPp Required input is therefore Y the yield strength for p 0 and the slope B There is no simple expression for the cavity stress p so it is calculated numerically It is very important to be aware that our use of the term Mohr Coulomb model is consistent with the literature and is therefore not the same as what is called Mohr Coulomb in Autodyn terminology Instead the Autodyn Mohr Coulomb is in this report referred to as Piecewise linear 2 3 4 Piecewise linear model The Piecewise linear yield model is the most advanced of the three and indeed the two other models are special cases of the Piecewise model Again the yield stress depends on the pressure The user has to input four pairs of p Y datapoints and the yield stress is then linearly interpolated For pressures larger than the highest input the yield stress is constant
14. lecting Save data or simply close the window and return to the main window since the target material model is now stored in memory 6 RUNNING SIMULATIONS After both the projectile and target data have been entered penetration simulations may be run There are several different simulation modes each of which is described below 6 1 Single simulation By pressing the button Run single simulation a penetration simulation is performed for the exact velocity entered in the menu A progress bar is displayed while the simulation is running When the simulation is complete new windows are opened showing the penetration depth velocity and acceleration as functions of time The variables vectors x v a t are further stored in memory and are accessible from Matlab to be manipulated as desired 16 The calculations stop either when the projectile has perforated the target or when it has come to rest inside the target If perforation occurs the exit velocity is displayed It may then be of interest to find the ballistic limit 1 e the minimum impact velocity that achieves perforation and the minimum target length that stops perforation Either of these is possible as described later 6 2 Multiple simulations By pressing the button Run multiple simulations the program performs several simulations in order to find the relationship between impact velocity and final penetration depth The maximum impact velocity is the o
15. ne entered in the menu system while the minimum impact velocity is 100 m s Twenty simulations are performed at constant intervals between maximum and minimum impact velocity This may take a few minutes depending on the computer speed When the simulations are complete a new window is opened with a plot of the relationship between impact velocity and final penetration depth Again these variables Xi and Vi are stored in memory 6 3 Ballistic limit The button Find ballistic limit is only relevant in the cases of finite targets On selecting this option the program searches for the ballistic limit through an iteration procedure using the following algorithm First the initial run is performed If the projectile perforates the target the initial velocity for the next iteration is determined by the following formula 02 rest _ i 2 Y VM v This comes from an empirical relationship in 6 If the projectile does not perforate the target the impact velocity for the next run is increased according to this formula ye L l yo 0 i 0 x where L is the target length and is the length of the projectile nose 6 4 Required thickness The button Find required thickness is also only relevant for finite targets It works in the same way as the search for ballistic limit except that the target length thickness is varied in each iteration instead of the impact velocity 17 If the projectile perforate
16. ration using Forrestal s formula FFI RAPPORT 99 04415 5 Littlefield D L Anderson Jr C E Partom Y Bless S J The penetration of steel targets finite in radial extent Int J Impact Engng Vol 19 No 1 pp 49 62 1997 6 Recht R F Ipson T W Ballistic perforation dynamics J Appl Mech 30 pp 384 390 1963 18 DISTRIBUTION LIST FFIBM Dato 27 November 2002 RAPPORTTYPE KRYSS AV RAPPORT NR REFERANSE RAPPORTENS DATO RAPP f Jnotar f ar 2002 04647 FFIBM 766 130 27 November 2002 RAPPORTENS BESKYTTELSESGRAD ANTALL SIDER Unclassified 18 RAPPORTENS TITTEL FORFATTER E MULTIFUNCTIONAL NUMERICAL TOOL FOR TELAND Jan Arild PENETRATION ANALYSIS FORDELING GODKJENT AV FORSKNINGSSJEF FORDELING GODKJENT AV AVDELINGSSJEF EKSTERN FORDELING INTERN FORDELING ANTALL EKSNR TIL ANTALL EKSNR TIL 1 Forsvarsbygg 9 FFI Bibl FFI ledelse FFIE FFISYS FFIBM FFIN Forfattereksemplar er Restopplag til Biblioteket v Leif Riis v Ragnar Bj rgaas v Helge Langberg v Tom Hermansen fi palts gt pal a NOR RR Re Elektronisk fordeling FFI veven Bjarne Haugstad BjH Svein Rollvik SRo Eirik Svins s ESv Henrik Sj l HSj Knut B Holm KBH Svein E Martinussen SEM John F Moxnes JFM Jan Arild Teland JTe FFI K1 Retningslinjer for fordeling og forsendelse er gitt i Oraklet Bind I Bestemmelser om publikasjoner for Forsvarets forskningsinstitutt pkt 2 og 5 Benytt ny side om n dvendig
17. s the target the target length is increased according to the following formula yo LY P 2 Y If the projectile does not perforate the target the target length is decreased according to this expression L x 2 where x is the penetration depth in the current cycle haa The program determines the least necessary target thickness required to stop the projectile Because of uncertainties surrounding the material models and since effects such as scabbing are not accounted for one should include an appropriate safety margin if the results are to be used for design purposes 7 SUMMARY A numerical tool based on 1 for calculating penetration of rigid projectiles into various materials has been developed The user can to a certain degree define his own target material models and how boundary effects are accounted for As new penetration theory is developed this can be implemented in the program A graphical user interface has been created so that the program is very easy to use References 1 Berthelsen P A Cavity expansion og penetrasjonsmekanikk Materialmodeller og numeriske losningsmetoder FFI RAPPORT 99 04260 2 Forsvarets Handbok for Vapenvirkninger 2002 3 Forrestal M J Altman B S Cargile J D Hanchack S J An empirical equation for penetration depth of ogive nose projectiles into concrete targets Int J Impact Engng Vol 15 No 4 pp 395 405 1994 4 Sjol H Teland J A Prediction of concrete penet
18. tudies whereas the one in 2 is more aimed at the novice who needs a quick result 2 GRAPHICAL USER INTERFACE The program can be run in two modes either using the new graphical user interface GUI or the older textbased user interface For a beginner the GUI is clearly the easiest way to access the main program while a more experienced user may prefer the other method El ure N File Edit View Insert Tools Window Help amp FFI Advanced Penetration Tool Analytical Numerical Cavity Expansion Analysis Select Projectile Create material Define new projectile Load material data Projectile mass Target material Projectile diameter Boundary option GF Projectile curvature Target diameter cm Impact velocity TS Target length cem Flat part diameter Cavity diameter cm Run single simulation Find ballistic limit Run multiple simulations Find required thickness I Figure 2 1 GUI of the Multifunctional Penetration Tool The GUI is shown in Figure 2 1 Most ofthe options should be self explanatory but for completeness we describe everything below In order to completely define a penetration problem it is necessary to define parameters related to both the projectile and the target In the left column of the GUI projectile parameters are defined while target parameters are input in the right column 2 1 Projectile parameters Some common projectiles can be selected from the
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