Localized Interaction Models (LIMs)

Transcription

1 Capter ocalized Interaction Models IMs Nomenclature a, a, a parameters determining te model; A, A, A parameters, Tale.6., Eqs ; tickness of te sield; B, B, B parameters, Eqs..6.5,.6.,.6.6,.7.4; BV allistic limit elocity; D drag force acting at te projectile; D function, Eqs..6. and.7.; D resistance of te flat part of te nose of actor; flat D a, D ~ aerage drag force; a DOP dept of penetration; e, e parameters, Eqs ; f, f functions, Eqs and.4.9; instantaneous dept of penetration, Fig...; H dept of penetration DOP; k tangent of semi-angle of a conical-saped nose of cone projectile; K function, Eq..7.; K calier radius ead of a ogie-nosed projectile, Eq..6.8; lengt of te nose of te actor, Fig...; = R ; lengt of a cylindrical part of te actor, Fig...; IM localized interaction model; 3

2 4 Hig-Speed Penetration Dynamics: Engineering Models and Metods m mass of projectile; n inner normal ector at te surface of projectile; Q function, Eq..4.; r radius of te flat luntness of te nose of actor; r = r R ; R radius of te sank of actor; t time; T Q, Eq..4.6; u, u function, Eq..3.; u n cos, also Eq..3.,.3.4; U function, Eqs..3.7 and.3.4; act initial elocity of projectile; instantaneous elocity of actor normal act; ector of te surface element elocity of te projectile; allistic limit elocity BV of projectile; l residual elocity of projectile; res V function descriing te dependence s. and, Eq..3.4;,, cylindrical coordinates associated wit te actor, Fig...; function in te epression for te drag force acting at te nose of actor, see Eqs ;, functions determining te actor-sield interaction surface, Figs...,..,..3,..4 and..5, Eqs... and..4; semi-erte angle of cone-nosed actor; parameter, Eq ; friction coefficient; fr function, Eq..4.9; function, Eq..4.34; see,, ; radius of te circle of ogie, Fig..6.; og tangent ector at te projectile surface;

3 ocalized Interaction Models IMs 5 angle etween te ector n and te ector, Fig...; see,, ; function, Eq..4.; function determining te sape of te actor, Fig...; or d d ; ; parameters, Eqs..7.6 and.7.8; parameters, Eqs..6.36,.6.39 and.6.4; n, functions determining te IM; function, Eq...6; Superscripts unit ector.. Basics of te ocalized Interaction Teory Many engineering models for penetration modeling elong to te category of te so-called localized interaction models IMs, in wic te integral effect of te interaction etween a sield and a penetrating projectile is descried as a superposition of te independent local interactions of te projectile surface elements wit te sield. Eery local interaction is primarily determined y te local elocity of te surface element and te angle etween te local surface elocity ector and te local normal ector to te projectile surface as well as y some gloal parameters tat take into account te integral caracteristics of te sield e.g., ardness, density, etc.. Te unified description of different ersions of IM wic are widely used in te penetration mecanics can e presented in te following form:

4 6 Hig-Speed Penetration Dynamics: Engineering Models and Metods [ n u, n u, ] ds df n, n ds if if if u * u u u u *.. u n / u,.. u n cos,..3 were see Fig... df is te force acting at te surface element ds of te projectile tat is in contact wit te sield, n and are te inner normal and tangent unit ectors at a gien location on te projectile surface, respectiely, is an unit ector of te surface element elocity of te projectile,, is te angle etween te ector n and te ector. Te non-negatie functions n u, and u,, te normal stress and te tangential stress, respectiely, determine te model of te projectile-sield interaction and depend also on te parameters tat caracterize, primarily, te properties of te sield. Te unit tangent ector lies in te plane of te ectors and n and is normal to te ector n ; its direction is cosen suc tat, i.e., te friction force is directed in te positie direction of te ector. ds n Fig.... Description of te IM.

5 ocalized Interaction Models IMs 7 Parameter u * u * determines te maimum magnitude of te angle, * cos u*, werey te actor still interacts wit te sield. It is assumed tat for * te contact etween te lateral surface of te actor and te sield is disrupted. Since te magnitude of * is not known, it is commonly assumed tat u *. Taking into account te possiility of caitation loss of contact etween a portion of te lateral surface of te penetrator and te sield during penetration into solid media is one of te researc directions to roe IMs Bazeno and Koto,,. Te first and te second formulas in Eqs... descrie interaction etween te actor and te sield upon teir contact wile te tird equation determines te condition wen tere is no contact. Te case wit u is descried separately y te second equation ecause te coice of te direction of te tangent ector in tis situation is undetermined. Te resultant force acting on te projectile at eac instant of time is determined y integrating df oer te surface of te projectile-sield contact at te same instant, S. et us now consider a normal act act elocity is normal to te acted plate of a rigid symmetric ody, werey projectile eecutes a translational motion under te effect of te drag force D, wic can e written in te general case as follows: D df df df,..4 S S perp S lat were S perp is a part of te contact surface S tat is normal to te actor elocity as a rule, te flat luntness and S lat is te lateral surface of te actor. Sustituting df from Eq... into Eq...4, we otain: were D, ds u, ds,..5 n S perp S lat

6 8 Hig-Speed Penetration Dynamics: Engineering Models and Metods u n, u u, u u,...6 Hereafter we consider normal penetration and use te following notations Fig.... Te coordinate, te instantaneous dept of penetration, is defined as te distance etween te nose of te actor and te front surface of te sield, and is te lengt of te nose of te actor. Te cylindrical coordinates,, are associated wit te actor, and te equation,, were is some function, determines te sape of te actor. Generally, we consider actors wit flat luntness and a cylindrical part of te lengt, and assume tat tis cylindrical part does not interact wit te sield. In oter words, all te aoe formulas refer to te nose of te actor wic is located etween te cross-sections and. In spite of te presence of te cylindrical part sometimes for reity we will use te term conical actor or cone instead of conicalnosed actor. History of deelopment of IT is sureyed in our earlier monograp Ben-Dor et al., 6a. Emergence and deelopment of IT in te field of gas dynamics is compreensiely descried in te monograps y Alekseea and Barantse, 976; Bunimoic and Duinsky, 995 and a Sield wit a finite tickness Semi-infinite sield,, Fig.... Te notations.

7 ocalized Interaction Models IMs 9 Mirosin and Kalido, 99,, as well as in te reiews of Bunimoic and Duinsky, 996, 997. Progress acieed y IT in gas dynamics stimulated attempts to etend tis approac also to penetration mecanics. Notaly, a numer of monograps pulised in te last decade of te t century e.g. Bunimoic and Duinsky, 995; Vederniko and Scepanosky, 995; Vederniko et al., 995; Ostapenko, 997 ae separate sections dealing wit application of te approimate models in ot fields, gas dynamics and penetration mecanics. A similar to IT concept forms te asis for te approac tat is known in te penetration mecanics as te differential area force law DAF. Tis model was proposed y te AVCO Corporation in te early 97s Hadala, 975; Heuzé, 99, Bernard and Creigton, 979. In tis metod te projectile is diided into a numer of small suregions, and te total force acting on te actor is otained y summing integrating te forces acting at te su-regions. Regarding te particular realization of tis procedure for penetration into soils actually it was deeloped for tis purpose it must e noted tat DAF differs significantly from IT: i epression for te local force eplicitly depends on te instantaneous dept of penetration and ii analytical epression for te local force is quite inoled. Tis is te reason tat DAF does not allow deriing eplicit formulas for te penetration dept, and is mainly applied in numerical simulations of penetration for te non-normal act Heuzé, 99. A similar to IT approac can e also found in te study y Rect, 99. Hereafter we consider compreensiely te two-term and tree-term models wic descrie penetration of rigid strikers aing te sape of odies of reolution. Additional formulas for te design of strikers aing different sapes and descriptions of different classes of models can e found in our earlier monograp Ben-Dor et al., 6a.

8 Hig-Speed Penetration Dynamics: Engineering Models and Metods. Impactor-Sield Interaction Surface.. Semi-infinite sield Te formalism of te description of te actor-sield interaction surface in te case of a semi-infinite sield is illustrated in Fig.... Generally, two stages of penetration can e considered. Te first stage, entry into te sield, occurs wen Fig...a. In tis case, te flat luntness of te actor if any and te part of its lateral surface etween te cross-sections and interact wit te sield. Te second stage Fig..., i.e., motion inside te sield, is caracterized y full immersion of te luntness and of te lateral surface of te actor into te sield and occurs wen. Terefore, te moing area of te actor-sield interaction can e descried as follows see Fig...: were,.. if.. if In some instances, it is conenient to define function also for te negatie alues of te ariale, assuming tat for. a Stage Stage Fig.... Two stages of penetration into a semi-infinite sield.

9 ocalized Interaction Models IMs * * Fig.... Penetration into a semi-infinite sield. Description of te area of actor sield interaction. *.. Sield aing a finite tickness Consider a sield wit a finite tickness,. First, let. In tis case perforation can e considered as a tree-stage process sown in Fig...3. At te first stage entry of te actor into te sield,, te flat luntness of te actor if any and te part of te lateral surface etween te cross-sections and interact wit te sield. At te second stage full immersion of te actor into te sield,, te flat luntness of te actor if any and te entire lateral surface interact wit te sield. At te tird stage emergence of te actor from te sield,, te flat luntness of te actor if any does not interact wit te sield, wile part of te lateral surface of te actor etween te cross-sections and interacts wit te sield. Te case is illustrated in Fig...4 and can e analyzed in a similar manner to te case wen. In ot cases, te moing area of te actor-sield interaction can e descried as follows see Fig...5:,..3 were function is defined y Eq... and

10 Hig-Speed Penetration Dynamics: Engineering Models and Metods if...4 if In some instances, it is conenient to define functions and for a sield aing a finite tickness as follows: for and for. Te latter definition lies tat. Terefore Eq...3 can e used as a unified description of te area of te actor-sield interaction, taking into account tat and for a semi-infinite sield and and is defined y Eq...4 for a sield aing a finite tickness. Te model can e slified if we do not take into account te stage were penetrator is only partially immersed in te sield. Since suc slification is used frequently, in Sections.5-.7 we consider tis suclass of te models. a Stage Stage c Stage 3 Fig...3. Tree stages of penetration into a sield aing a finite tickness. Te case.

11 ocalized Interaction Models IMs 3 Stage a Stage c Stage 3 Fig...4. Tree stages of penetration into a sield aing a finite tickness. Te case. a Fig...5. Penetration into a sield aing a finite tickness. Description of te area of te actor-sield interaction.

12 4 Hig-Speed Penetration Dynamics: Engineering Models and Metods.3 General Relationsips for 3-D Impactors.3. Drag force. Equation of motion Using te adopted system of coordinates and notations allows us to derie an epression for te drag force. Since u, u, n, ds u, dd,.3. u, u,, u,,.3. Eq...5 can e transformed into te following form: n nose u,, u, d, D,, d.3.3 were nose is te nose area of te actor, nose d,..3.4 Friction etween te actor and te sield is taken into account as follows: fr n,.3.5 were fr is a friction coefficient. Taking into account Eqs...6, and.3.5 we can rewrite Eq..3.3 as follows: u,, U, d, D, n, nose d n.3.6 were U, [ u fr u ] u fr..3.7

13 ocalized Interaction Models IMs 5 Function in Eq..3.6 for te total drag force descries te resistance force acting at te nose of te actor:.3.8 in te case of penetration into a semi-infinite sield, and if if,.3.9 if wen an actor penetrates into a sield aing a finite tickness. Clearly, we do not consider ere penetration penomena tat are accompanied y plug formation. Equation of motion of te actor aing mass m, wit te initial conditions, m D,,.3.,,.3. allows us to determine te dept of penetration and te elocity of te actor as a function of time t and act elocity. Since te rigt-and side of Eq..3. does not depend on time in te eplicit form, te order of tis differential equation can e decreased. Considering as a function,, and taking into account tat we can rewrite Eq..3. as follows: et d d d d dt dt d dt.3. d m D,..3.3 d

14 6 Hig-Speed Penetration Dynamics: Engineering Models and Metods V ;.3.4 e te solution of Eq..3.3 wit te initial condition..3.5 Te dept of penetration DOP into a semi-infinite sield for a gien act elocity, H, tat is defined as te dept at wic te actor slows down to zero elocity, can e found from te following equation: V H;..3.6 Once te function V ; is determined, te law of motion of te actor can e found as te solution of te differential equation for wit te appropriate initial condition: d t V ;,..3.7 dt Te latter equation can e written as follows: dz t,..3.8 V z; Equation.3.8 allows us to deduce te inerse dependence, s. t, if required. Terefore, Eqs..3.4 and.3.8 determine te function t in a parametric form..3. Residual and allistic limit elocities. Dept of penetration In te case of a sield aing a finite tickness, te allistic limit elocity BV, l, is usually defined in analytical models as te initial elocity of te actor tat is required for te actor to emerge from te sield wit zero elocity. Terefore, l is determined from te following equation:

15 ocalized Interaction Models IMs 7 V ;..3.9 Te epression for te residual elocity, te elocity of te actor wit wic te actor emerges from te sield, reads: res l V ;,..3. In te case of a semi-infinite sield te dept of penetration DOP, H, for te known act elocity,, is found from te equation: l V H;..3. Terefore, general caracteristics of penetration can e otained troug soling a first order ordinary differential equation..3.3 Impactor Haing a Sape of Body of Reolution.3.3. General formulas If te actor is a ody of reolution ten Consequently,..3. nose r,.3.3 were r is radius of te flat luntness of te actor nose, and Eqs..3. and.3.7 ly tat u u, U U..3.4 fr Using Eqs te epression for te drag force, Eq..3.6, can e rewritten as follows: D, n, r n, fr d..3.5

16 8 Hig-Speed Penetration Dynamics: Engineering Models and Metods.3.3. Cone-nosed actor. Finite tickness sield If te nose of te actor as te sape of a sarp cone wit semi-angle ten and Eq..3.5 can e written as follows: tan.3.6 sin, tan fr tan D, d..3.7 n For a finite tickness sield, Eq..3.3 wit a drag force D gien y Eq..3.7 yields: ~ d~ tan fr tan ~,.3.8 sin, m were res n ~ ~ d d..3.9 ~ Sustituting and into Eq..3.8 we otain te relationsip etween and res : were d tan fr tan,.3.3 sin, m res n dd d d..3.3 We canged te order of integration see Fig...5 wen was calculated.

17 ocalized Interaction Models IMs 9 Equation.3.3 lies a formula for l wen res and. l.4 Projectiles Haing a Sape of Bodies of Reolution. Two-Term Models.4. Aritrary ody of reolution In tis section we consider te following class of models: n u, u a, u, u,,.4. were function u and parameter a determine te model. Along wit tis general model we also consider two particular models: and u u, u a n a u a.4. au au u u, a n,.4.3 u u were two parameters, a and a, depend on te mecanical properties of te sield material. Tese two models are often used in penetration mecanics. In particular, Eq..4. and Eq..4.3 are related wit sperical and cylindrical two-term caity epansion models, respectiely see Capter. Sustituting n from Eq..4. into Eq..3.5 we otain: were in a general case D, f f,.4.4 f a r fr d,.4.5 fr n

18 Hig-Speed Penetration Dynamics: Engineering Models and Metods fr d r f..4.6 In te particular case of te model gien y Eq..4. fr d r a f,.4.7 wile in te case of te model gien y Eq..4.3 for sarp actors witout flat luntness fr d a f..4.8 Equation of motion of te actor, Eq..3.3, wit, D gien y Eq..4.4, is a linear ordinary differential equation wit respect to : f f d m d..4.9 Te solution of tis equation wit te initial condition.4. reads: d Q f m Q ~ ~ ~,.4. were d f m ep Q..4. In te case of a semi-infinite sield, Eq..4. yields an equation for te DOP, H, after sustituting H :

19 ocalized Interaction Models IMs H m f Q d,.4.3 were Eqs..4.5 and.4.7 are used wit. In te case of a sield aing a finite tickness, te BV, l, can e calculated from Eq..4. y sustituting, and l : Assuming tat l l m f Q d..4.4, we can write Eq..4. for te residual elocity, as follows: res res T m f Q d,.4.5 were T Q..4.6 It is assumed tat Eq..3.9 is used for calculation of in te case of sield of a finite tickness. Eqs..4.4 and.4.5 ly te following sle relationsip etween te act elocity, te residual elocity and te BV: l T res l, l Sarp conical-saped actor.4.. General equations If a nose of projectile as a sape of a conical ody of reolution wit semi-angle tan k, ten cone k.4.8 cone

20 Hig-Speed Penetration Dynamics: Engineering Models and Metods and formulas for functions f and f in Eqs..4.5 and.4.6 can e written as follows:, e f e f,.4.9 were in general case:,,,, cone cone cone fr cone k k z z a k k z e e e.4. ].5[ d..4. In te particular case of te model gien y Eq..4. 3,, cone cone cone fr cone cone k k a a k k k a e e e,.4. wile in te case of te model gien y Eq..4.3.,, 3 cone cone fr cone k a a k k a e e e.4.3 Taking into account Eq..4.9 te deriatie of te function Q in Eq..4. can e written as follows: ep Q f me e Q f m d f m f m d dq.4.4 and, consequently, ~ ~ ~ ~ ~ ~ Q m d d dq m d Q f..4.5

21 ocalized Interaction Models IMs 3 Eq..4., tat determines te dependence s., reads: Q,.4.6 Q were ˆ e Q ep d ˆ d m ˆ..4.7 e ep [ ˆ ˆ] dˆ m.4.. Semi-infinite sield In te case of a semi-infinite sield,, is determined y Eq... and te epression for Q can e written as follows: e Q ep,.4.8 m were 3 ˆ dˆ 3 if 3 3 if ˆ dˆ ˆ dˆ. dˆ if if.4.9 Sustituting and H into Eq..4.6 we otain te equation for te dept of penetration, H : Q H,.4.3

22 4 Hig-Speed Penetration Dynamics: Engineering Models and Metods wic as te following solution: H 3m 3 ln e m ln e 3 if if ˆ ˆ,.4.3 were 3 e ˆ ep m Eq..4.6 descries te dependence s., were Q is gien y Eq Sield aing a finite tickness In te case of a sield aing a finite tickness, te functions and are determined y Eqs...4 and.., respectiely; te epression for Q, Eq..4.7, can e written as follows: e Q ep [,,.4.33 m were is determined y Eq..4.9 wile 3 ˆ ˆ d ˆ ˆ d if. 3 if if if.4.34 Te dependence s. can e otained y sustituting Q from Eq into Eq..4.6 and taking into account Eqs..4.9 and.4.34.

23 ocalized Interaction Models IMs 5 Since 3 3, 3 3,.4.35 ten Eq lies tat m Q T, T ep e.4.36 and Eq..4.6 allows to determine te epression for te residual elocity: res [ T ] T,,.4.37 l were l T.4.38 is te BV ecause res wen l..5 Aeraged IMs. General Approac.5. Introduction Te drag force depends upon te instantaneous penetration dept ecause te projectile-sield contact area aries at te stage of te incomplete immersion of te projectile in te sield. Tis dependence renders te model quite inoled. Te latter sortcoming is usually eliminated y replacing te ariale integration limits y constant integration limits in te epression for te drag force, and, altoug suc slification can air te accuracy of te model i et al., 4. Ostapenko et al., 994 alidated te following approimate estimations: te stage of incomplete immersion of te projectile in te semi-infinite sield can e disregarded if R for penetration into soil and 3 R for penetration into metal sields. In Section.5 we sow for strikers aing a sape of odies of reolution tat suc slification can e interpreted as a result of

24 6 Hig-Speed Penetration Dynamics: Engineering Models and Metods application of a certain aeraging procedure. In Section.6 we present a compreensie description of two-term models wic are used ereafter. In some cases we use formulas pertinent to te tree-term model for penetration into semi-infinite sields; tese formulas can e found in Section.7. We start from te Eq..3.5 for te drag force D were te function is aritrary and n u, u,, u..5. Eq..3.5, can e rewritten as follows: were fr n D,, r, d,.5. n, n, fr Sield aing a finite tickness Assume tat te actor perforates te sield. et us calculate te spatial aerage oer of te drag force D, gien y Eq..5.. Te epression for te aerage drag force, D, reads: D a d n, r D, d, d a d..5.4 Taking into account te definition of te function gien y Eq..3.9 we otain:

25 ocalized Interaction Models IMs 7 d d..5.5 Te doule integral in rigt side of Eq..5.4 can e transformed y canging te order of integration see Fig...5: d, d, d d, d..5.6 Using Eqs we can rewrite Eq..5.4 as follows: were D a D,.5.7 D n, r, d..5.8 Te equation of motion of actor, md d D a, can e written as te following differential equation wit separale ariales: d m D..5.9 d Te solution of Eq..5.9 wit te initial condition reads: d ~ ~ D ~..5. m Sustituting, and l in Eq..5., we otain te epression for te BV: l d ~ ~ D ~ m..5.

26 8 Hig-Speed Penetration Dynamics: Engineering Models and Metods In order to otain te epression for te residual elocity let us sustitute and res in Eq..5.: d ~ ~ D ~ res m..5. It can e easily sown tat formulas for te BV and for te residual elocity coincide for te eact and aeraged models if te actor as te sape of a sarp cone..5.3 Semi-infinite sield Aeraging D, gien y Eq..5. oer H, were H, as efore, is te DOP and, we otain te epression for te aerage drag force, D ~ a : ~ D a H, r n H D, d H H d, d..5.3 Assuming tat H and canging te order of integration in te integral in Eq..5.3, we arrie at te following relation see Fig...: H H d, d H H, d, d, d. H d.5.4 Terefore we proed tat D ~ a D.5.5

27 ocalized Interaction Models IMs 9 and te equation of actor motion can e written as follows: d m D..5.6 d Te solution of Eq..5.6 wit te initial condition, d ~ ~ D ~ m,.5.7 allows us to determine te epression for te DOP setting and H : d ~ ~ D ~ H m Aeraged Two-Term Models.6. General two-term model Te equation of te general two-term model reads: n u, u u, u, u,,.6. were u and u are te known functions wic determine te model. After sustituting n from Eq..6., epression for D from Eq..5.8 can e rewritten as follows: were A D A A,.6. r fr d,.6.3 fr n

28 3 Hig-Speed Penetration Dynamics: Engineering Models and Metods A r fr d..6.4 Te contriution of te flat part of te nose of actor, total drag force is as follows : D flat, in te D flat B B, B r, B r,.6.5 In Sections we derie formulas, ased on te general twoterm model and its su-models, for te finite widt and semi-infinite sields..6. Sield aing a finite tickness.6.. General two-term model Eqs yield te following relations wen D is gien y Eq..6.: A A A A ep,,.6.6 m A A A l ep,.6.7 A m A A A res A ep m A A l ep, l m.6.8 were A and A are determined in Eqs..63 and.6.4, correspondingly.

29 ocalized Interaction Models IMs Su-model n u, u a In te case wen u a a is a constant, n u, u a, u, u,,.6.9 Eqs..6.4 and do not cange, wile Eqs..6.3 and.6.5 can e written as follows: A a r fr d,.6. fr n D flat B B, B r a, B r..6. If friction is not taken into account Eq..6. can e written in te following form: A ar,.6. taking into account tat fr d d d d R r, R Su-model n u, au a In te case of te su-model u, a u a,.6.4 Eqs..6. and do not cange wile n A fr a r d,.6.5 D flat B B r a, B, B r a..6.6

30 3 Hig-Speed Penetration Dynamics: Engineering Models and Metods If, ten taking into account Eq..6.3 we can rewrite fr Eq..6.5 in te following form: A a r 3 d a wile Eq..6. remains alid for A. R.6..4 Su-model n u, a [ u u ] a In te case of te su-model gien y Eq..4.3, wen d,.6.7 n u, a [ u u ] a,.6.8 te formulas in Eqs , remain alid wit A a fr d,.6.9 A a d..6. If fr ten Eq..6. for A remains alid. Since, te model is applicale only for sarp actors wen r or for calculation of te drag force acting te lateral surface of te actor. fr.6.3 Semi-infinite sield Eqs..5.7 and.5.8 yield te following formulas for te general two-term model determined y Eq..6.: A A,.6. m A A A ep

31 ocalized Interaction Models IMs 33 m A H ln,.6. A A were A and A are determined y Eqs and.6.4. In te case of te su-models considered aoe, A and A are calculated using te formulas in te corresponding capters..6.4 Ogie-saped actors.6.4. Description of te ogie sape Particular attention in penetration mecanics as een gien to ogienosed actors. Te generatri of te nose of ogie-nosed projectiles is an arc of te circle see Fig..6. wic is descried in te general case of truncated ogie-nosed actor y te following equation: og R,,.6.3 were og is te radius of te circle wit te center at te line. Eq..6.3 descries te increased cone upwards arc of te circle. Since tis arc must intersect te semi-ais of positie ordinate, tis lies te following constraint: og,.6.4 og og r R..6.5 Depending on te relationsip etween te parameters, og, and R, tere are seeral possile ersions of te nose sape. If Eq..6.4 is satisfied and og R ten te constraint gien y Eq..6.5 is satisfied and r, i.e., te actor as a flat luntness truncated ogie and as te form sown in Fig..6.a. If R ten te solution of te inequality in Eq..6.5 reads: og og

32 34 Hig-Speed Penetration Dynamics: Engineering Models and Metods Since * * R og og, og..6.6 R R og,.6.7 R te constraint gien y Eq..6.4 is satisfied in tis case. Te sape of te generatri is sown in Fig..6.c. In particular, r, i.e. projectile * as a sarp ogie sape see Figs..6.,c. wen og og. Te projectiles aing te sape tat corresponds to te case wen og R in particular, for r are most often found in te literature on penetration mecanics. a R r og R r og c R og Fig..6.. Generatri of an ogie-nosed actor.

33 ocalized Interaction Models IMs 35 Using te dimensionless parameters K og,, R R r r R,.6.8 were K is te so called calier radius ead, te aoe analysis sows tat te projectile as a sape sown in Fig..6.a if te sape sown in Fig..6. if.5 K.5,.6.9 K ma[.5,.5 ], or K.5 wile.6.3 and te sape sown in Fig..6.c if K.5 wile..6.3 Te alues, K. 5 and, correspond to te emisperical nose. Terefore te admissile alues of K and must satisfy te following constrains:.5 K.5 K.5 if if..6.3 Te domain tat is determined y tese constrains is sown in Fig..6.. If te inequalities in Eq..6.3 are satisfied ten Eq..6.5 yields te following epression for r : r K 4K n fr.6.4. Su-model u, a u a, For te su-model

34 36 Hig-Speed Penetration Dynamics: Engineering Models and Metods K CHR K CHR.5.5 a c Hemispere. Fig..6.. Domain on te plane K, corresponding to te sapes a-c of te nose of actor wic are sown in Fig..6.. n fr u, a u a,,.6.34 in te general case of truncated actor, integral in Eq..6.7 can e calculated and te epression for A can e written as follows: were ar A,.6.35 r [3r r 8K ] K For a non-truncated ogie-saped nose wen r, K.5,.6.37 te constrains in Eq..6.3 are satisfied for and Eq yields, or K K K

35 ocalized Interaction Models IMs 37 n fr Su-model u, a [ u u ] a, For te su-model u, a [ u u ] a,,.6.4 n fr in te case of a non-truncated ogie-saped nose wen te constrains gien y Eqs and.6.38 are satisfied, te integral in Eq..6. can e calculated and te result is gien y Eq..6.35, were ln K K K,.6.4 K and K Summary of two-term models Tale.6. Summary of two-term models BR aritrary ody of reolution, ON - ogie-nosed ody of reolution. Model numer p Impactor s sape Friction A A u u BR Eq..6.3 Eq..6.4 u a 3 au a a u 4 u a BR Eq..6. Eq..6.4 BR fr Eq..6. Eq..6.4 BR Eq..6. Eq..6.5 BR fr Eq..6. Eq..6.7 ON fr Eq..6. ON, r fr Eq..6. Eq..6.35, Eq Eq..6.35, Eq BR Eq..6.9 Eq..6. BR fr Eq..6. Eq..6. ON, r fr Eq..6.9 Eq..6.35, Eq..6.4

36 38 Hig-Speed Penetration Dynamics: Engineering Models and Metods For conenience te main formulas for A and A in Sections are summarized in Tale.6.. Caracteristics l and res for a finite widt sield and H for a semi-infinite sield are calculated in all cases using Eqs..6.7,.6.8 and.6., correspondingly. Blank in te column Friction indicates tat te model is presented for a general case,. fr.7 Aeraged Tree-Term Model et us consider an aeraged tree-term model witout friction, n fr u, a u a u a,..7. For semi-infinite sield, te relationsip D A A A.7. is alid instead Eq..6., were A and A are determined y Eqs..6. and.6.7 wile te formula for A reads: fr A a r d..7.3 Te aeraged resistance force acting at te flat part of te nose of a projectile, D, is as follows: flat D flat B B B, Bi r ai, i,,..7.4 In te case of truncated ogie-saped actors, integral in Eq..7.3 can e calculated fr and te epression for A can e written in te following form: were ar A,.7.5

37 ocalized Interaction Models IMs 39 r [8K K K K 6K sin K r r K.7.6 r is determined y Eq and inequalities in Eq..6.3 are satisfied. For a non-truncated ogie-saped nose wen Eq..7.6 yields: r, 4K, K.5,.7.7 K 4K 6K K K sin 4K 4K K..7.8 Equation of motion of te actor lies te following formula for te DOP: H mk, A, A,,.7.9 A were K, g, g, g d g g g..7. Integral in Eq..7. can e found analytically see also Zook, 977: ln g if g g K, g, g, g.7. ln if g

38 4 Hig-Speed Penetration Dynamics: Engineering Models and Metods were g g ln g g tan tan. g g g g, g g, 4 if if.7., g g g Oerslified Models IMs yield te dependencies for te instantaneous drag, D, s. te instantaneous elocity,, and in te case of te non-aeraged models, s. te instantaneous penetration dept,. Coefficients in tese relations depend upon te sape of a penetrator and can e calculated using te known formulas. It is assumed tat te model wic determines te local interaction etween a sield and a penetrator is known. Along wit te descried aoe approac, a different slified approac is used. In tis approac te local interaction etween a sield and a striker is not modeled, and a particular dependence, tat includes as a rule empirical coefficients, of te drag force acting at te striker s. and less common s., is assumed. Suc approac is useful for determining eperimental dependencies and analyzing general laws goerning penetration. Heimdal and Sculz, 986 studied te motion of an actor for an aritrary function D. A numer of inestigators ae proposed power law dependences for different media Mileiko and Sarkisyan, 98; Mileiko et al., 994; Forrestal et al., 984; Forrestal et al., 986. Various approaces to determine te drag force acting on te ody as a function of its elocity or elocity and penetration dept were considered y Stone, 994; Zook, 977; Bet, 946; Allen et al., 957a; Bernard, 978; Den, 979, 986, 987. A rief analysis of suc models and references to te early studies can e found in Goldsmit, 96 and Backman and Goldsmit, 978.