SIMULATION OF WAVE PROPAGATION IN AN HETEROGENEOUS ELASTIC ROD

Transcription

1 SIMUATION OF WAVE POPAGATION IN AN HETEOGENEOUS EASTIC OD ogéro M Saldanha da Gama Unversdade do Estado do o de Janero ua Sào Francsco Xaver 54, sala 5 A 559-9, o de Janero, Brasl e-mal: rsgama@domancombr Carlos Danel B G Barroso Unversdade do Estado do o de Janero ua Sào Francsco Xaver , o de Janero, Brasl Abstract A contnuous mathematcal model for descrbng the dynamcal response of a heterogeneous lnear-elastc rod, left n a nonequlbrum state s presented n ths work The problem s represented by a set of two hyperbolc partal dfferental equatons that, n general, does not admt contnuous solutons In addton, the rod s assumed to be composed by N dfferent materals, gvng rse to N- statonary dscontnutes n the stran feld The phenomenon s descrbed n the reference confguraton n a lnear elastcty context, gvng rse to N dfferent propagaton speeds The (generalzed) soluton presents shock waves, even for cases nvolvng contnuous ntal data Smulatons nvolvng boundary condtons (not usual for hyperbolc problems) are consdered n order to provde a way for descrbng the dynamcs of fnte rods Keywords wave propagaton, heterogeneous elastc rods, emann problem Introducton Some technques for studyng the mechancal response of materals are based on the propagaton of stress waves Ths provdes nformaton about the way solds behave when the forces actng on them are no longer n statc equlbrum In ths paper we shall dscuss the dynamcal response of a pecewse homogeneous (heterogeneous) elastc-lnear rod left n a nonequlbrum state In other words, our objectve s to obtan the stran, the stress and the velocty felds startng from a gven ntal data and subjected (or not) to some boundary condton Mathematcally, ths one-dmensonal phenomenon s represented, n the reference confguraton, by a lnear hyperbolc system of partal dfferental equatons whose egenvalues depend on the poston In fact, these egenvalues are pecewse constant, snce the rod s assumed to be pecewse homogeneous (composed by N dfferent materals) As t wll be shown later, ths hyperbolc system wll not admt (n general) a soluton n the Classcal sense So, t wll be necessary to work wth the jump condtons assocated wth the set of equatons n order to deal wth dscontnuous functons (generalzed solutons of the problem) A composton of these dscontnuous functons wll gve rse to the complete soluton of the problem Governng and consttutve equatons From Contnuum Mechancs (Bllngton and Tate, 98) we have that, for the one-dmensonal phenomenon under study here, ε = σ ρ = () where ρ s the mass densty n the reference confguraton (pecewse constant here), v s the velocty, σ s the normal component of the Pola-Krchhoff tensor and ε s the stran The frst equaton above represents a geometrcal compatblty whle, the second one, represents the lnear momentum balance n the reference confguraton In both the equatons t represents the tme whle X represents the poston (n the reference confguraton) The stran feld ε s defned as x ε = ()

2 n whch x represents the poston n the current confguraton (spatal poston) In ths work t wll be assumed (lnear elastcty) that the Pola-Krchhoff normal stress σ s a pecewse lnear functon of the stran ε In other words, σ = cε, for X < X < X (3) where cs a postve constant The mass densty ρ wll be assumed constant n X < X < X ρ = ρ = constant, for X < X < X (4) 3 The assocated emann problem and ts generalzed soluton et us consder now the followng ntal data problem (named emann problem) ε = σ ρ = ( ε, v) = ( ε, v ) for X < X ( ε, v) = ( ε, v ) for X > X (5) n whch ε, ε, v and v are known constants The soluton of Eq (5) conssts of connectng the left state ( ε, v) to the rght state ( ε, v) by means of rarefactons (contnuous solutons) and/or shocks (dscontnutes satsfyng the entropy condtons) Two states are connected by a rarefacton f, and only f, between these states, the correspondng egenvalue s an ncreasng functon of the rato ( X X)/ t (Smoller, 983; ax, 97 and John, 974) The egenvalues assocated to the hyperbolc system are gven, n crescent order, by / / / / σ ' c σ ' c λ = = andλ = = for X < X < X ρ ρ ρ ρ (6) So, f we assume that X and that X, the soluton of Eq (5) wll depend only on the rato ( X X )/ t and, snce the egenvalues are constant, the soluton (generalzed) wll be dscontnuous In other words, the left state ( ε, v ) wll be connected to an ntermedate state ( ε, v ) by a dscontnuty (called -shock or back shock) whle the rght state ( ε, v ) wll be connected to ths ntermedate state by another dscontnuty (called -shock or front shock) Snce λ < < λ we have, from the entropy (Keyftz and Kranzer, 978; Callen, 96) condtons that the shock speed s (back shock speed) s always negatve whle s (front shock speed) s always postve The ntermedate state ( ε, v ) s obtaned from the ankne-hugonot jump condtons gven, for ths hyperbolc system, by (Slattery, 97) [ v] [ ε] [ σ] [ ρv] = = s (7) where s denotes the shock speed whle the brackets denote the jump From the equatons represented n Eq (7) we have that v v σ σ = = s ε ε ρ( ) v v v v σ σ = = s ε ε ρ( v ) v (8)

3 Ths set of equatons gves rse to the followng (generalzed) soluton ε ε for < ( X X)/ t < s ε for ( )/ ε for s < ( X X)/ t < v for < ( X X)/ t < s for ( )/ v for s < ( X X)/ t < = s < X X t < s v = v s < X X t < s (9) where ε v s s v v ε ε = c / ρ v v ε ε = = = c ρ c ρ c / ρ () It s to be notced that the -shock and the -shock are contact dscontnutes, snce s = λ and s = λ So, there s no entropy generaton assocated wth these shocks (ax, 97) Fgure () shows the above soluton n the plane X t Fgure () shows the stran and the velocty as functons of the rato ( X X)/ t Now, let us suppose that X, X = X and X Ths case represents an nfnte rod composed by two homogeneous parts In such a case the soluton wll present a statonary shock at the (reference) poston X The (generalzed) soluton of Eq (5) wll depend only on the rato ( X X)/ t too Nevertheless the -shock (left) and the -shock (rght) speeds have dfferent absolute values Fgure The soluton of the emann problem (Eq (5)) represented n the plane X and X, It s to be notced that s = s t for the case n whch X

4 STAIN ε s s (X-X )/t VEOCITY v s s (X-X )/t Fgure The soluton of the emann problem (Eq (5)), represented as a functon of ( X X)/ t, for a case n whch v = v and ε > ε Snce σ / ε s a constant for < X <, the stress behaves lke the stran Snce there exsts an statonary shock at X = X = X, we conclude, from the jump condtons across ths shock, that velocty and stress do not jump at ths pont So, only the stran ε jumps across the statonary shock and, snce [ σ ] =, we have that (Keyftz and Kranzer, 978) cε = c ε wth ε = lm ε and ε = lm ε () X X, X< X X X, X> X In ths case, the jump condtons gve rse to the followng set of equatons v v σ σ = = s ε ε ρ ( ) v v v v σ σ = = s ε ε ρ ( ) v v () and the complete soluton s gven by ε for < ( X X )/ t < s ε for s < ( X X )/ t < ε = ε for < ( X X )/ t < s ε for s < ( X X )/ t < v for < ( X X )/ t < s v= v for s < ( X X )/ t < s v for s < ( X X )/ t < (3) where

5 ε ε c ( v v ) c ( ε c / ρ ε c / ρ ) = c c / ρ c c / ρ c c / ρ c c / ρ c ( ) ( / / ) v v c ε c ρ ε c ρ = c c / ρ c c / ρ c c / ρ c c / ρ c c v cρ v c ε ε ρ v = cρ c ρ c ρ c ρ s s = = c ρ c ρ (4) It s remarkable that Eq () conssts of a partcular case of Eq (4), obtaned when c = c and ρ = ρ In ths case there s no statonary jump at X and ε = ε = ε even for ( X X )/ t = Fgure (3) presents the soluton, obtaned wth the ad of Eq (3), n the plane X t, for a case n whch X, X = X and X The representaton n the plane X t does not depend on the ntal data ( ε, v) and ( ε, v ), once the propagaton speeds do not depend on the states ( ε, v) Fgure 3 The soluton of the emann problem (Eq (5)) represented n the plane X X = X and X t for a case n whch X, 4 The assocated emann problem for cases n whch X X for any Now we shall consder the cases n whch the nterface between two dfferent materals (placed at any poston X ) does not concde wth the jump n the ntal data (placed at X ) Here, the soluton of the emann problem wll no longer depend on ( X X)/ t In fact, the soluton wll depend on ( X X)/ tonly untl a shock (front or back) reaches a statonary shock At ths pont, a new emann problem arses, centered at the poston of the statonary shock, havng as ntal tme the tme n whch the shock nteracton has been So, let us consder the problem defned by Eq (5) assumng that X < X < X Snce X s dfferent from any X, we have the -shock and the -shock centered at X and a statonary shock at each X The soluton wll depend on the rato ( X X)/ t whle there s no shock nteracton between shocks When the -shock reaches the statonary shock at X = X, the soluton changes ts behavor In any case, the ntermedate state becomes a new ntal data (whch respect to the tme n whch the shock nteracton occurred) gvng rse to a new emann problem The soluton

6 of ths new emann problem has always the same structure of Eq (3) and, therefore, enable us to solve Eq (5) for any pecewse ntal data In order to llustrate the soluton procedure, let us suppose a partcular case n whch (nfnte rod composed by three dfferent homogeneous parts) X, X = 7, X =, X3 = 3 and X4 wth c/ ρ = 3 c / ρ and c3/ ρ3 = 3 c / ρ Startng from the ntal data ( ε, v) = ( ε, v ) for X < X ( ε, v) = ( ε, v ) for X > X (5) we have the ntermedate state (, v ) ε (called here ( ε, v )) gven by Eq () At the pont a the front shock (wth speed c / ρ ) reaches the statonary shock placed at X 3 So, a new emann problem, centered at a arses, havng as ts left state ( ε, v) and at ts rght state ( ε, v) For ths emann problem, the ntermedate states ( ε, v ) and ( ε, v ) are gven by Eq (4) epeatng ths procedure, we construct Fg (4) below n whch the soluton s presented n the plane X t Fgure 4 The soluton of the emann problem represented n the plane X X =, X3 = 3 and X4 wth c/ ρ = 3 c / ρ and c3/ ρ3 = 3 c / ρ t for X, X = 7,

7 Table () shows the relatonshp between Eq (4) and each one of the states presented n Fg (4) Table The states,, 3, 4, 5, 6, 7, 8, 9,,, and 3 and ther relaton wth Eq (4) emann problem centered at EFT STATE INTEMEDIATE STATE - Eq(4) INTEMEDIATE STATE Eq(4) a 3 b 5 4 c d e f 7 9 g 3 IGHT STATE Snce all the propagaton speeds are prevously known and constant, t s very easy to determne the tme assocated wth each shock nteracton (a, b, c, d, e, f and g) For nstance, the pont a s reached when t c = 3 ρ / The pont e s reached when t 7 ρ/ c 5 Fnte rods problems nvolvng boundary condtons = The pont c s reached when t = ρ / c The tools presented up to ths pont are suffcent for descrbng wave propagaton n rods n whch an edge s assumed to be fxed ( v = ) or free ( σ = and ε = ) Such boundary condtons are automatcally satsfed by means of the ntroducton of artfcal states beyond the actual rod In other words, for mposng a fxed edge at X, t s suffcent to assume the exstence of a rod at the left ( X < X), wth a state such that v = For mposng a fxed edge at X N, t s suffcent to assume the exstence of a rod at the rght ( XN > X ), wth a state such that v = On the other hand, for mposng a free edge boundary condton, t s suffcent to consder the artfcal rod wth a state such that ε = Ths can be done n an easy way too The choce of the state n the artfcal extenson of the rod s done based on Eq (), once the materal of the artfcal extenson can be the same of the actual rod For nstance, let us consder a problem n whch X =, X = 4, X3 =, X = 7 and c/ ρ = c / ρ Two dstnct stuatons wll be smulated: A rod fxed at both edges (wth ρ = ρ ); A rod fxed at the left edge, wth the rght edge free (wth ρ = ρ) Table () and Fg(5) present some results assocated wth the above cases The soluton procedure s based on the employment of the Eq (4) after each shock nteracton Whle Tab () presents quanttatve results for specfed left and rght states as well as gven boundary condtons, Fg (5) presents results, whch do not depend on the ntal data or boundary condton Ths s a feature of ths knd of hyperbolc system n whch the egenvalues (speeds of propagaton) do not depend on the states In Fg (5) the red dots (at the left) are assocated wth the center of emann problems constructed n order to satsfy the boundary condton at the left The blue dots play the same role, at the rght The black dots ndcate the shock nteracton wthn a same materal whle, the green dots, ndcate the nteracton between a (front and back) shock and a statonary one

8 Table Some results obtaned for cases and, assumng the rod at rest for t = Here w= v ρ / c case ε ε ε w ε w ε3 w3 ε4 w4 ε5 w5 ε6 w Fnal emarks The tools presented n ths paper allow, n a very smple way, the smulaton of any ntal data problem (even wth boundary condtons) nvolvng lnear-elastc rods For cases n whch the mass densty and the rato stress/stran are not pecewse constant functons (that were not consdered here) the prevously presented results are avalable too, provded these felds be approxmated by pecewse constant ones Ths can be done, for nstance, choosng, between X and X, the mean value of the mass densty and the mean value of the rato stress/stran So, our orgnal problem defned by ε = σ ~ ~ ρ =, wth σ = cx ( ) ε and ρ = ρ( X) ( ε, v) = ( ε, v ) for X < X ( ε, v) = ( ε, v ) for X > X and boundarycondtons (6) s replaced by the followng one ε = σ ρ =, wth σ = cε and ρ = ρ for X < X < X ( ε, v) = ( ε, v ) for X < X ( ε, v) = ( ε, v ) for X > X and boundarycondtons (7) where X ~ ~ X c = cxdx ( ) and ρ = ρ( X) dx X X X X X (8) X

9 Fgure 5 The soluton (for cases and ) represented n the plane X t, for a fnte rod wth length

10 7 Acknowledgements The author M Saldanha da Gama gratefully acknowledges the fnancal support provded by the agency CNPq through grant 346/84 8 eferences Bllngton, E W and Tate, A, 98 The Physcs of Deformaton and Flow, McGraw-Hll, New York, USA, 69p Callen, H B, 96 Thermodynamcs John Wley & Sons, New York, 376p John, F, 974 Formaton of Sngulartes n Onedmensonal Nonlnear Wave Propagaton Comm Pure Appl Math, vol7, pp Keyftz, B and Kranzer,H,978 Exstence and Unqueness of Entropy Solutons to the emann Problem for Hyperbolc Systems of Two Nonlnear Conservaton aws, J Dff Eqns, vol 7, pp ax, P, 97 Shock Waves and Entropy, Contrbutons to Nonlnear Functonal Analyss Academc Press, New York, pp Slattery, J C, 97 Momentum, Energy and Mass Transfer n Contnua McGraw-Hll Kogakusha, Tokyo, Japan, 679p Smoller, J, 983 Shock Waves and eacton-dffuson Equatons, Sprnger-Verlag, New York, USA, 58p 9 Copyrght Notce The authors are the only responsble for the prnted materal ncluded n hs paper