Simulation of wave propagation in a piecewise homogeneous elastic rod

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1 INTENATIONA JOUNA OF STUCTUA CHANGES IN SOIDS Mechancs and Applcatons Volume 3, Number,, pp Smulaton of ave propagaton n a pecese homogeneous elastc rod Mara aura Martns-Costa, ogéro M. Saldanha da Gama aboratory of Theoretcal and Appled Mechancs, Mechancal Engneerng Program, Unversdade Federal Flumnense, ua Passo da Pátra 56, 4-4, Brazl Mechancal Engneerng Department, Unversdade do Estado do o de Janeroua São Francsco Xaver, 54, 55-3, Brazl Abstract Ths artcle deals th modelng and smulaton of the ave propagaton phenomenon n a heterogeneous lnear elastc rod treated as pecese homogeneous. Employng a reference confguraton approach the pecese homogeneous elastc rod s mathematcally descrbed by a lnear hyperbolc system of partal dfferental equatons. Its generalzed soluton s obtaned by connectng ntermedate states by contact shocks thus allong analytcal solutons for some ntal value problems even hen boundary condtons are consdered. Keyords: emann problem, lnear elastcty, pecese homogeneous materal.. Introducton Stress aves propagaton n solds s an mportant tool for studyng the mechancal response of materals. Wave propagaton provdes nformaton about the ay solds behave hen the forces actng on them are no longer n statc equlbrum. Some phenomena such as scatterng, dsperson and attenuaton, hch strongly nfluence ave propagaton, affectng the thermomechancal response of the materals, may be attrbuted to nonlneartes lke materal heterogenety, ave characterstcs and loadng condtons (Chen et al., 4). Zhuang et al. (3) have performed a systematc expermental nvestgaton of the nfluence of nterface scatterng on fnte-ampltude shock aves hch affects shock aves dsspaton and dsperson by consderng shock ave propagaton n perodcally layered compostes and have observed that these materals can support steady structured shock aves. These authors noted that ave propagaton through layered materals composed by sotropc layers allos nvestgatng the effect of heterogeneous materals under shock loadng. Chen et al. (4) presented an approxmate analytcal soluton to one-dmensonal ave propagaton n layered heterogeneous materals subjected to hgh velocty plate mpact loadng condtons, based on Floquet s theory, usng nether a snusodal ave loadng nor unt step loadng, hch, accordng to these authors, have been used n prevous analytcal orks. Berezovsky et al. (6) consdered a pecese homogeneous meda, accountng for the ave dstorton of nonlnear elastc aves. The authors performed numercal smulatons of one-dmensonal ave propagaton n layered nonlnear heterogeneous solds, employng a fnte volume approxmaton for hyperbolc problems, n hch the emann problem s solved at each nterface beteen dscrete elements. They have consdered fnte ampltude nonlnear ave propagaton to study scatterng, dsperson and attenuaton of shock aves, employng a ave propagaton algorthm accountng for thermodynamc consstency, ntroduced by Berezovsky and Maugn (), for the to-dmensonal problem n meda th rapdly varyng propertes. The dsperson effects due to the mcrostructure n nonlnear deformaton aves have been consdered by Engelbrecht et al. (7). Ths artcle presents a dscusson concernng the dynamcal response of a specfc class of heterogeneous lnear elastc rods hch s pecese homogeneous. Ths elastc rod s left n a nonequlbrum state. Startng from a gven Emal: laura@mec.uff.br ; rsgama@terra.com.br 47

2 48 Internatonal Journal Of Structural Changes In Solds, 3(),, ntal data, the technque proposed n ths paper allos obtanng analytcal solutons for the stran, the stress and the velocty felds. Eventually ths ntal value problem may be subjected to some boundary condtons. The pecese homogeneous lnear elastc rod consdered n the present ork gves rse to a one-dmensonal problem n the reference confguraton hch s mathematcally descrbed by a lnear hyperbolc system of partal dfferental equatons th egenvalues dependng on the poston. In fact, these egenvalues are pecese constant, snce the rod s assumed to be pecese homogeneous beng composed by N dfferent materals. Several problems n Mechancs are represented by hyperbolc systems, hch permt very realstc descrptons, snce the propagaton of any quantty or nformaton n real natural phenomena s characterzed by a fnte speed. Hoever they may not admt a regular soluton, requrng a larger space of admssble solutons. The hyperbolc system descrbng the pecese homogeneous lnear elastc rod consdered n ths artcle does not admt n general a soluton n the classcal sense, requrng an enlargement of the space of admssble solutons allong orkng th the jump condtons assocated th the set of equatons n order to deal th dscontnuous functons. The generalzed solutons of the problem are obtaned by connectng ntermedate states by contact shocks (ax, 97; Smoller, 983). A contact shock may be veed as the lmtng case of a rarefacton n hch the rarefacton fan s reduced to a sngle lne; namely a dscontnuty th assocated egenvalue correspondng exactly to the shock speed. Unlke ordnary shocks, the contact shock s reversble, thout any assocated entropy generaton (Saldanha da Gama, 99). The reference confguraton approach employed to descrbe the pecese homogeneous rod gves rse to a problem characterzed by deformaton jumps at any to dstnct materals nterface, represented by statonary shocks n other ords, the nterface poston s not modfed. An adequate composton of these dscontnuous functons gves rse to the complete analytcal soluton of the ntal value problem even th boundary condtons. The study of elastc ave equatons accountng for ther scatterng n non-unform meda s mportant for acoustc problems. Approxmate methods can lead to convenent smplfcatons, for eak nhomogenetes. (Tenenbaum and Zndeluk, 99a) present an exact algebrac soluton for the scatterng of acoustc aves n onedmensonal elastc meda, consderng the ave propagaton pattern n layered meda, vald for strong nhomogenetes. In a subsequent artcle (Tenenbaum and Zndeluk, 99b) the same authors propose a sequental algorthm th arbtrary nlet pulse to solve nverse acoustc scatterng problems for plane aves n elastc nonhomogeneous meda, consstng of a mathematcal nverson of the prevous drect problem. It s orth notng that the methodology presented n ths ork could be extended to nonlnear homogeneous problems by employng the approxmate emann solver developed by oe (997), consderng approxmate solutons hch are exact solutons to an approxmate problem. Essentally, ths scheme allos each emann problem hch s to be solved for each to consecutve steps, n order to mplement a dfference scheme lke Glmm s one (see (Gama and Martns-Costa, 9) and references theren) to approxmate an orgnally homogeneous nonlnear system by a heterogeneous lnear one. The procedure for mplementng ths scheme s brefly descrbed by an example n n hch a nonlnear elastc rod s consdered.. Mechancal model The one-dmensonal phenomenon consdered s ths ork s mathematcally descrbed, from a contnuum mechancs vepont (Bllngton and Tate, 98) as v t X () v t X here represents the mass densty n the reference confguraton (pecese constant here), v the velocty, s the normal component of the Pola-Krchhoff tensor and s the stran. The frst equaton above represents a geometrcal compatblty hle the second one represents the lnear momentum balance n the reference confguraton. In both the equatons t represents the tme hle X represents the poston (n the reference confguraton). The stran feld s defned as x () X n hch x represents the poston n the current confguraton (spatal poston). Snce a lnear elastcty hypothess s assumed n ths ork the Pola-Krchhoff normal stress s a pecese lnear functon of the stran. In other ords,

3 Martns-Costa and Saldanha da Gama / Smulaton of ave propagaton n elastc rods 49 c,for X X X (3) here c s a postve constant. The mass densty s assumed constant n X X X. constant, for X X X (4) 3. The assocated emann problem and ts generalzed soluton Assocated th equatons ()-(4) there s an ntal value problem named assocated emann problem, gven by v t X (, v) (, v) for X X (5) v (, v) (, v) for X X t X n hch,, v and v are knon constants. The soluton of the hyperbolc system represented by Eq. (5) s reached by connectng the left state (, v ) to the rght state (, v ) by means of rarefactons (contnuous solutons) and/or shocks (dscontnutes satsfyng the entropy condtons). To states are connected by a rarefacton f, and only f, beteen these states, the correspondng egenvalue s an ncreasng functon of the rato ( X X)/ t (Smoller, 983; ax, 97; John, 974). The egenvalues assocated to the hyperbolc system descrbed by Eq. (5) are gven, n ncreasng order, by / / / / ' c ' c and for X X X (6) n hch ' represents the frst dervatve of the normal component of the Pola-Krchhoff tensor th respect to the deformaton. When X and X, the soluton of Eq. (5) depends on the rato ( X X)/ t only and, snce the egenvalues are constant, the generalzed soluton s dscontnuous. In other ords, the left state (, v ) s connected to an ntermedate state (, v ) by a dscontnuty (called -shock or back shock) hle the rght state (, v ) s connected to an ntermedate state ( *, v * ) by another dscontnuty (called -shock or front shock). Snce, the entropy condtons (Keyftz and Kranzer 978; Callen 96) ensure that the shock speed s (back shock speed) s alays negatve hle s (front shock speed) s alays postve. The ntermedate state (, v ) s obtaned from the ankne-hugonot jump condtons, hch for the hyperbolc system (5) are gven by (Smoller 983) v s (7) v here s denotes the shock speed and the brackets denote the jump. Equaton (7) allos concludng that v v s ( v v ) (8) v v s ( v v) The set of equatons (5)-(8) gves rse to the follong (generalzed) soluton for the stran and the velocty for ( X X)/ t s v for ( X X)/ t s * * for s ( X X) / t s v v for s ( X X) / t s (9) for s ( X X) / t v for s ( X X) / t n hch

4 5 Internatonal Journal Of Structural Changes In Solds, 3(),, * v v * v v, v c /, c / () c c s and s It s mportant to note that, snce s and s, both the -shock and the -shock are called contact dscontnutes and no entropy generaton s assocated th these shocks (ax 97). No supposng that X, X X and X, an nfnte rod composed by to homogeneous parts s represented. In such a case the soluton presents a statonary shock at the (reference) poston X and the generalzed soluton of Eq. (5) also depends only on the rato ( X X)/ t. Nevertheless the -shock (left) and the -shock (rght) speeds have dfferent absolute values. Snce there exsts a statonary shock at X X X, t may be concluded, from the jump condtons across ths shock, that velocty and stress do not jump at ths pont. So, only the stran jumps across the statonary shock and, snce, t comes that (Keyftz and Kranzer, 978) c c th lm and lm () XX, XX XX, XX In ths case, the jump condtons gve rse to the follong set of equatons v v s ( v v ) v v s ( v v) and the complete soluton s gven by for s v for * s for s * v v for s * s for s v for s for s here ( X X )/ t and () (3) c ( v v ) * c c / c c / c c / c c / * c ( c / c / ) * c ( ) ( / / ) v v c c c c c / c c / c c / c c / v c c c c v c v c c c s c / and s c / (4) emark: Equaton () conssts of a partcular case of Eq. (4), obtaned hen c c and. In ths case * there s no statonary jump at X and even for ( X X )/ t. Fgure () presents the soluton, obtaned by employng Eq. (3) n the plane X t, for a case n hch X, X X and X. It s orth notng that the representaton n the plane X t presented n Fgure does not depend on the ntal data (, v ) and (, v ), once the propagaton speeds do not depend on the states (, v).

5 Martns-Costa and Saldanha da Gama / Smulaton of ave propagaton n elastc rods 5 Fgure : emann problem (Eq. (5)) n the plane X t for X, X X and. 4. The assocated emann problem hen X X for any Ths secton studes problems n hch the nterface beteen to dfferent materals (placed at any poston X ) s not concdent th the jump n the ntal data (placed at X ). In ths case the soluton of the emann problem no longer depends on ( X X)/ t. In fact, the soluton depends on ( X X)/ t only untl a shock (ether front or back) reaches a statonary shock, characterzng a shock nteracton. At ths pont, a ne emann problem arses, centered at the poston of the statonary shock, havng as ntal tme the tme n hch the shock nteracton has taken place. No, consderng the problem defned by Eq. (5) and assumng X X X, snce X s dfferent from any X, both the -shock and the -shock are centered at X, hle a statonary shock s present at each X. It s mportant to note that hle there s no shock nteracton beteen shocks comng from dstnct ponts, the soluton depends on the rato ( X X)/ t only. When the -shock reaches the statonary shock at X X, the soluton behavour s changed. In any case, the ntermedate state becomes ne ntal data (th respect to the tme n hch the shock nteracton occurred) gvng rse to a ne emann problem. The soluton of ths ne emann problem has alays the same structure of Eq. (3) enablng Eq. (5) to be solved for any pecese constant ntal data. In order to llustrate the soluton procedure, a partcular case s no consdered: an nfnte rod composed by three X, X.7, X, X3.3 X th dfferent homogeneous parts such that: and 4 c/ 3 c/ and c3/ 3.3 c /. Startng from the ntal data (, v) (, v), for X X and (, v) (, ) v, for X X, the ntermedate state (, v ) hch ll be treated as a left state (, v ) s gven by Eq. (). At the pont a the front shock (th speed c / ) reaches the statonary shock placed at X, gvng rse to a ne emann problem, centered at a, 3 characterzed by the left state (, v) and the rght state (, v ). The ntermedate states (, v ) and (, v ) for ths ne emann problem, are gven by Eq. (4). epeatng ths procedure, a soluton n the plane X t may be constructed, as depcted n Fgure. Table () relates Eq. (4) and each of the states presented n Fgure. Snce all the propagaton speeds are prevously knon and constant, the tme assocated th each shock nteracton (a, b, c, d, e, f and g) s easly determned. For nstance, pont a s reached hen t.3 / c ; pont e hen t.7 / c and pont c hen t / c. emann problem centered at Table : States to 3 and ther relaton th Eq. (4). X IGHT STATE EFT STATE INTEMEDIATE STATE Eq.(4) INTEMEDIATE STATE Eq.(4) a 3 b 5 4 c d e f 7 9 g 3

6 5 Internatonal Journal Of Structural Changes In Solds, 3(),, Fgure. emann problem soluton for X, X.7, X, X3.3 and X 4 ; th c/ 3 c / and c3/ 3.3 c /. 5. Fnte rods problems nvolvng boundary condtons The tools presented up to ths pont are suffcent for descrbng ave propagaton n rods n hch one edge s assumed to be fxed ( v ) and the other s ether fxed ( v ) or free ( and ). Such boundary condtons are automatcally satsfed by ntroducng artfcal states beyond the actual rod. In other ords, for mposng a fxed * edge at X, t suffces to assume the exstence of a rod at the left ( X X ), th a state such that v hle a fxed edge at X N s mposed by assumng the exstence of a rod at the rght-sde ( X * N X ), th a state such that v. On the other hand, for mposng a free edge boundary condton, t suffces to consder an artfcal rod th a state such * that. Ths can be done n an easy ay too. The choce of the state n the artfcal extenson of the rod s done based on Eq. (), assumng the same materal for both the artfcal extenson and the actual rod. For nstance, n ths ork a problem n hch X, X 4, X 3, X 7 and c/ c / s consdered for to dstnct smulated stuatons namely stuaton () representng a rod fxed at both edges and stuaton () representng a rod fxed at the left edge, th the rght edge free. Some selected results assocated th the cases defned above as () and () are presented n Table (), assumng the rod at rest for t and defnng v / c. The soluton s reached by employng Eq. (4) after each shock nteracton. The quanttatve results are presented for specfed left and rght states as ell as gven boundary condtons. Table : Some results for cases () and (). case

7 Martns-Costa and Saldanha da Gama / Smulaton of ave propagaton n elastc rods An applcaton to nonlnear elastcty In ths secton a nonlnear elastc rod s consdered. In ths case, the Pola-Krchhoff normal stress may assume, for nstance, the follong nonlnear consttutve equaton f, for X X X (5) A convenent redefnton s no consdered for the constant c, before each advance n tme c f, here (6) X X X X gvng rse to the follong approxmaton for the stress c,for X X X (7) It s mportant to note that, s ths case, the problem les thn the range of the procedure proposed n ths ork, beng reduced to a pecese lnear functon of the stran. The assocated emann problem, gven by equaton (5) s no obtaned from equatons (), (), (7) and (4) and all the prevously descrbed steps after equaton (5) reman unchanged. 7. Fnal emarks Although ths artcle presents a dscusson concernng pecese homogeneous lnear elastc rods, the results can be extended to any lnear heterogeneous rod. Ths extenson s performed by approxmatng the heterogeneous rod by a pecese homogeneous one. Ths latter could be composed by any number of dfferent materals, for nstance,, or materals could be consdered, accordng to the requred accuracy. Also, takng advantage of the scheme proposed by oe (997), hch approxmates nonlnear homogeneous problems by lnear heterogeneous ones, t could be drectly extended to pecese nonlnear elastc rods, as brefly stated n secton 6. In addton, Glmm s scheme allos buldng an approxmaton for hyperbolc problems subjected to any arbtrary ntal data. It suffces to approxmate the arbtrary ntal condton by pecese constant ntal data. In the sequence, a emann problem an ntal value problem characterzed by a step functon ntal condton s to be solved for each to consecutve steps. The man dea behnd the method s to approprately gather the soluton of as many emann problems as desred to successvely march from a gven tme nstant t tn to the successve tme nstantt n t n t. Acknoledgements The authors gratefully acknoledge the fnancal support provded by the Brazlan agences CNPq and FAPEJ. eferences Berezovsk, A., Berezovsk, M. and Engelbrecht, J., 6 Numercal smulaton of nonlnear elastc ave propagaton n pecese homogeneous meda, Materals Sc. Engng. A, 48, Berezovsk, A., and Maugn, G. A., Smulaton of Thermoelastc ave propagaton by means of a composte ave-propagaton algorthm, J. Comput. Physcs, 68, Bllngton, E. W. and Tate, A., 98 The Physcs of Deformaton and Flo, McGra-Hll. Callen, H. B., 96 Thermodynamcs, John Wley. Chen, X. Chandra, N. and ajendran, A. M., 4 Analytcal soluton to plate mpact problem of layered heterogeneous materal systems, Int. J. Solds Structures, 4, Engelbrecht, J., Berezovsk, A. and Salupere, A., 7 Nonlnear aves n solds and dsperson, Wave Moton, 44, John, F., 974 Formaton of sngulartes n onedmensonal nonlnear ave propagaton, Comm. Pure Appl. Math., 7, Keyftz, B. and Kranzer,H., 978 Exstence and unqueness of entropy solutons to the emann problem for hyperbolc systems of to nonlnear conservaton las, J. Dff. Eqns., 7, ax, P., 97 Shock aves and entropy. In Contrbutons to Nonlnear Functonal Analyss, pp , Academc Press. oe, P.., 997 Approxmate emann solvers, parameter vectors and dfference schemes, J. Comput. Phys, 35, 5-58.

8 54 Internatonal Journal Of Structural Changes In Solds, 3(),, Saldanha da Gama. M., Martns-Costa M.., 8 An alternatve procedure for approxmatng hyperbolc systems of conservaton las. Nonlnear Anal.: eal World Appl., 9, 3-3. Smoller, J., 983 Shock Waves and eacton-dffuson Equatons, Sprnger-Verlag. Tenenbaum,. and Zndeluk, M., 99 An exact soluton for the one-dmensonal elastc ave equaton n layered meda, J. Acoust. Soc. Am., 9, Tenenbaum,. and Zndeluk, M., 99 A fast algorthm to solve the nverse scatterng problem n layered meda th an arbtrary nput, J. Acoust. Soc. Am., 9, Zhuang, S., avchandran, G. and Grady, D. E., 3 An expermental nvestgaton of shock ave propagaton n perodcally layered compostes, J. Mech. Phys. Solds, 5,