EQUATION CHAPTER 1 SECTION 1STRAIN IN A CONTINUOUS MEDIUM

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1 EQUTION HPTER SETION STRIN IN ONTINUOUS MEIUM ontent Introdcton One dmensonal stran Two-dmensonal stran Three-dmensonal stran ondtons for homogenety n two-dmensons n eample of deformaton of a lne Infntesmal deformatons Rotatonal Sesmology t s necessary to have three components of translaton s components of stran and three components of rotaton to flly characterze the change n the medm arond pont (Sryanto 6) bt n practce we do not consder the rotatonal component (k and Rchards. INTROUTION In the most general case we can consder contnos deformaton to be the reslt of a translaton rotaton and stretchng along the maor aes. In the classcal dervaton of sesmc-elastc stran we do not consder the rotatonal components. Sesmc deformaton reslts by sbectng a rock to stress wth a mnmal amont of temporary deformaton. ecase we are consderng the elastc wave feld we wll consder deformatons that too small to reach the yeld strength of the rock and change shape or sze permanently. It s helpfl to descrbe a general deformaton feld as a fncton of poston n three-dmensonal space: ( ) We choose to eamne the deformatonal behavor of a very small porton of rock say abot a few mneral grans at a pont P P ). ( We refer to deformaton at ths pont by (P ). y redcng the sze of or problem to short dstances we are less lkely to err n appromatng a general fncton of deformaton by one that s

2 only lnear. Ths s analogos to epandng a fncton sng a Taylor seres epanson bt only sng the frst two terms of the epanson. One-dmensonal stran In the one-dmensonal case deformaton ( ) can be calclated n the vcnty of the orgn O (where.e. Taylor Seres) as a fncton of the dstance away from the orgn: ( O) ( O) d d ( ) ( O) ( O) O d! d Sbstttng for a specfc vale of a we have ( a) ( ) f we only consder the frst term of the epanson. ( a ) d d If the hgher-order terms are assmed nsgnfcant the deformaton can be descrbed as a smple lnear fncton. The deformaton at any pont along the as s a fncton of a small deformaton at the orgn pls an addtonal vale proportonal to dstance from the orgn.... rosly there s a connecton between homogeneos lnear stran (above) and the epanson of the nverse. If we assme that the nverse s epandng at the same throghot (homogeneos) and that t s proceedng at a constant rate ths s eqvalent to sayng that stran s homogeneos. In a - sense whle the rate of deformaton s homogeneos the accmlated effect of the stran as seen from the orgn s greater at greater dstances from the observer where the cmlatve velocty of the epanson s lnear wth offset between the observer and the target (sensor n or case). The constant rate of lnear deformaton of the nverse s known as the Hbble constant. The Nobel Prze n physcs was awarded to those (Sal Perlmtter ran P. Schmdt dam G. Ress) who dscovered that the rate of epanson of the nverse s actally ncreasng and not constant.

3 ( Two-dmensonal stran Two-dmensonal case (; )

4 Three-dmensonal stran Note that I wll ntentonally se the symbol n order to remark on ts dfference f only temporarly from the gradent of a scalar feld that we have already seen. Ths symbol does not represent the gradent of a scalar feld bt the gradent tensor of the dsplacement vector feld. ompare the smple gradent of a scalar feld: () ˆ ˆ ˆ to the gradent tensor of the dsplacement feld: ( P) ( P) ( P) ( P) ˆ ˆ ˆ ( P ) ( P ) ( P) ( P) ˆ ( P ) ( P ) ( P ) ( P) ˆ ( P ) ( P ) ( P ) ( P) ˆ where ll the components of the dsplacement contrbte to the gradent of the deformaton where ( P ) means the second component of the dsplacement vector for pont P. Ths can be epressed as: ( P) ( P) ( P) ( P) ( P) ( P) ( P ) ( P) ( ) P ( P) ( P) ( P) ()

5 For the - stran shown n the last fgre above there s a Taylor seres epanson for fnctons wth three varables. In three-dmensons by analogy to the one-dmensonal condtons the deformaton fncton can be appromated by the frst two terms of the epanson n Taylor seres: P O P ( ) ( ) ( ) where s here defned as the dscrete dfferental of the fncton vald for the very small volmes of rock we consder bt whose lmt for a contnos medm s the dfferental d. Remember that (vector) descrbes the behavor of deformaton n all -dmensons at once. The total deformaton wll have contrbtons from all three drectons. We can eamne contrbtons to deformaton from each of these three drectons ( P) ( P) ( P) ( P) ˆ ˆ ˆ ( P) For eample term s read as the partal dfferental of the fncton wth respect to the ˆ bass vector drecton. The vertcal bar wth the sb-ndces sgnfes that drng

6 dfferentaton any of the other two components of n ether the ˆ or ˆ drectons are constant and dfferentate to zero. Note that becase s tself a fncton of a three-varable fncton of poston n the ˆ drecton can have terms that relate to the and vale of P. In ths fgre above we see that there are two types of three-dmensonal vectors sed to descrbe the deformaton. One vector relates poston of any pont n the ndeformed state P ( ) wth respect to the orgn n the ndeformed state (O ). nother vector relates the new poston of the pont P ( ) after the deformaton wth respect to the new orgn ( O ). The dfference n the deformaton between that whch occrs at the orgn (O ) and at P s the deformaton at P relatve to the new deformed coordnate system. The dscrete dfferental at P s a three-dmensonal vector wth the followng components: ( P ) ( ( P ) ( P ) ( P ) ) where ( P) ( ( P )( P )( P) ) s the general deformaton fncton that we appromate n a lnear fashon. fter deformaton O O and P P. Three-dmensonal case (; ) P s the vector that descrbes the deformaton wth respect to the new orgn O that s we have eperenced a coordnate transformaton. O s related to O throgh the dsplacement vector O O ( ) and P s related to P throgh the dsplacement vector: P O P () ( ) ( ) ( ) smlar to the Taylor epanson for the one-dmensonal case. s the spatal change of the dsplacement vector. In the most general case each component of the dsplacement vector s a fncton of three other components: f( )

7 If we want to epress the deformaton n the deformed co-ordnate system we can se () and the llstraton to determne that: P ( P) P P ( O) ( P) P P P P O ( ) ( ) We can epress the dsplacements as a gradent tensor where each dsplacement component has contrbtons from the other three dmensons (see bo): y assmng that P s a vector fncton (!) and n ndcal notaton From () and () we have that the deformed pont located wth respect to the orgn n the deformed system s: P ' P ' etc. Usng ndcal notaton we can rewrte the above epressons as: P Usng matr notaton we can also wrte ths as: ()

8 ondtons for homogenety n two-dmensons We can now revert to two-dmensons to convey the same concept bt wth less notaton than the three-dmensonal case; (4) eformaton s homogeneos throghot a space when the followng two condtons are met: () Parallel materal lnes n the ndeformed space reman parallel followng the deformaton and () f straght lnes n the ndeformed space reman as straght lnes followng the deformaton. These two condtons are eqvalent mathematcally to statng that (or n ndcal notaton) does not change throgh space and can be consdered to be of constant vale: say a b c d. That s we can wrte (4) as: d c b a (5) where a b c and d ctally the assmpton of homogeneos stran s scale-dependent. For a sffcently small scale we can always consder the deformaton to be homogeneos. eyond the scale of homogenety we mst start to deal wth nhomogeneos deformaton also known as non-homogeneos or heterogeneos deformaton. In order to deal wth the nhomogenety we wold epand the eqaton () to nclde more terms. However when dealng wth elastcty sch as the propagaton of waves normally we have that the deformatons are very small a few percent at most and

9 (or n ndcal notaton) <<. In geologcal applcatons these qanttes can be qte large and the vales for a b c d are not nfntesmal. n eample of deformaton of a lne We can now demonstrate the condton of homogenety by showng that a straght lne remans straght throghot the deformaton. onsder a lne pror to deformaton: m k (6) From (5) and (6) we have that a b a b( m k c d c d( m k y replacng the vale of n the second eqaton wth the vale of taken from the frst bk eqaton we have: a bm bk bk c dm dk a bm a bm bc bdm a bm c dm k k kd a bm a bm a bm a bm a bm c dm bc bd ad bd k (7) a bm a bm c dm ad bc k a bm a bm m k We have shown that the deformed straght lne s also another lne! ecase m does not depend on k two parallel lnes (same m bt a dfferent k ) wll reman parallel after the deformaton. They wll have the same slope m. We can also demonstrate that a crcle wll transform nto an ellpse and that one ellpse wll transform nto another ellpse. seres of ncremental homogeneos transformatons can be descrbed by a sngle fnte-deformaton ellpse. (T: ondtons for prncpal stran aes: The case of general transformaton: etenson and contracton along two perpendclar aes pls a rotaton) ) )

10 Let s consder the homogeneos deformaton: (8) or T where T cos sn sn a cos b b d (9) T s matr for deformaton va a rotaton transformaton s the matr of another deformaton and s the resltant general deformaton matr. Note that matr s symmetrc so that t can be dagonalzed. In ths deformaton two perpendclar lnes est that reman perpendclar after the deformaton n the same drecton as before the deformaton (egenvectors). long these drectons there s ether only etenson or only shortenng. These drectons are the prncpal drectons of deformaton along whch the deformaton s ether mamm or mnmm. We have that : acos bsn bcos d sn asn bcos bsn d cos s a reslt the rotaton angle s related to the general deformaton components by: tan (9 ) Once we arrve at a vale for we can mltply the left-hand sde of eqatons for and by sn and cos to obtan a cos sn b sn cos We also mltply the left hand sde of eqatons for and by to get: sn and cos respectvely

11 d sn cos (9 ) In decomposng from eqaton (9) the for components are shown to be trgonometrc fnctons. We cold now st fnd the egenvectors and show they are at rght angles to each other (See eercses)! However an alternatve approach s to show that f two perpendclar lnes est n the deformed system they are also the prncpal deformaton aes. Let s start by consderng two perpendclar lnes: m () m fter applcaton of the deformaton matr as gven by: () a b b d we have after sbsttton by () a b bm dm and a b b m d m whch solve as: md b mb d and respectvely. We want to know whether mb a ma b md b m and mb a mb d? m ma b These condtons can be shown as tre by sbstttng nto the dentty so that: m. m From the eqaton above we obtan that the deformed lnes are also at rght angles f the followng combnaton s met: ( d ) TOO: Graphcally ths can be epressed as: m b m a b ()

12 Qeston: Show that the homogeneos deformaton of a crcle: 4 y s an ellpse and draw the reslt. Infntesmal deformatons We now want to demonstrate that the stran theory we have looked at so far for fnte geologcal deformatons can also be appled to the very small deformatons that are sed n wave propagaton theory and sesmology. Form eqaton (9) we have; d b b a cos sn sn cos T where and ecase deformatons are very small recall that << and that hgher order dervatves wll also be neglgble. From eqaton (9 ) we have now: s well sn tan and cos (4a) tan

13 and to a frst order eqaton (9 ) becomes b a (4b) d For ( ) P P P n the two-dmensonal case we have: cos sn sn cos d b b a (5) and n the nfntesmal case sng appromatons () and (4): c (6) If we dsregard those second-order terms we get: Smlarly:

14 ( ) ( ) Then: y or f we se eqaton and 4(a): (7) The frst term on the rght hand of the eqaton corresponds to the nfntestmal nternal deformaton of the rock body whereas the second term corresponds to an nfntesmal clockwse rotaton by angle.

15 Fnally we arrve at the eqaton for the nfntesmal stran tensor e : In ndcal notaton we can wrte the nfntesmal stran tensor as: e or for the general case: ( ) e