Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown n Fg...a s a body B wth materal partcle. One dstngushes between ths body and the space n whch t resdes and through whch t travels. Shown n Fg...b s a certan pont n Eucldean pont space E. χ B E (a) (b) (c) Fgure..: (a) a materal partcle n a body, (b) a place n space, (c) a confguraton of the body By fng the materal partcles of the body to ponts n space, one has a confguraton of the body χ, Fg...c. A confguraton can be epressed as a mappng of the partcles to the pont, χ (..) A moton of the body s a famly of confguratons parametersed by tme t, χ t, (..) At any tme t, Eqn... gves the locaton n space of the materal partcle, Fg.... these partcles are not the dscrete mass partcles of Newtonan mechancs, rather they are very small portons of contnuous matter; the meanng of partcle s made precse n the defntons whch follow 0
Secton. t t Fgure..: a moton of materal The Reference and Current Confguratons Choose now some reference confguraton, Fg.... The moton can then be measured relatve to ths confguraton. The reference confguraton mght be the confguraton occuped by the materal at tme t 0, n whch case t s often called the ntal confguraton. For a sold, t mght be natural to choose a confguraton for whch the materal s stress-free, n whch case t s often called the undeformed confguraton. However, the choce of reference confguraton s completely arbtrary. Introduce a Cartesan coordnate system wth base vectors E for the reference confguraton. A materal partcle n the reference confguraton can then be assgned a unque poston vector E relatve to the orgn of the aes. The coordnates,, of the partcle are called materal coordnates (or Lagrangan coordnates or referental coordnates). Some tme later, say at tme t, the materal occupes a dfferent confguraton, whch wll be called the current confguraton (or deformed confguraton). Introduce a second Cartesan coordnate system wth base vectors e for the current confguraton, Fg.... In the current confguraton, the same partcle now occupes the locaton, whch can now also be assgned a poston vector e. The coordnates,, are called spatal coordnates (or Euleran coordnates). Each partcle thus has two sets of coordnates assocated wth t. The partcle s materal coordnates stay wth t throughout ts moton. The partcle s spatal coordnates change as t moves. 0
Secton. reference confguraton χ current confguraton t E e e E Fgure..: reference and current confguratons In practce, the materal and spatal aes are usually taken to be concdent so that the base vectors E and e are the same, as n Fg...4. Nevertheless, the use of dfferent base vectors E and e for the reference and current confguratons s useful even when the materal and spatal aes are concdent, snce t helps dstngush between quanttes assocated wth the reference confguraton and those assocated wth the spatal confguraton (see later). E,e,, E,e, Fgure..4: reference and current confguratons wth concdent aes In terms of the poston vectors, the moton.. can be epressed as a relatonshp between the materal and spatal coordnates, χ, t),,, t Materal descrpton (..) or the nverse relaton (, χ, t),,, t Spatal descrpton (..4) (, If one knows the materal coordnates of a partcle then ts poston n the current confguraton can be determned from... Alternatvely, f one focuses on some locaton n space, n the current confguraton, then the materal partcle occupyng that poston can be determned from..4. Ths s llustrated n the followng eample. 0
Secton. Eample (Etenson of a Bar) Consder the moton t t,, (..5) These equatons are of the form.. and say that the partcle that was orgnally at poston s now, at tme t, at poston. They represent a smple translaton and unaal etenson of materal as shown n Fg...5. Note that at t 0. χ confguraton at t 0 confguratons at t 0 Fgure..5: translaton and etenson of materal Relatons of the form..4 can be obtaned by nvertng..5: t t,, These equatons say that the partcle that s now, at tme t, at poston was orgnally at poston. Convected Coordnates The materal and spatal coordnate systems used here are fed Cartesan systems. An alternatve method of descrbng a moton s to attach the materal coordnate system to the materal and let t deform wth the materal. The moton s then descrbed by defnng how ths coordnate system changes. Ths s the convected coordnate system. In general, the aes of a convected system wll not reman mutually orthogonal and a curvlnear system s requred. Convected coordnates wll be eamned n.0... The Materal and Spatal Descrptons Any physcal property (such as densty, temperature, etc.) or knematc property (such as dsplacement or velocty) of a body can be descrbed n terms of ether the materal coordnates or the spatal coordnates, snce they can be transformed nto each other usng..-4. A materal (or Lagrangan) descrpton of events s one where the 04
Secton. materal coordnates are the ndependent varables. A spatal (or Euleran) descrpton of events s one where the spatal coordnates are used. Eample (Temperature of a Body) Suppose the temperature of a body s, n materal coordnates, (, t) (..6) but, n the spatal descrpton, (, t). (..7) t Accordng to the materal descrpton..6, the temperature s dfferent for dfferent partcles, but the temperature of each partcle remans constant over tme. The spatal descrpton..7 descrbes the tme-dependent temperature at a specfc locaton n space,, Fg...6. Dfferent materal partcles are flowng through ths locaton over tme. moton of ndvdual materal partcles Fgure..6: partcles flowng through space In the materal descrpton, then, attenton s focused on specfc materal. The pece of matter under consderaton may change shape, densty, velocty, and so on, but t s always the same pece of materal. On the other hand, n the spatal descrpton, attenton s focused on a fed locaton n space. Materal may pass through ths locaton durng the moton, so dfferent materal s under consderaton at dfferent tmes. The spatal descrpton s the one most often used n Flud Mechancs snce there s no natural reference confguraton of the materal as t s contnuously movng. However, both the materal and spatal descrptons are used n Sold Mechancs, where the reference confguraton s usually the stress-free confguraton... Small erturbatons A large number of mportant problems nvolve materals whch deform only by a relatvely small amount. An eample would be the steel structural columns n a buldng under modest loadng. In ths type of problem there s vrtually no dstncton to be made 05
Secton. between the two vewponts taken above and the analyss s smplfed greatly (see later, on Small Stran Theory,.7)...4 roblems. The densty of a materal s gven by and the moton s gven by the equatons, t, t. (a) what knd of descrpton s ths for the densty, and what knd of descrpton s ths for the moton? (b) re-wrte the densty n terms of what s the name gven to ths descrpton of the densty? (c) s the densty of any gven materal partcle changng wth tme? (d) nvert the moton equatons so that s the ndependent varable what s the name gven to ths descrpton of the moton? (e) draw the lne element jonng the orgn to (,,0 ) and sketch the poston of ths element of materal at tmes t and t. 06