17/0/017 Lecture 16 (Refer the text boo CONTINUUM MECHANICS by GEORGE E. MASE, Schaum s Outlnes) Knematcs of Fluds Last class, we started dscussng about the nematcs of fluds. Recall the Lagrangan and Euleran way of analyzng flud moton. As the contnuum encompasses of several partcles, the Lagrangan analyss deals wth each partcles. To qucly descrbe certan quanttes, consder a contnuum n moton. Intally at tme t=to, the slope of the contnuum s as shown and s referred wth the orthogonal coordnates OX1XX3. Fg. 1: Representaton of the statement above (Source: Schaum s outlne of theory and problems of contnuum mechancs by George Mase) At a mathematcal pont P0, a partcle of the contnuum s assocated at tme t=t0. The poston vector of ths partcle s X X1I ˆ ˆ ˆ 1 X I X 3I3 The above poston vector s called materal coordnate. As the contnuum s movng, at a later tme t=t, the poston of the same partcle mght have changed. Even the contnuum also deforms shfts ts poston.
Let the poston at tme t be gven as n the fgure x x1eˆ ˆ ˆ 1 xe x3e3 Ths new poston descrpton s spatal coordnates. The same partcle that was present at P0 at tme t0 s now dsplaced and the dsplacement vector s gven as s. You can also see that the coordnates also shfted by a vector b. From vector algebra: s b x X If the coordnates OX1XX3 and ox1xx3 are merged, you get b 0 Hence, s x X (Ths means, x s the poston vector of the partcle at tme t, whose ntal poston s X ). In ndex notaton: s x X When the contnuum s n moton and deformaton, the partcles poston may be expressed n the form: x = x(x1,x,x3,t) or x x( X, t) You now, x present locaton of the partcle that occuped the pont (X1, X, X3) at tme t=to. (Ths s mappng the ntal confguraton wth the current confguraton). Such type of moton descrpton s Lagrangan formulaton. The dependent quantty s x and ndependent quantty s X. If the moton or deformaton s represented by the form: X = X(x1,x,x3,t) or X X ( x, t) where ndependent varable s x and t. Ths s Euleran formulaton. Ths descrpton provdes you the tracng of orgnal poston of the partcle that now occupes the spatal coordnate or locaton (x1, x, x3). The Lagrangan and Euleran mappngs are therefore nverse functons. For the nverse functons to exst, the necessary requrement s that the Jacoban must exst. Jacoban:
J x x x X X X 1 1 1 x x x x X X X X x x x X X X 3 3 3 If ths Jacoban (determnant) s zero, then unque nverse does not exst. From x = x(x1,x,x3,t), the Lagrangan form, you can form materal deformaton gradent by partally dfferentatng t wth X.e., x x x X X X x x x x X X X X x x x X X X F 1 1 1 3 3 3 Note: Ths s not a determnant. It s a tensor From X = X(x1,x,x3,t) The Euleran form, you get spatal deformaton gradent X X X x x x X X X X x x x x X X X x x x H 1 1 1 3 3 3 You can also form Materal Dsplacement Gradent. s x F X X (Recall s=x-x) As obvous, the materal dsplacement gradent s also a tensor. In smlar lnes, spatal dsplacement gradent tensor. can also be formulated as follows: s X H x x
The Deformaton Tensors To now about deformaton, the procedure s to see how much change s there between postons of two partcles from ther ntal confguraton (at t=t0) and later confguraton (at t=t). Consder the fgure below where the materal coordnates OX1XX3 and spatal coordnates ox1xx3 are merged. Fg. : The deformaton tensor representaton (Source: Schaum s outlne of theory and problems of contnuum mechancs by George Mase) There are two neghborng partcles that occupy postons P0 and Q0 ntally at tme t=t0. The dfferental elemental element length between two partcles s dx as per vector algebra. After a certan tme, at the nstant t=t, the contnuum has moved as well as deformed. The postons of those partcles are gven n spatal coordnates x and x dx The square of the dfferental element length between P0 and Q0 s: ( dx ). dx dx In ndex notaton, ( dx ). dx dx From X=X(x1,x,x3,t), you have seen:
X x dx ( ) F X x dx X dx dx dx x x X x C dx dx X X dx dx x Where C Cauchy s deformaton tensor. In the deformed confguratons, where the partcles are at postons P and Q, ( dx). dx dx ( dx) dxdx Also from Lagrangan expresson, x = x(x1,x,x3,t) x dx dx X ( ) dx dxdx dx dx X X ( dx) G dx dx x x Where G Green s deformaton tensor. The measure of deformaton s evaluated based on the dfference (dx) (dx) for the two neghborng partcles. ( ) ( ) dx dx dx dx dx dx X X x x dx dx X X L dx dx x x Here L Lagrangan or Green s fnte stran tensor L 1 x X x X
In a smlar way, you can form Euleran stran tensor E 1 X x X x Wth ths bref bacground nformaton on: 1. Materal coordnates OX1XX3. Spatal coordnates ox1xx3 x 3. Materal dervatve gradent X X 4. Spatal dervatve gradent x s x 5. Materal dsplacement gradent X X s X 6. Spatal dsplacement gradent x x X X 7. Cauchy s deformaton tensor x x x x 8. Green s deformaton tensor dxdx X X 9. Lagrangan s fnte stran tensor, etc. For fluds, we can descrbe propertes n Euleran or Lagrangan way..e., For example, the densty n the materal descrpton wll be: ρ = ρ(x1,x,x3,t).e. ρ = ρ(x, t) Ths wll be the densty of the flud partcle at the poston (X1,X,X3). In Euleran form : ρ= ρ(x(x1,x,x3,t), t) = ρ(x(x,t),t) = ρ * (x,t)