2.13 Variation and Linearisation of Kinematic Tensors

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1 Section.3.3 Variation an Linearisation of Kinematic ensors.3. he Variation of Kinematic ensors he Variation In this section is reviewe the concept of the variation, introce in Part I, 8.5. he variation is efine as follows: consier a fnction (, with ( a secon fnction which is at most infinitesimally ifferent from ( at every point x, Fig..3. ( ( ( Figre.3.: the variation x hen efine ( ( he Variation (.3.) he operator is calle the variation symbol an is calle the variation of (. he variation of ( is nerstoo to represent an infinitesimal change in the fnction at x. Note from the figre that a variation of a fnction is ifferent to a ifferential. he orinary ifferentiation gives a measre of the change of a fnction reslting from a specifie change in the inepenent variable (in this case. Also, note that the inepenent variable oes not participate in the variation process; the variation operator imparts an infinitesimal change to the fnction at some fixe x formally, one can write this as x 0. he Commtative Properties of the variation operator () (.3.) 37

2 Section.3 Proof: ( ) ( x x () ( ( (.3.3) Proof: x ( x ( x ( x x ( ( ( Variation of a Fnction Consier A, a scalar-, vector-, or tensor-vale fnction of, variation to,, A changes to A A. When we apply a. he variation of A is then efine as A (, ) A ( ) A ( ) (.3.4) (in the limit as 0 ). his can be expresse sing the concept of the irectional A, so that erivative in the sal way (see.6.): consier the fnction A A an 0 A A0 A/, or 0 Setting then gives Eqn..3.4; ths A A. A aylor expansion gives A A A (.3.5) 0 A( ) A( ) A[ ] (.3.6) where A[ ] is the irectional erivative of A in the irection ; the irectional erivative in this context is the variation of A: A, A[ ] A 0 (.3.7) 38

3 Section.3 For example, consier the scalar fnction P : E, where P an E are secon orer tensors. hen E, E E[ E] P : E E P : E 0 (.3.8) he secon variation is efine as A A A[ ] A 0 (.3.9) For example, for a scalar fnction of a vector, the chain rle an Eqn..3. give, 0 0 (.3.0) Variation of Fnctions of the Displacement In what follows is iscsse the change (variation) in fnctions A () when the isplacement (or velocity) fiels nergo a variation. hese ieas are sefl in formlating variational principles of mechanics (see, for example, 3.9). Shown in Fig..3. is the crrent configration frozen at some instant in time. he isplacement fiel is then allowe to nergo a variation. his change to the isplacement fiel eviently changes kinematic tensors, an these changes are now investigate. Note that this variation to the isplacement inces a variation to x, x, bt X remains nchange, X 0. X reference configration ( x crrent configration Figre.3.: a variation of the isplacement 39

4 Section.3 o evalate the variation of the eformation graient F, F x X an Eqn...43, F Gra I isplacement fiel, note that the efinition.3.7,,, where is the. One has, from F F F, 0 0 Gra F I Gra (.3.) Noting the first commtative property of the variation,.3., this can also be expresse as, Gra Note that is completely inepenent of the fnction. F (.3.) Here are some other examples, involving the inverse eformation graient, the Green- Lagrange strain, the inverse right Cachy-Green strain an the spatial line element: { Problem -3} F C F E F εf F gra εf (.3.3) where ε is the small strain tensor, Eqn One also has, sing the chain rle for the irectional erivative, Eqn..5.8, the irectional erivative for the eterminant, Eqn..5.3, the trace relation.0.0e, Eqn...8b, J, et F, et F F et F F et FGra F et FF : Gra (.3.4) JtrGra F Jtrgra J iv 330

5 Section.3 Example o pt some of the above concepts into a simple an less abstract setting, consier the following scenario: a bar over 0 X is extene, as illstrate in Fig..3.3, accoring to: x X 3 X x 3 (.3.5) he eformation graient is F Gra 4X (.3.6) So, for example, in the initial configration (A), an infinitesimal line element at X 0 oes not stretch ( F 0 ) whereas a line element at X stretches by 4. he inverse eformation graient is F grax 8 x 3 (.3.7) his implies that, in the crrent configration (B), an infinitesimal line element at x 3 is the same size as its conterpart in the initial configration ( F 0 ) whereas a line element at x 5 shrinks by a factor of 4 when retrning to the initial configration F ( ) 4 X A A B X F 4 C C 4 X Figre.3.3: a motion an a variation 33

6 Section.3 Now introce a variation, which moves the bar from configration B to configration C: x X 3 (.3.8) he point at 3 moves to 3 3 an the point at 5 moves to 5 5. (his variation happens to be a simple linear fnction of x, bt it can be anything for or prposes here.) he variation is plotte below as a fnction of X an x. 5 3 x 5 3 Figre.3.4: the variation as a fnction of x an X Differentiating Eqns..3.9, the graients of the variations are X X Gra 4 gra (.3.0) which are the slopes in Figre.3.4. o calclate the F associate with the new variation configration, i.e. F points X have now move to: an so 3 3, note that X X (.3.) 33

7 Section.3 F Gra X 3 4X4X (.3.) his says that an infinitesimal line element at X 0 oes not stretch when moving to configration C ( F 0 ) whereas a line element at X stretches by 4 4. Sbtracting Eqn..3.7 form Eqn..3.: 4 From Eqn..3.0, this verifies Eqn..3., that F F F X (.3.3) F Gra (.3.4) We col also calclate the variation of F by moving irectly from configration B to configration C. he movement of the particles from B to C is given by Eqn..3.9: 3 F Gra X 3 4X. X an so, base on this motion, o calclate the that the new crrent position x is (Eqn..3.): F associate with the new variation configration, i.e. F, note x X X C 3 3 xc X 3 (.3.5) his means that the point 3 3 in configration C correspons to X 0 an the point 5 5 correspons to X. hen, F xc 3 C 8 3 xc (.3.6) So an element at the point 3 3 in configration C oes not change in size as it is mappe back to the initial configration, whereas an element at the point 5 5 shrinks back to the / 4 4, as inicate in Fig initial configration by a factor of Alternatively, since xc xx, this can be written as F 8 x 3 (.3.7) 333

8 Section.3 Sbtracting Eqn..3.8 from Eqn..3.7, the variation of the inverse eformation graient is then F F F x 3 x x (.3.8) Using a series expansion, ),, for small (neglecting terms of orer F 8 x 3 (.3.9) From Eqns..3.8 an.3.0, this verifies the relation.3.3: A formla for the inverse eformation graient is F I/ x, bt that F F gra δ (.3.30) F I x. F Igra. However, note that / C he Lie Variation he Lie-variation is efine for spatial vectors an tensors as a variation holing the eforme basis constant. For example, i j b L a aijg g (.3.3) he object is first plle-back, the variation is then taken an finally a psh-forwar is carrie ot. For example, analogos to..66, a a (.3.3) L, For example, consier the Lie-variation of the Eler-Almansi strain e. First, from..56b, b b e E e E F. From..40a,. hen.3.3b gives εf b e b b F εf ε L e, (.3.33) 334

9 Section.3.3. Linearisation of Kinematic Fnctions Linearisation of a Fnction As for the variation, consier A, a scalar-, vector-, or tensor-vale fnction of. If nergoes an increment, then, analogos to.3.4, A A[ ] he irectional erivative A[ ] linearization of A with respect to is efine to be A (.3.34) in this context is also enote by A, A A,,. he L A (.3.35) Using exactly the same metho of calclation as was se for the variations above, the linearization of F an E, for example, are L F L E where ε gra gra, F F F Gra, E E E F εf is the linearise small strain tensor ε. (.3.36) Linearisation of Variations of a Fnction One can also linearise the variation of a fnction. For example,, A, A, he secon term here is the irectional erivative his leas to an expression similar to vector, L A (.3.37) A[, ] A[ ] A (.3.38) A 0. For example, for a scalar fnction of a (.3.39) 335

10 Section.3 Consier now the virtal Green-Lagrange strain,.3.b, E F εf. o carry ot the linearization of E, it is convenient to first write it in the form E F εf F gra Gra gra F F F Gra (.3.40) hen E E Gra F F Gra (.3.4) Recall that the variation is inepenent of ; this eqation is being linearise with respect to, an is naffecte by the linearization (see Fig..3.3 below). However, the motion, an in particlar F, are affecte by the increment in. hs { Problem 4} E sym Gra Gra (.3.4) reference configration crrent configration Figre.3.3: linearisation As with the variational operator, one can efine the linearization of a spatial tensor as involving a pll back, followe by the irectional erivative, an finally the psh forwar operation. hs, a a (.3.43).3.3 Problems 336

11 Section.3. Use Eqn..., E F F I GravF, Eqn..3., F Gra,, an Eqn...8b, grav, to show that E F εf, where ε is the small strain tensor, Eqn Use.3. to show that the variation of the inverse eformation graient F is F F gra. [Hint: ifferente the relation F F I by the proct rle an then se the relation vf grav Gra for vector v.] 3. Use the efinition C F F to show that C F εf. sym A A A to show that 4. Use the relation Gra F F Gra sym Gra E Gra 5. Use e ε gra gra to show that the e e sym Gra sym gra Gra gra 337

THE DISPLACEMENT GRADIENT AND THE LAGRANGIAN STRAIN TENSOR Revision B

THE DISPLACEMENT GRADIENT AND THE LAGRANGIAN STRAIN TENSOR Revision B HE DISPLACEMEN GRADIEN AND HE LAGRANGIAN SRAIN ENSOR Revision B By om Irvine Email: tom@irvinemail.org Febrary, 05 Displacement Graient Sppose a boy having a particlar configration at some reference time