Handbook of Fluid Dynamics and Fluid Machinery. Vol 1: by Allen E. Fuhs, Joseph A. Schetz

By Allen E. Fuhs, Joseph A. Schetz

Handbook Of Fluid Dynamics And Fluid Machinery

Volume One basics Of Fluid Dynamics

Joseph A. Schetz And Allen E. Fuhs

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R. Soc. Edinburgh, 102 A, 291- 303. [23] Dacorogna , B. (1989 ). Direct methods in the calculus of variations. Applied Mathemati cal Sciences, Springer, 78. [24] Luskin, M. ( 1996). On the computation of crystalline microstructure. Acta Num erica, 5, 191-257. A . J. Strong Convergen ce of Numerical Solution s to Degener ate Variational Problems. Math. , 64, 117-127. [26] Ortiz, M. A. ( 1999). Nonconvex energy minimi sation and dislocation in ductile single crystals. Journal of the Mechanics and Physics ofSolids, 47, 397--462.

A, + <5, . . 3(,) := (a1' a2, .. , a, <5, . . , an) are in D(a ). Inequ ality (9) yields and the boundedness of ok yields the boundedness of sk E IR n. Thu s sk --4 So E IR n for a subsequence . We have So E £ (a) and it is also easy to see that ak + 'l/Ja,sk --4 ao + W n,so on D(a ) where ao = V1(a) . 'I is So . (ii): By iterating the inequality V J ~ RJ we obtain Vk J ~ RJ :.. Since the sequence Vk J is nonin creasing and bounded from below the limitl of the right-hand side of (8) exists and satisfies l = V l .

Rate-independency is obtained by assuming homogeneity in i of degree I, namely l( x , z , o:i) = o:l(x , z, i ) for 0: 2: 0. Furthermore, we assume that l( x, z , ·) : T zZ f--+ [0,00] is convex and that l satisfies l( x , z, v) 2: clvl for some c > O. Considering a process z : [0, T ] x n f--+ Z the dissipation on an interval [to , it] is then Diss(z; [to , tl]) = J~~l For each x E Z via ~ In ~( x , z(t , x) , i(t, x ))dxdt. ) defines a distance metric on D(x ;zo, zI) = inf{ l ~ 10 ~(x , z(s) , i( s))d s I z E Cl ([0,1], Z) , z(O) = zo, z(l) = Zl }.

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