By Chia-Shun Yih
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Extra resources for Advances in Applied Mechanics, Vol. 14
Cclyl):*ir6 u + ... 37) wherein the leading term (&‘LI,6u = 0 has been dropped, and the omitted terms in the second bracket are all proportional to 2, or 2:. 38) , where p = l):*;ul/i&u:. 41 ) for small enough values of 5 and l. 41) represents a relation snapping loads 4 . +, = 1 - 2 ( - & d / i c ) 1 ’ 2 . 9) for the simple model. 41) does not display a local maximum in A, but in the vicinity of 1” = A, gives A-t relations like those shown in Fig. 5b for the simple model. 41) can be matched smoothly to solutions I ( ( ; for srmll A.
8), wherein 6u can be chosen as u, . 21) and expressions for the successive coefficients i3, i4, . . 16) can also be found in a straightforward way. It is seen that the post-buckling coefficient 2 , depends only on the buckling mode i i , , but the calculation of 2 , would generally require the determination of u2 as well, and so on. I t is now evident that the relations between iand found for the bifurcated path are entirely similar to those illustrated in Fig. 2 for the k( relation of the simple model.
In earlier work, the notation CJ . 61; has been used for the inner product representing internal virtual work, but it has become clear that the dot is really not needed. ] We can, if we wish, assert Eq. 4), and no potential energy functional. However, in the present elastic case, the link to energy is desirable, in order to derive and exploit certain symmetry properties embedded in the stress-strain relations. “:I 6 C l 61:, = a’“:](SC2 he1, 61; 1 it follows that t”[C] and since 8“ is a symmetric bilinear operator, so must (T’ be.