inserted into the equations of motion and by identifying equal powers of ξ and solving for the highest order expansion functions recursion relations are derived as wkþ2 ¼ 1 R2ðk þ1Þðk þ2Þðλ þ2μÞ ρðR2∂ 2 t wk þ2R∂ 2 t wk 1 þ ∂2 t wk 2Þ ðλþμÞð2Rðk þ1Þ2wkþ1 þðk þ2Þðk 1ÞwkÞ μ∇2 awk Rðk þ1Þðλ þμÞ∇a Sk þ1 ððk 1Þ λ þðk 3ÞμÞ∇a SkÞ, ð4:13Þ Skþ2 ¼ 1 R2μðk þ1Þðk þ2Þð ρðR2∂ 2 t Sk þ2R∂ 2 t Sk 1 þ ∂2 t Sk 2Þ μð2Rðk þ1Þ 2Skþ1 þkðk þ1ÞSk þ ∇2 aSkþ1Þ ð λ þμÞ∇að∇a SkÞ ðλ þμÞðk þ1ÞR∇awk þ1 ððk þ2Þ λþðk þ4ÞμÞ∇awk , ð4:14Þ where k = 0, 1, . . .. The advantage with these recursion relations is that all the higher order expansion functions can be expressed in terms of the four lowest order ones w0, w1, S0, and S1 by using these equations recursively. It is noticed that the procedure so far does not depend on any assumption about the thickness of the shell. The recursion relations can be used also to get a good representation of the displacements in the shell for other purposes. To obtain the shell equations the boundary conditions on the shell are now applied. The relevant stress components in spherical coordinates are as follow. The radial stress, i.e. σrr is written σrr ¼ðλ þ2μÞ∂rwþλ 2w r þ ∇a Sk r , ð4:15Þ and the shear stress, i.e. t is written t ¼μ ∂rS S r þ ∇aw : ð4:16Þ The expansions of the displacement components are inserted into the stress components σrr and t which yield σrr ¼ 1 RþξXk¼0 ðλ þ2μÞ ðk þ1ÞRwk þ1 þkwk ð Þþ λð2wk þ∇a SkÞ ð ξ k, ð4:17Þ t ¼ μ RþξXk¼0 ðk þ1ÞRSkþ1 þðk 1ÞSk þ ∇awk ξ k: ð4:18Þ The boundary conditions at ξ = h are then applied, usually this means that both two stress components are zero. It is convenient to take the sum and difference between the equations at the two boundaries. By combining the stresses according toPσ ¼σðh, θ, tÞþσð h, θ, tÞ¼0andΔσ ¼σðh, θ, tÞ σð h, θ, tÞ¼0andusing the expansions of σrr and t and the recursion relations, these four boundary conditions can be written in terms of the four displacements w0, w1, S0, and S1 which deliver four partial differential equations. These are given as an expansion in h, which can, in principle, be given to any order. Finally, by lengthy manipulations which have been carried out using the commercial program Mathematica, the four shell equations including terms up to h2 become 32 R. Okhovat and A. Bostr€om
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