TAILIEUCHUNG - Elasticity Part 13

Tham khảo tài liệu 'elasticity part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Sadd Elasticity Final Proof 2 55pm page 349 In Cartesian coordinates these expressions would give 1 df 1 df 1 df 2m dx 2m dy 2m dz 1 d2f 1 d2f e 2m ỡx2 ey 2m @y2 _of _ @2f _ @2f Sx dx2 ơy dy2 T x dxdy Thus for this case any harmonic function can be used for Lame s potential. Typical forms of harmonic functions are easily determined and some examples include x2 y2 xy rn cos nd log r log R z with r px2 y2 6 tan 1 y R px2 y2 z2 Galerkin Vector Representation In the previous sections the displacement vector was represented in terms of first derivatives of the potential functions f and w. Galerkin 1930 showed that it is also useful to represent the displacement in terms of second derivatives of a single vector function. The proposed representation is given by 2m 2 1 - v r2V - V where the potential function V is called the Galerkin vector. Substituting this form into Navier s equation gives the result r4 V -T r 1 V Note that for the case of zero body forces the Galerkin vector is biharmonic. Thus we have reduced Navier s equation to a simpler fourth-order vector equation. By comparing the representations given by with that of the Helmholtz potentials can be related to the Galerkin vector by f -2m V X w 2 12 V r2 V Notice that if V is taken to be harmonic then the curl of w will vanish and the scalar potential f will also be harmonic. This case then reduces to Lame s strain potential presented in the previous section. With zero body forces the stresses corresponding to the Galerkin representation are given by Displacement Potentials and Stress Functions 349 TLFeBOOK Sadd Elasticity Final Proof 2 55pm page 350 Sx 2 1 - V @ r2Vx nr2 - @2 V ơx y ơx2J Sy 2 1 - V @ r2 Vy fvV2 - V ơy ơy2J s 2 1 - V @ r2Vz fnr2 - 22 V ơz y @z2 txy 1 - V ị-r2Vx Ị-V V --@ - V yơy ơx J ơxơy tyz 1 - V fẳ r2Vy @ v2iỳ -J - V ơz ơy ơyơz tzx 1 - V -@- r2Vz @ r2Vx V ơx ơz ơzơx As previously mentioned for no body forces the Galerkin .

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