tailieunhanh - Elasticity Part 10

Tham khảo tài liệu 'elasticity part 10', 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 56pm page 259 Infinite Domains For the region shown in Figure 10-7 c the general form of the potentials is determined in an analogous manner as done in the previous case. The logarithmic terms in may be expanded in the region exterior to a circle enclosing all m contours Ck to get log z - zk r log z log 1 - z r log z - z 2 zx 4--------- r log z arbitrary analytic function Combining this result with the requirement that the stresses remain bounded at infinity gives the general form for this case g z c z m s . s logz z g z 2p 1 k 4 m k 2-i Fk S1 s1 2it1 .JJ .logz - c z 2p 1 k 2 where si S1 t1 are the stresses at infinity and z and P z are arbitrary analytic functions outside the region enclosing all m contours. Using power series theory these analytic functions can be expressed as r z r XX anz-n n 1 c z r XX bnz-n n 1 An examination of the displacements at infinity would indicate unbounded behavior unless all stresses at infinity vanish and SFk r SFk r 0. This fact occurs because even a bounded strain over an infinite length will produce infinite displacements. Note that the case of a simply connected infinite domain is obtained by dropping the summation terms in . Circular Domain Examples We now develop some solutions of particular plane elastic problems involving regions of a circular domain. The process starts by developing a general solution to a circular region with arbitrary edge loading as shown in Figure 10-8. The region 0 r R is to have arbitrary boundary loadings at r r R specified by sr r f1 e Tre r f2 e which can be written in complex form as f r f1 e if2 ff r Sr - The fundamental stress combinations and displacements in polar coordinates were given in relations . The tractions given by may be expressed in polar form as Tx iTy r -id g z zg z c z IrrR x y ds Complex Variable Methods 259 TLFeBOOK Sadd Elasticity Final Proof 2 56pm page