tailieunhanh - Introduction to Elasticity Part 9

Tham khảo tài liệu 'introduction to elasticity part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | where i ự 1. An analytic function f z is one whose derivatives depend on z only and takes the form a ip y. It is easily da dp dy dx shown that a 21 and p satisfy the 22 f z where a and p are real functions of x and Cauchy-Riemann equations da dp dx dy If the first of these is differentiated with respect to x and the second with respect to y and the results added we obtain P2 Ẽ sv2a 0 23 dx2 dy2 This is Laplace s equation and any function that satisfies this equation is termed a harmonic function. Equivalently a could have been eliminated in favor of p to give v2p 0 so both the real and imaginary parts of any complex function provide solutions to Laplace s equation. Now consider a function of the form xp where p is harmonic it can be shown by direct differentiation that v4 xp 0 24 . any function of the form xp where p is harmonic satisfies Eqn. 12 and many thus be used as a stress function. Similarly it can be shown that yp and x2 y2 p r2p are also suitable as is p itself. In general a suitable stress function can be obtained from any two analytic functions p and X according to p Re x - iy p z x z 25 where Re indicates the real part of the complex expression. The stresses corresponding to this function p are obtained as ơx ơy 4 Re p0 z N. - Vx 2 iTxy 2 zp z x z 26 where the primes indicate differentiation with respect to z and the overbar indicates the conjugate function obtained by replacing i with -i hence z x - iy. Stresses around an elliptical hole In a development very important to the theory of fracture Inglis5 used complex potential functions to extend Kirsch s work to treat the stress field around a plate containing an elliptical rather than circular hole. This permits crack-like geometries to be treated by making the minor axis of the ellipse small. It is convenient to work in elliptical a p coordinates as shown in Fig. 4 defined as x c cosh a cos p y c sinh a sin p 27 . Inglis Stresses in a Plate Due to the Presence of Cracks and Sharp Corners .