Handbook of Elasticity Solutions by Mark L. Kachanov, B. Shafiro, I. Tsukrov

By Mark L. Kachanov, B. Shafiro, I. Tsukrov

This instruction manual is a suite of elasticity strategies. a few of the effects offered right here can't be present in textbooks and come in clinical articles in simple terms. a few of them have been bought within the closed shape relatively lately. The suggestions were completely checked and decreased to a "user pleasant" shape. each attempt has been made to maintain the booklet freed from misprints. the idea of elasticity is a mature box and loads of suggestions are ava- capable. We needed to make offerings in choosing fabric for this publication. The emphasis is made on effects correct to common good mechanics and fabrics technological know-how appli- tions. ideas with regards to structural mechanics (beams, plates, shells, etc.) are passed over. The content material is restricted to the linear elasticity. we're thankful to B. Nuller for a number of clarifications in regards to the touch pr- lem and to V. Levin for feedback on Eshelby's challenge. We additionally get pleasure from a n- ber of feedback and reviews made by means of L. Germanovich, I. Sevostianov, O. Zharii and R. Zimmerman. we're quite indebted to E. Karapetian for a considerable assist in placing the fabric together.

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X - g, Ri = Xi - ~i (i = 1,2,3). Components of the displacement vector u at point x can be written as Ui = Uij(X; g)Fj. Pan & Chou (1976) gave Cartesian components of Green's tensor Uij (x; g) in the following form: 43 Point forces and systems of point forces j = 1,2 U22(X;~) = L. 2 { ,=1 I (Ai - I ( Vi 2) + 2v;Bi ( -*1 - Vi R2 B i ) ---:- - - 3 ~ D. , where VI = J C13 - C13JCI3 + C13 + 2C44 2JC33C44 J C13 + C13JC13 - C13 2JC33C44 2C44 +~------~==---------- v2 = JCI3 - CI3JC 13 + CD + 2C44 2JC33 C44 J C13 + C13J C13 - C13 2JC33 C44 Di = JRr + R~ + z; 2C44 I D.

The Cartesian components of displacements at point x = XI el + X2e2 + X3e3 are given by uI I+V{[XIX3 =- - - - (1-2V)XI] F3 + 2(1-v)R+X3 FI 2n E R3 R(R + X3) R(R + X3) + [2R(vR+X3)+xj]XI } R3(R+X3)2 (xIFI +X2 F2) I+V{[X2X3 (1-2V)X2] U2 = - - - - F3 2n E R3 R(R + X3) + U3 = where R = + 2(1-v)R+X3 F2 R(R + X3) [2R(vR + X3) + xj]X2 } R3(R+X3)2 (xIFI+X2 F2) ~;; ([2(1; v) + ~qF3 + [R;R-::3) + ~~ }XIFI +X2 F2)} Ixl = jx? +xi +xj. 1. Force normal to the boundary (Boussinesq's problem) The solution is due to Boussinesq (1885).

8. , V! and G2, V2, occupying the regions X3 > 0 and X3 < 0, correspondingly, are joined along the plane X3 = o. e! + F2e2 + F3e3 is applied at point (0,0, c), where c > O. e! + X2e2 + X3e3 is expressed in terms of Papkovitch-Neuber's harmonic vector B = B! e! + B2e2 + B3e3 and harmonic scalar Bo as follows: u(x; c) = B - 1 4(1 - v) V(x· B + Bo) or, in Cartesian components: u! (3 - 4v) = 4(1 B! - v) 1 4(1 - v) (aB! aB2 aB3 X ! - +X2- +X3aX! aX! aX! +aBO) aX! U2= (3 - 4v) 1 (aB! -+X2-+X3-+4(1 - v) 4(1 - v) aX2 aX2 aX2 aX2 = (3 - 4v) 1 (aB!

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