By Mark L. Kachanov, B. Shafiro, I. Tsukrov
This guide is a set of elasticity strategies. a few of the effects provided the following can't be present in textbooks and come in medical articles in basic terms. a few of them have been got within the closed shape really lately. The ideas were completely checked and diminished to a "user pleasant" shape. each attempt has been made to maintain the publication freed from misprints. the idea of elasticity is a mature box and loads of ideas are ava- capable. We needed to make offerings in deciding on fabric for this publication. The emphasis is made on effects correct to basic good mechanics and fabrics technology appli- tions. recommendations on the topic of structural mechanics (beams, plates, shells, etc.) are skipped over. The content material is proscribed 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 via 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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1. Force normal to the inteiface Point force F = F3e3 is normal to the interface between the half-spaces and is applied at point (0,0, c), where c > O. Papkovitch-Neuber's functions Band Bo are as follows. :... G1 + (3 - 4v])Gz Rz R~ G] - G2 Rt + G] + (3 - 4V1)G2 [C(3 - 4v]) Rz 4(1 - v] )G] [(1 - 2V1)(3 - 4vz) - 2G2(V] - V2)/(Gj - G2)] G2 x In(Rz + (3 - 4V2)G] + X3 + C)] } where Ri = xi + x~ + (X3 - c)z and R~ = xl + x~ + (X3 + c)z. 1). 2. Force parallel to the interface Point force F = Fle1 is applied at point (0, 0, c), where c > 0.
2. Force tangential to the boundary Point force F = FI e I is tangential to the boundary of the transversely isotropic halfspace. 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!
They also gave expression for Green's function in the case of cubic symmetry. For the case of general anisotropy, they gave representations, in terms of the Stroh eigenvalues, that are less usable. REMARK 2. Both Pan & Chou (1976) solution, given above, and Ting & Lee (1997) solution have the advantage, as compared with the pioneering work of Lifshitz & Rozentsveig (1947), that they are valid for all combinations of elastic constants and, in particular, do not degenerate in the case of isotropy.