Download Advances in Doublet Mechanics by Mauro Ferrari, Vladimir T. Granik, Ali Imam, Joseph C. PDF

By Mauro Ferrari, Vladimir T. Granik, Ali Imam, Joseph C. Nadeau

The lately proposed, totally multi-scale thought of doublet mechanics, provided the following in a self-contained shape, deals unprecented possibilities to reconcile the discrete and continuum representations of solids whereas protecting an easy analytical structure and entire compatibility with lattice dynamics and continuum mechanics. Its purposes comprise micro-electro-mechanical structures (MEMS), granular and particulate media, nanotubes and peptide arrays. Novel effects are pronounced, together with the identity of a brand new type of dispersive floor waves, and the presentation of tools for the experimental choice of the fundamental microstructural parameters. The relationships among doublet mechanics, lattice dynamics, and continuum theories are tested.

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48) Taking into account eqns. 49) M(M) n'~. ij In the first approximation (M = 1) eqns. 48) describe the motion of Cosserat's continuum with asymmetric force macrostresses (Cosserat and Cosserat 1909). Under some conditions (see, below, Sect. 2 the couple macrostresses Mij vanish. 47) to the equations of motion of the "symmetric" classical continuum. 49)2 become identities. ) eqns. 48) include different force macrostresses: u~f) in eqn. 47) and uW in eqn. 48). 48) at M > 1 become quite different from the equations of motion of both classical and Cosserat continua.

55) and the identities QijQik = tSjk ovt = QimQjk~, 8v m UXj . 56) where v is any vector, it can be deduced from eqn. 57) and from eqns. 59) where €ijk is the permutation tensor. 57) it is found that + [( rPl + "2,urPOtI 1 It )Q. 60 ) "lOti =+ (QilT~IT~q 1]Ot with no further algebraic simplification apparent. We note from eqns. 61) is an identity for all Q given that the determinant of Q is unitary (Shames and Cozzarelli 1991). 36 2. Doublet Thermomechanics (K. Mon and M. Ferrari) By eqns. 61) it is seen that the axial, torsional, and shear microstrains are properly invariant under the finite rigid motions considered in eqns.

In this reference frame the above vectors and the operator \J are expressed as T~ = T~i ei r=xiei = Ui ei R=Xiei U cP = (jJi ei } \l=ei 8 / 8x i. 6) We adopt the convention that repeated Roman indices imply summation from 1 to 3. This convention does not apply to Greek subscripts. In view of eqn. 7) V x E yo. In order to derive the basic kinematic relations for the three microstrains, we consider an arbitrary doublet (A, BOt) with axis ,~ in the initial region Yo. In the deformed region V of the current configuration this axis is mapped into the axis (Ot (see Fig.

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