Continuous Media with Microstructure
The contributions to the book concern various aspects of extension of classical continuum models. These extensions are related to the appearance of microstructures both natural as well as these created by processes. To the first class belong various therm
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Abstract. The paper presents the mixed structural mechanics - molecular approach to analysis of stresses and deformations at the nanoscale range. Plots of Young modulus, for the individual bonds, as well as for the bulk nanotubes are given, taking molecular interactions into account. Numerical calculations for carbon nanotubes of different chiralities are performed.
1 Introduction Investigating and modeling of materials and devices in the nanoscale range constitutes an up to date topic in mechanics and material science. In recent years different approaches have been proposed to describe their mechanical phenomena and properties. Among them the idea to couple the molecular models with continuum and structural mechanics description is one of the most effective and promising methods of analysis. The continuum models with some equivalent molecular properties have been proposed (see e.g.[5], [6]). They lead mostly to mechanical models (rods, plates, shells etc) with equivalent, but constant, mechanical parameters. It contradicts the molecular situation, in which the potentials are essentially nonharmonical, and hence the stiffness parameters are non-constant. In the present paper an alternative approach will be presented, with constitutive model taking directly molecular interactions into account. Our considerations include potentials depending not only on the intermolecular distances, but also on the angles between the bonds. The paper is organized as follows: we start with the short description of the discrete molecular system, followed by the the concept of the mixed continuum-molecular model, Gwidon Szefer · Dorota Jasi´nska Cracow University of Technology, ul. Warszawska 24, 31-155 Krak´ow e-mail: {szefer,jasinska}@limba.wil.pk.edu.pl
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G. Szefer and D. Jasi´nska
called nanocontinuum. This model is applied to carbon nanotubes (CNT), for which the modified Morse potential and Tersoff-Brenner potential are used. The role of the bond-angles is shown in details. Numerical examples for different CNTs are presented.
2 Field Quantities in the Discrete Systems To introduce field quantities in the discrete molecular systems let us consider a set of material points Ai , with position vectors ri , i = 1, ..N referred to a fixed Cartesian frame {0xk }, k = 1, 2, 3. Let the initial position vector be denoted by roi . The system is subjected to the action of resulting from the given potential forces U(r1 , ..rN ) = U(ri j , θi jk ), where ri j = rij , rij = rj − ri and θi jk - angle between adjacent bonds. The influence of the angle θi jk and possibly further environment plays an important role in bonds of many materials (like carbon, silicon etc.). It concerns many devices in modern nanoengineering, like carbon nanotubes and silicon wafers. The gradient ∂U 1 · eij ; eij = · rij , (1) fij = ∂ ri j ri j describes the interaction forces between molecules, which induce the motion of the molecules according to the equations mi · r¨ ij = ∑ fij + fext i ; i = 1..N,
(2)
j
where mi - mass of the molecule Ai , and fext i
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