Euler elastic properties

We have discussed the value of struts or columns in structural mechanics and described their linear elastic properties. They have another characteristic that is not quite so obvious. When columns are subject to a compressive load, they are subject to buckling. A column will compress under load until a critical load is reached. Beyond this load the column becomes unstable and lateral deformations can grow without bound. For thin columns, Euler showed that the critical force that causes a column to buckle is given by... 

The molecular theory is similar to Cauchy s description of the elastic theory of solids [1] and utilizes additive local molecular pair interactions to describe elasticity. The latter approach was taken by Oseen [2], who was the first to establish an elastic theory of anisotropic fluids. Oseen assumed short-range intermolecular forces to be the reason for the elastic properties, and he derived eight elastic constants in the expression for the elastic free energy density of uniaxial nematic phases. Finally, he retained only five of them, which enter the Euler-Lagrange equations describing equilibrium deformation states of the nematic mesophase, and omitted the other three. 

All the experiments measming mechanical properties of nano-composites ai e performed using macro-scale specimens, however, attempts are made to interpret these data in terms of nano-structural features. One has to be very cautious m doing so, since the traditional continuum mechanics approach has to be modified substantially in order to account for the discontinuous nature of matter at the nano-scale. It has also been shown that for length scales below 5 nm, Ber-noulli-Euler continuum elasticity becomes not valid and higher ordei- elasticity has to be applied [11-13]. In order to find suitable means for bridging the gap between the models based on the mechanics of continuum matter at the microscale and mechanics of discrete matter at the nano-scale, molecular dynamics approach combined with higher order continuum elasticity can be used. 

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