Stability of rod-shaped nanoparticles embedded in an elastic matrix

In many biological tissues as well as in some technical materials we find nano-sized rod-shaped particles embedded in a relatively soft matrix. Loss of stability of equilibrium, i.e. buckling, is one of the possible failure modes of such materials. In the present paper different kinds of load transfer between matrix and reinforcing particles, which are typical for rod-shaped nanostructures in biological tissues, are considered with respect to stability of equilibrium. Two regimes of matrix stiffnesses leading to different modes of buckling, and a transition regime in between, have been found: soft matrix materials leading to the so-called ‘flip mode’ (also called ‘tilt mode’) and hard matrix materials resulting in ‘bending mode’ buckling. The transition regime is of particular interest for biological tissues. Numerical and semi-analytical as well as asymptotic concepts are employed leading to results for estimating the critical load intensities both in the form of closed form solutions and diagrams. The analytical solutions are compared with results of finite element analyses. From these comparisons indications are gained for deciding which of the different analytical approaches should be chosen for a particular nanostructure configuration in terms of the associated buckling modes.