Growth-induced axial buckling of a slender elastic filament embedded in an isotropic elastic matrix

Author-Highlights
• A theory of a growing filament in an elastic matrix is considered.
• The critical value of the growth parameter at which axial buckling occurs. computed.
• An asymptotic analysis and a numerical analysis provide a complete picture of the problem.
• A rod theory MODEL of the same problem is also considered.
• By comparing both approaches, an estimate for the foundation modulus parameter of a Winkler foundation is obtained.

We investigate the problem of an axially loaded, isotropic, slender cylinder embedded in a soft, isotropic, outer elastic matrix. The cylinder undergoes uniform axial growth, whilst both the cylinder and the surrounding elastic matrix are confined between two rigid plates, so that this growth results in axial compression of the cylinder. We use two different modelling approaches to estimate the critical axial growth (that is, the amount of axial growth the cylinder is able to sustain before it buckles) and buckling wavelength of the cylinder. The first approach treats the filament and surrounding matrix as a single 3-dimensional elastic body undergoing large deformations, whilst the second approach treats the filament as a planar, elastic rod embedded in an infinite elastic foundation. By comparing the results of these two approaches, we obtain an estimate of the foundation modulus parameter, which characterises the strength of the foundation, in terms of the geometric and material properties of the system.