A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method

•A size dependent finite element model is developed.•Nonlocal elasticity theory is used.•Finite element solution is presented for buckling of microtubules.•The effect of elastic matrix surrounded by microtubules is modeled.

A simple nonlocal beam model is proposed to study buckling response of protein microtubules. The size-effect for buckling model of microtubules is considered by using the nonlocal continuum theory. Finite element procedure is used for solution of nonlocal differential equation of microtubules for elastic stability. The influence of the small length scale on the buckling value is examined for different geometric parameters. The effect of elastic matrix surrounded of microtubules is also examined and some benchmark results are presented.