Time-fractional dependence of the shear force in some beam type problems with negative Young modulus

•We study the fractional dynamics of the shear force in a beam with negative Young modulus.•We describe the shear force in terms of the fractional derivative of order 3/2 of the deflection with respect to time.•We consider different type of boundary conditions: simply supported, cantilever and doubly-clamped.•The results have been illustrated by means of several numerical examples. Under the Euler–Lagrange equation, a time-fractional dependence for the shear force in a beam in the absence of a transverse load is analysed. The main assumption is that the beam is composed by a material with a negative Young modulus. The existence of such materials is documented in several recent studies. A strategy adopting analytical calculations involving Laplace transforms concludes that, up to a known constant, the shear force can be computed as the Riemann–Liouville time-fractional derivative of order 32 of the deflection of the beam. This identity presents an interesting connection with the Bagley–Torvik equation.