Quaternionic Elasticity Theory with Local Fractional Derivatives

Abstract

We develop a quaternionic formulation of linear elasticity in which the classical derivative is replaced by Yang's local fractional derivative and the classical Cauchy--Riemann operator by a fractal Cauchy--Riemann operator. We formulate fractal kinematics, a constitutive law with a material length scale and the equilibrium equation, and show that, for displacement fields with polynomial components, the resulting homogeneous fractal Lamé system is equivalent to the monogenicity of an explicit quaternionic field. On the polynomial sector, that is, for finite combinations of normalized fractal monomials, we prove a fractal Kolosov--Muskhelishvili representation of the displacement field in terms of two monogenic quaternionic potentials, together with its completeness, by a direct construction that uses neither the Galerkin nor the Papkovich--Neuber potentials. The position term of the classical formula has to be replaced by a corrected position operator, which satisfies the exact Weyl relation with the partial local fractional derivatives. Finally a fractal Papkovich-Neuber representation is derived.