On Λ-fractional elastic solid mechanics

After defining the fractional Λ-derivative, having all the requirements for corresponding to a differential, the fractional Λ-strain is established. Contrary to the common strain, that has a local character, fractional strain access a non-local character, quite important for expressing deformations in non-homogeneous media with microcracks and inhomogeneities, that may change during deformation. The purpose of the present work is the establishement of the principles and laws of the non-linear Λ-fractional Elasticity. The Λ-fractional non-linear stress–strain relations are derived. The restriction into the linear fields is presented. Further, fractional deformation of a fractal bar is discussed. The Fractional deformations and fractional elastic problems are set up with the definition of stresses and displacements in the initial space. Further, the Λ-fractional analysis with its conjugate Λ-fractional space is presented, considering fractional derivatives of both sides in the bending of a cantilever beam under uniform continuously distributed loading.