Fractional continuum mechanics is the generalization of classical mechanics utilizing fractional calculus. Contrary to classical theory, the obtained description is non-local, which is inherently the consequence of the fractional derivative definition based on the interval. So, all fields obtained in the framework of this new formulation, such as temperature, thermal stresses, total stresses, displacements, etc., at the specific point of interest, depend on the information from its surroundings. The dimensions of these surroundings and the ways of influencing the results are governed by the fractional differential operator applied. In this article, the application of the fractional continuum mechanics to thermoelasticity is presented. A classical solution is obtained as a special case.

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In this paper the classical EULER-BERNOULLI beam (CEBB) theory is reformulated utilising fractional calculus. Such generalisation is called fractional Euler-Bernoulli beams (FEBB) and results in non-local spatial description. The parameters of the model are identified based on AFM experiments concerning bending rigidities of micro-beams made of the polymer SU-8. In experiments both force as well as deflection data were recorded revealing significant size effect with respect to outer dimensions of the specimens. Special attention is also focused on the proper numerical solution of obtained fractional differential equation.

The objective of the present article is to show the formulation for elastic-viscoplastic material model accounting for intrinsic anisotropic microdamage. The strain-induced anisotropy is described by the evolution of the intrinsic microdamage process — defined by the second-order microdamage tensor. The first step of the possibility of identification procedure (calibration of parameters) are also accounted and illustrated by numerical examples.

Abstract. In this paper, a generalisation of previous author's formulation of fractional continuum mechanics for the case of anisotropic non-locality is presented. The discussion includes a review of competitive formulations available in literature. The overall concept is based on the fractional deformation gradient which is non-local due to fractional derivative definition. The main advantage of the proposed formulation is its structure, analogous to the general framework of classical continuum mechanics. In this sense, it allows to define similar physical and geometrical meaning of introduced objects. The theoretical discussion is illustrated by numerical examples assuming anisotropy limited to single direction. NotationSection 2and finallywhere] denotes the interval of non-local interaction, and Γ is the Euler gamma functionNext, based on non-standard definition of motion by Eq. (4) the authors define the deformation gradient F α aswhere e a and E A are base vectors in coordinate systems {x a } and {X A }, respectively.

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