The state of fractional hereditary materials (FHM)

The widespread interest on the hereditary behavior of biological and bioinspired materials motivates deeper studies on their macroscopic "minimal" state. The resulting integral equations for the detected relaxation and creep power-laws, of exponent β, are characterized by fractional operators. Here strains in SBV_loc are considered to account for time-like jumps. Consistently, starting from stresses in L^r_loc, r ∈ [1, β^-1], β ∈ (0, 1) we reconstruct the corresponding strain by extending a result in [42]. The "minimal" state is explored by showing that different histories delivering the same response are such that the fractional derivative of their difference is zero for all times. This equation is solved through a one-parameter family of strains whose related stresses converge to the response characterizing the original problem. This provides an approximation formula for the state variable, namely the residual stress associated to the difference of the histories above. Very little is known about the microstructural origins of the detected power-laws. Recent rheological models, based on a top-plate adhering and moving on functionally graded microstructures, allow for showing that the resultant of the underlying "microstresses" matches the action recorded at the top-plate of such models, yielding a relationship between the macroscopic state and the "microstresses".