From non-local Eringen’s model to fractional elasticity

Abstract

Eringen's model is one of the most popular theories in nonlocal elasticity. It has been applied to many practical situations with the objective of removing the anomalous stress concentrations around geometric shape singularities, which appear when the local modelling is used. Despite the great popularity of Eringen's model in mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in the research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels, and for the paradigmatic case of the Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen's model to spatially heterogeneous material distributions.

Fractional Piola identity and polyconvexity in fractional spaces

“…Another interesting approach related to our investigation, and going from a nonlocal peridynamic framework to a nonlocal fractional situation in the linear case, appears in the recent references [21,32,33]. It is also worth mentioning [14], where the well-posedness for a fractional linearly elastic equation is shown.…”

Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

“…The existence and uniqueness of solutions to three-dimensional wave equation with the Eringen model as a constitutive equation is studied in [36] and it is found that the problem is in general ill-posed in the case of smooth kernels and well posed in the case of singular, nonsmooth kernels. Considering the longitudinal and shear waves propagation in non-local medium, the influence of geometric nonlinearity is investigated in [37].…”

On a Computational Approach to Micro- and Macro-modelling of Damage in Brittle and Quasi-brittle Materials

“…However, for more complicated material structures such transparent theory is not available; e. g. for practical computational simulations of behaviour of fibrereinforced concrete structures under mechanical loads [39] recommends the "generalized Mazars model" with several heuristic parameters, respecting anisotropy together with different behaviour under tension and pressure like [9] and [40], inspired by [41], [42] and [21]. Fortunately the recent result [43] on the ill-possedness of the nonlocal approach [24], referring to the existence analysis [44], for boundary conditions significant in practical applications is not addressed to our formulation, as explained in [26]. Therefore we are ready to work with σ = A(σ) with values from R 3×3 sym (or its natural modification, as mentioned above) and to compute…”