2018/06/11 by Anton Evgrafov, Evgrafov, Anton, José C. Bellido +1
Engineering · Materials Science · Mathematics · #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Nonlocal and gradient elasticity in micro/nano structures #Numerical methods in engineering #math.AP
paper · pdf · doi:10.48550/arxiv.1806.03906
arxiv created 2018/06/11 · openalex publication_date 2018/06/11 · arxiv updated 2018/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.