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Nonlocal strain gradient elastic beam models with two-step differential approach and decoupling of standard and extra boundary conditions, I

Castrenze Polizzotto, Paolo Fuschi and Aurora Angela Pisano

Vol. 10 (2022), No. 3, 205–231

Within the framework of Eringen’s nonlocal elasticity, a theory of coherent nonlocal strain gradient (NSG) beam models, i.e., admitting equivalent integral and differential approaches, is reported. Making reference to shear undeformable Euler–Bernoulli (EB) beams in bending, it is shown that the coherence requisite requires that the constitutive equations incorporate a pair of two-phase local/nonlocal models of which one is driven by strain and the other by strain gradient. The governing integrodifferential equation is found to be of sixth differential order, accompanied by four standard plus two gradient BCs, whereas the conjugate differential equation is of order eight, that is, two orders higher, two being the number of the nonlocality BCs, herein also determined. A two-step differential approach is presented in which ODEs of fourth order are addressed with decoupling of the BCs.

nonlocal elasticity, strain gradient elasticity, size effects, equivalent integral/differential approaches
Mathematical Subject Classification
Primary: 74B99, 74E99
Received: 20 January 2022
Revised: 22 March 2022
Accepted: 28 April 2022
Published: 4 December 2022

Communicated by Francesco dell'Isola
Castrenze Polizzotto
Dipartimento di Ingegneria
Università degli Studi di Palermo
Paolo Fuschi
Dipartimento Patrimonio Architettura Urbanistica
Università degli Studi “Mediterranea”
Reggio Calabria
Aurora Angela Pisano
Dipartimento Patrimonio Architettura Urbanistica
Univ degli Studi “Mediterranea”
Reggio Calabria