On nonlocal lam strain gradient mechanics of elastic rods

Numerous contributions can be found in the recent literature exploiting the nonlocal strain gradient model, introduced in consequence of unification of the differential relation (consequent but not equivalent to Eringen nonlocal integral law) and strain gradient elasticity. In the present paper, Eringen nonlocal integral and Lam modified strain gradient theories are coupled to formulate a nonlocal Lam strain gradient model of elasticity. Three scale parameters, describ-ing nonlocality, dilatation, and stretch gradient, are utilized to significantly estimate size-dependent responses of 1D nanocontinua. The governing constitutive law is established via a variationally consistent approach, based on suitably selected test fields, projected for formulating well-posed static and dynamic problems of engineering interest. The non-local Lam strain gradient model, developed for nanorods, provides axial force fields in terms of integral convolutions involving elastic axial strain fields. The integral law, equivalent to an expedient set of constitutive differential and boundary conditions, is exploited for studying static and free vibration behaviors of simple nanostructural schemes. Exact analytical solutions are gotten in terms of nonlocal and gradient characteristic parameters. Validation of the proposed strategy is carried out by comparing the contributed results with those obtained by the modified nonlocal strain gradient theory.