1,721,022 research outputs found
Genesis and progress of virtual power principle
No full textThe virtual power principle (VPP) of continuum mechanics states a celebrated variational equality between external and internal virtual powers for any virtual velocity field conforming with linear kinematic constraints. The topic is here addressed to investigate how the original ideas born in the early XIX century are modelled by modern formulations based on Functional Analysis and Differential Geometry. These notions are able to provide an effective mathematical context for proving existence of Lagrange multipliers associated with the constraint of rigidity on velocity fields. The VPP stands as privileged tool for giving to stress fields a consistent definition based on duality with conforming virtual stretching fields. By complementarity, the VPP generates a variational condition for integrability of stretching fields, with self-equilibrated stresses as test fields. Progress is got by the formulation of the rate virtual power principle (RVPP) by time derivation of the VPP along the motion, with internal virtual power integrated per unit mass. The basic distinction between spatial and material fields according to the geometric paradigm is prompted to replace the one previously adopted in the literature. The need for a non-redundant implicit formulation of the rigidity constraint is emphasised to contrast degeneracy. This logical demand avoids proliferation of multipliers, in the spirit of Ockham’s Razor, a celebrated philosophical motto with multiform applications. The shining mathematical theory set out by Leonhard Euler, Jean-Baptiste Le Rond d’Alembert, Joseph Louis Lagrange, and Augustin Cauchy is in this respect a point of optimality. A geometric rate theory of elasticity meets the call for no-dissipation in push-closed elastic cycles, with non need of any finite strain elastic energy functional, thus leading to a proper statement of rate equilibrium problems, basilar for computational formulations and for investigations about instability phenomena and post-critical behaviours
Multiscale innovative materials and structures (MIMS)
No full text: Increasing attention is growing towards advanced multiscale metamaterials and nanostructures, due to recent developments in nanoscience and nanotechnology [...]
Axial and flexional behaviour of elastic nano-beams by stress-driven two-phase elasticity
No full textSize-dependent structural behaviour of axially functionally graded nanobeams with non-uniform cross-section under axial and transversal loads is investigated by two-phase integral stress-driven elasticity. An effective nonlocal model is used by introducing a convex combination of the purely nonlocal integral stress-driven relation with a local phase. The stress-driven nonlocal model does not show ill-posedness behaviours such as the Eringen strain-driven model and leads to well-posed elastostatic nonlocal problems in all cases of technical interest. In particular, the integral convolution of the two-phase mixture is obtained by considering the bi-exponential kernel. A nanocantilever subject to an axial or a transversal force at the tip is considered. The nonlocal integral stress-driven model is solved and transversal and axial displacements are evaluated
Carichi critici per nano-travi in elasticita non locale
No full textQuesto contributo presenta alcuni risultati per i carichi critici di compressione in esempi paradigmatici di nano-travi. Il modello usato per descrivere la cinematica delle nano-travi e quello di Bernoulli-Euler. Per ovviare ai comportamenti paradossali messi in evidenza in letteratura, connessi alla relazione costitutiva elastica non-locale differenziale derivata dalla proposta di Eringen, qui si adotta una relazione integrale presentata di recente cosiddetta stress-driven. Contrariamente al modello integrale strain-driven proposto da Eringen, l'approccio stress-driven conduce a problemi elasto-statici per nano-travi che sono ben posti. Si effettuano anche confronti con i risultati ottenibili da modelli di elasticita a gradiente
A consistent variational formulation of Bishop nonlocal rods
No full textThick rods are employed in nanotechnology to build modern electromechanical systems. Design and optimization of such structures can be carried out by nonlocal continuum mechanics which is computationally convenient when compared to atomistic strategies. Bishop’s kinematics is able to describe small-scale thick rods if a proper mathematical model of nonlocal elasticity is formulated to capture size effects. In all papers on the matter, nonlocal contributions are evaluated by replacing Eringen’s integral convolution with the consequent (but not equivalent) differential equation governed by Helmholtz’s differential operator. As notorious in integral equation theory, this replacement is possible for convolutions, defined in unbounded domains, governed by averaging kernels which are Green’s functions of differential operators. Indeed, Eringen himself, in order to study nonlocal problems defined in unbounded domains, such as screw dislocations and wave propagation, suggested to replace integro-differential equations with differential conditions. A different scenario appears in Bishop rod mechanics where nonlocal integral convolutions are defined in bounded structural domains, so that Eringen’s nonlocal differential equation has to be supplemented with additional boundary conditions. The objective is achieved by formulating the governing nonlocal equations by a proper variational statement. The new methodology provides an amendment of previous contributions in the literature and is illustrated by investigating the elastostatic behavior of simple structural schemes. Exact solutions of Bishop rods are evaluated in terms of nonlocal parameter and cross section gyration radius. Both hardening and softening structural responses are predictable with a suitable tuning of the parameters
On the regularity of curvature fields in stress-driven nonlocal elastic beams
No full textElastostatic problems of Bernoulli–Euler nanobeams, involving internal kinematic constraints and discontinuous and/or concentrated force systems, are investigated by the stress-driven nonlocal elasticity model. The field of elastic curvature is output by the convolution integral with a special averaging kernel and a piecewise smooth source field of elastic curvature, pointwise generated by the bending interaction. The total curvature is got by adding nonelastic curvatures due to thermal and/or electromagnetic effects and similar ones. It is shown that fields of elastic curvature, associated with piecewise smooth source fields and bi-exponential kernel, are continuously differentiable in the whole domain. The nonlocal elastic stress-driven integral law is then equivalent to a constitutive differential problem equipped with boundary and interface constitutive conditions expressing continuity of elastic curvature and its derivative. Effectiveness of the interface conditions is evidenced by the solution of an exemplar assemblage of beams subjected to discontinuous and concentrated loadings and to thermal curvatures, nonlocally associated with discontinuous thermal gradients. Analytical solutions of structural problems and their nonlocal-to-local limits are evaluated and commented upon
Nonlocal gradient mechanics of nanobeams for non-smooth fields
No full textNonlocal continuum theories are investigated in the case of non-smooth fields representing the most general condition in mechanics of nanobeams. The treatment starts from the general formulation of elasticity provided by the abstract form of nonlocal gradient theory for nanobeams. The equivalent differential problem is then derived to reverse the constitutive law. Such a formulation requires prescription of non-classical interface conditions at discontinuity abscissae that play a fundamental role to close the relevant differential problem. In this paper, the simplest constitutive interface conditions not involving spatial convolutions are established, thus providing a significant improvement of the treatment contributed in Caporale et al. (2022) in which interface conditions are complexly formulated in terms of spatial convolutions. The developed differential scheme is fundamental for theoretical and computational purposes and plays a key role for strain-driven based models for which inversion of the constitutive law is essential to explicitly get the unknown solution fields. Exemplar continuum problems are finally analyzed and discussed to show merits and effectiveness of the proposed formulation in comparison with previous treatments in literature
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