Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

Failla, Giuseppe; Santini, Adolfo; Zingales, Massimiliano

doi:10.1016/j.mechrescom.2009.09.006

Cited by 25 publications

(19 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By analogy to the mechanicallybased model of non-local bar proposed by Di Paola et al [42,45,46], a linear dependence on the product of the volumes of the interacting beam segments through appropriate attenuation functions governing the spatial decay of non-local effects is also included.…”

Section: Long-range Resultantsmentioning

confidence: 99%

“…3). Thus, from a mechanical point of view, the proposed non-local beam model is conceptually equivalent to the non-local bar built by Di Paola et al [42,45,46], where the longrange volume axial forces are represented, within the context of a discrete model, as linearly-elastic springs of distance-decaying stiffness connecting non-adjacent volume elements. In this regard it worth remarking that, if the long-range volume transverse forces/moments were taken as depending on the relative transverse displacement and not on the pure shear deformation (10), long-range volume transverse forces/moments would erroneously arise from a relative transverse displacement induced, for instance, by a rigid rotation of the beam.…”

Section: Long-range Resultantsmentioning

confidence: 99%

“…The purpose of this paper is to illustrate a new nonlocal TM beam model, derived within the framework of a mechanically-based non-local elasticity theory recently proposed by the authors [39][40][41][42][43][44][45][46], where nonlocal effects are modelled as long-range interactions resulting from the relative displacement between volume elements. Consistently with typical engineering beam theories, where the equilibrium of a beam segment is set in an average (weak) sense based on the stress resultants acting on the cross section, in the proposed non-local beam model the long-range interactions are modelled as volume forces and moments, mutually exerted by non-adjacent beam segments, that contribute to the equilibrium of any beam segment along with the local stress resultants.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A new displacement-based framework for non-local Timoshenko beams

2015

View full text Add to dashboard Cite

In this paper, a new theoretical framework is presented for modeling non-locality in shear deformable beams. The driving idea is to represent non-local effects as long-range volume forces and moments, exchanged by non-adjacent beam segments as a result of their relative motion described in terms of pure deformation modes of the beam. The use of these generalized measures of relative motion allows constructing an equivalent mechanical model of non-local effects. Specifically, long-range volume forces and moments are associated with three spring-like connections acting in parallel between couples of nonadjacent beam segments, and separately accounting for pure axial, pure bending and pure shear deformation modes. The variational consistency of the proposed non-local beam model is demonstrated by minimization of an appropriate total potential energy functional. Numerical results concerning the static behavior for different boundary and loading conditions are presented. It is shown that the proposed nonlocal beam model is able to capture experimental data on the static deflection of micro-beams, available in the literature.

show abstract

Section: Long-range Resultantsmentioning

confidence: 99%

Section: Long-range Resultantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new displacement-based framework for non-local Timoshenko beams

2015

View full text Add to dashboard Cite

show abstract

“…This approach has been used in several context of physics and engineering yielding the so-called fractional-order Fourier transport equation ([13], [14], [15], [16]) or the non-local fractional-order thermodynamics ( [17], [18], [19], [20], [21]). …”

Section: Introductionmentioning

confidence: 99%

An exact thermodynamical model of power-law temperature time scaling

Zingales¹

2016

Annals of Physics

View full text Add to dashboard Cite

In this paper a physical model for the anomalous temperature time evolution (decay) observed in complex thermodynamical system in presence of uniform heat source is provided. Measures involving temperatures T with power-law variation in time shows a different evolution of the temperature time rate T˙ (t) with respect to the temperature time-dependence T (t). Indeed the temperature evolution is a power-law increasing function whereas the temperature time rate is a power-law decreasing function of time. Such a behavior may be captured by a physical model that allows for a fast thermal energy diffusion close to the insulated location but must offer more resistance to the thermal energy flux as soon as the distance increases. In this paper this idea has been exploited showing that such thermodynamical system is represented by an heterogeneous onedimensional distributed mass one with power-law spatial scaling of its physical properties. The model yields, exactly a power-law evolution (decay) of the temperature field in terms of a real exponent as that is related to the power-law spatial scaling of the thermodynamical property of the system. The obtained relation yields a physical ground to the formulation of fractional-order generalization of the Fourier diffusion equation. R1) The discrete thermodynamic system described in eqs.(3,4) yields an approximation of powerlaw temporal evolution of the temperature defined by beta as the thermal conductivity c(V) and the specific heat coefficient C(V) varies along masses mj with the relation defined by alpha. The eqs. (3) and (4) do not explicitely contain the parameters alpha and beta. Is that an assumption? or is it a statement ?A1) The balance equations, in eq.(3) for the generic mass and in eq. (4) for the boundary mass m1, have been written in a generic form involving the non-homogeneity of the conductor in the thermal conductivity χ j and in the specific heat C j . Their specific variation, as power-laws of alpha, along the thermodynamic system provided in eqs.(5 a,b) corresponds to a power-law decay of the temperature with beta. It is a challenging problem whether other functional class of parameter variation may correspond to other measured time-varying evolution of the temperature fields. Therefore I decided not to specify the functional class of parameter variation in the generic balance equations. R2) Eq. (5a) and (5b) Why different denominators are different, it should be explained. A2) Thanks for the observation, I kept the same denominator and re-performed derivations in the revised manuscript. R3) Text after Eq.(5a) and (5b) "the real exponent alpha belongs to the interval -1<=alpha<=1". This interval should be -10 and b2<2 should be consider...

show abstract

“…Being capable of interpolating among the integer-order operators of classical differential calculus, fractional operators have been used indeed in several contexts of physics and engineering . They have proved suitable for reproducing long-memory effects and long-range spatial effects [20][21][22], with successful applications to viscoelasticity [23][24][25][26][27][28] and non-local elasticity [29][30][31][32][33][34][35][36]. Generalized transport equations have also been derived from fractional statistical mechanics [37].…”

Section: Introductionmentioning

confidence: 99%

The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors

Zingales

Failla

2015

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

a b s t r a c tIn a non-local fractional-order model of thermal energy transport recently introduced by the authors, it is assumed that local and non-local contributions coexist at a given observation scale: while the first is described by the classical Fourier transport law, the second involves couples of adjacent and non-adjacent elementary volumes, and is taken as proportional to the product of the masses of the interacting volumes and their relative temperature, through a material-dependent, distance-decaying power-law function. As a result, a fractional-order heat conduction equation is derived. This paper presents a pertinent finite element method for the solution of the proposed fractional-order heat conduction equation. Homogenous and non-homogeneous rigid bodies are considered. Numerical applications are carried out on 1D and 2D bodies, including a standard finite difference solution for validation.

show abstract

Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

Cited by 25 publications

References 19 publications

A new displacement-based framework for non-local Timoshenko beams

A new displacement-based framework for non-local Timoshenko beams

An exact thermodynamical model of power-law temperature time scaling

The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors

Contact Info

Product

Resources

About