Fractional Generalizations of Gradient Mechanics

This short chapter provides a fractional generalization of gradient mechanics, an approach (originally advanced by the author in the mid 80s) that has gained world-wide attention in the last decades due to its capability of modeling pattern forming instabilities and size effects in materials, as well as eliminating undesired elastic singularities. It is based on the incorporation of higher-order gradients (in the form of Laplacians) in the classical constitutive equations multiplied by appropriate internal lengths accounting for the geometry/topology of underlying micro/nano structures. This review will focus on the fractional generalization of the gradient elasticity equations (GradEla) an extension of classical elasticity to incorporate the Laplacian of Hookean stress by replacing the standard Laplacian by its fractional counterpart. On introducing the resulting fractional constitutive equation into the classical static equilibrium equation for the stress, a fractional differential equation is obtained whose fundamental solutions are derived by using the Greens function procedure. As an example, Kelvins problem is analyzed within the aforementioned setting. Then, an extension to consider constitutive equations for a restrictive class of nonlinear elastic deformations and deformation theory of plasticity is pursued. Finally, the methodology is applied for extending the authors higher-order diffusion theory from the integer to the fractional case.Comment: arXiv admin note: text overlap with arXiv:1808.0324

Towards Fractional Gradient Elasticity

An extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe power-law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one-dimension. The second involves the Riesz fractional derivative in three-dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case it is shown that stress equilibrium in a Caputo elastic bar requires the existence of a non-zero internal body force to equilibrate it. In the second case, it is shown that in a Riesz type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.Comment: 10 pages, LaTe

A Rich Example of Geometrically Induced Nonlinearity: From Rotobreathers and Kinks to Moving Localized Modes and Resonant Energy Transfer

We present an experimentally realizable, simple mechanical system with linear interactions whose geometric nature leads to nontrivial, nonlinear dynamical equations. The equations of motion are derived and their ground state structures are analyzed. Selective ``static'' features of the model are examined in the context of nonlinear waves including rotobreathers and kink-like solitary waves. We also explore ``dynamic'' features of the model concerning the resonant transfer of energy and the role of moving intrinsic localized modes in the process

Stochastically forced dislocation density distribution in plastic deformation

The dynamical evolution of dislocations in plastically deformed metals is controlled by both deterministic factors arising out of applied loads and stochastic effects appearing due to fluctuations of internal stress. Such type of stochastic dislocation processes and the associated spatially inhomogeneous modes lead to randomness in the observed deformation structure. Previous studies have analyzed the role of randomness in such textural evolution but none of these models have considered the impact of a finite decay time (all previous models assumed instantaneous relaxation which is "unphysical") of the stochastic perturbations in the overall dynamics of the system. The present article bridges this knowledge gap by introducing a colored noise in the form of an Ornstein-Uhlenbeck noise in the analysis of a class of linear and nonlinear Wiener and Ornstein-Uhlenbeck processes that these structural dislocation dynamics could be mapped on to. Based on an analysis of the relevant Fokker-Planck model, our results show that linear Wiener processes remain unaffected by the second time scale in the problem but all nonlinear processes, both Wiener type and Ornstein-Uhlenbeck type, scale as a function of the noise decay time τ. The results are expected to ramify existing experimental observations and inspire new numerical and laboratory tests to gain further insight into the competition between deterministic and random effects in modeling plastically deformed samples

Thermoplastic deformation of silicon surfaces induced by ultrashort pulsed lasers in submelting conditions

A hybrid 2D theoretical model is presented to describe thermoplastic deformation effects on silicon surfaces induced by single and multiple ultrashort pulsed laser irradiation in submelting conditions. An approximation of the Boltzmann transport equation is adopted to describe the laser irradiation process. The evolution of the induced deformation field is described initially by adopting the differential equations of dynamic thermoelasticity while the onset of plastic yielding is described by the von Mise's stress. Details of the resulting picometre sized crater, produced by irradiation with a single pulse, are then discussed as a function of the imposed conditions and thresholds for the onset of plasticity are computed. Irradiation with multiple pulses leads to ripple formation of nanometre size that originates from the interference of the incident and a surface scattered wave. It is suggested that ultrafast laser induced surface modification in semiconductors is feasible in submelting conditions, and it may act as a precursor of the incubation effects observed at multiple pulse irradiation of materials surfaces.Comment: To appear in the Journal of Applied Physic

Facile Method to Prepare Superhydrophobic and Water Repellent Cellulosic Paper

Silica nanoparticles (7 nm) were dispersed in solutions of a silane/siloxane mixture. The dispersions were applied, by brush, on four types of paper: (i) modern, unprinted (blank) paper, (ii) modern paper where a text was printed using a common laser jet printer, (iii) a handmade paper sheet detached from an old book, and (iv) Japanese tissue paper. It is shown that superhydrophobicity and water repellency were achieved on the surface of the deposited films, when high particle concentrations were used (≥1% w/v), corresponding to high static (θS ≈ 162°) and low tilt (θt < 3°) contact angles. To interpret these results, scanning electron microscopy (SEM) was employed to observe the surface morphologies of the siloxane-nanoparticle films. Static contact angles, measured on surfaces that were prepared from dilute dispersions (particle concentration <1% w/v), increased with particle concentration and attained a maximum value (162°) which corresponds to superhydrophobicity. Increasing further the particle concentration did not have any effect on θS. Colourimetric measurements showed that the superhydrophobic films had negligible effects on the aesthetic appearance of the treated papers. Furthermore, it is shown that the superhydrophobic character of the siloxane-nanoparticle films was stable over a wide range of pH

Two-temperature dual-phase-lags theory in a thermoelastic solid half-space due to an inclined load

This article addresses the thermoelastic interaction due to inclined load on a homogeneous isotropic half-space in context of two-temperature generalized theory of thermoelasticity with dual-phase-lags. It is assumed that the inclined load is a linear combination of both normal and tangential loads. The governing equations are solved by using the normal mode analysis. The variations of the displacement, stress, conductive temperature, and thermodynamic temperature distributions with the horizontal distance have been shown graphically. Results of some earlier workers have also been deduced from the present investigation as special cases. Some comparisons are graphically presented to estimate the effects of the two-temperature parameter, the dual-phase-lags parameters and the inclination angle. It is noticed that there is a significant difference in the values of the studied fields for different value of the angle of inclination. The method presented here maybe applicable to a wide range of problems in thermodynamics and thermoelasticity