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

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

Cytosolic Sequestration of Prep1 Influences Early Stages of T Cell Development

Objective: Prep1 and Pbx2 are the main homeodomain DNA-binding proteins of the TALE (three amino acid loop extension) family expressed in the thymus. We previously reported reduced Pbx2 expression and defective thymocyte maturation in Prep1 hypomorphic mice. To further investigate the role of this homeodomain DNA-binding protein in T cell development, we generated transgenic mice expressing the N-terminal fragment of Pbx1 (Pbx1NT) under the control of the Lck proximal promoter. Principal Findings: Pbx1NT causes Prep1 cytosolic sequestration, abolishes Prep1-dependent DNA-binding activity and results in reduced Pbx2 expression in developing thymocytes. Transgenic thymi reveal increased numbers of CD4 2 CD8 2 CD44 2 (DN3 and DN4) thymocytes, due to a higher frequency of DN2 and DN4 Pbx1NT thymocytes in the S phase. Transgenic thymocytes however do not accumulate at later stages, as revealed by a normal representation of CD4/CD8 double positive and single positive thymocytes, due to a higher rate of apoptotic cell death of DN4 Pbx1NT thymocytes. Conclusion: The results obtained by genetic (Prep1 hypomorphic) and functional (Pbx1NT transgenic) inactivation of Prep

High order amplitude equation for steps on creep curve

We consider a model proposed by one of the authors for a type of plastic instability found in creep experiments which reproduces a number of experimentally observed features. The model consists of three coupled non-linear differential equations describing the evolution of three types of dislocations. The transition to the instability has been shown to be via Hopf bifurcation leading to limit cycle solutions with respect to physically relevant drive parameters. Here we use reductive perturbative method to extract an amplitude equation of up to seventh order to obtain an approximate analytic expression for the order parameter. The analysis also enables us to obtain the bifurcation (phase) diagram of the instability. We find that while supercritical bifurcation dominates the major part of the instability region, subcritical bifurcation gradually takes over at one end of the region. These results are compared with the known experimental results. Approximate analytic expressions for the limit cycles for different types of bifurcations are shown to agree with their corresponding numerical solutions of the equations describing the model. The analysis also shows that high order nonlinearities are important in the problem. This approach further allows us to map the theoretical parameters to the experimentally observed macroscopic quantities.Comment: LaTex file and eps figures; Communicated to Phys. Rev.