On fractional order generalized thermoelasticity with micromodeling

This paper presents the theory of fractional order generalized thermoelasticity with microstructure modeling for porous elastic bodies and synthetic materials containing microscopic components and microcracks. Built upon the micromorphic theory, the theory of fractional order generalized micromorphic thermoelasticity (FOGTE_mm) is firstly established by introducing the fractional integral operator. To generalize the FOGTE_mm theory, the general forms of the extended thermoelasticity, temperature rate dependent thermoelasticity, thermoelasticity without energy dissipation, thermoelasticity with energy dissipation, and dual-phase-lag thermoelasticity are introduced during the formulation. Secondly, the uniqueness theorem for FOGTE_mm is established. Finally, a generalized variational principle of FOGTE_mm is developed by using the semi-inverse method. For reference, the theories of fractional order generalized micropolar thermoelasticity (FOGTE_mp) and microstretch thermoelasticity (FOGTE_ms) and the corresponding generalized variational theorems are also presented.     

Dynamics of semigroup actions of linear fractional transformations

Positive continued fractions are viewed as the orbit of 1 under the action of the semigroup of functions generated by r(x) = 1/x and s(x) = x + 1. The fixed points of the elements of this semigroup contain purely periodic continued fractions (which are the reduced surds by Lagrange's theorem on continued fractions). In this article, we study the fixed points of the elements of semigroups generated by general pairs of linear fractional transformations and show that topological transitivity of a non-commutative semigroup action implies that the fixed point set is dense. We also prove a similar result for two-generator semigroups of affine transformations on the real line.     

Lattice model of fractional gradient and integral elasticity: Long-range interaction of Grünwald–Letnikov–Riesz type

Lattice model with long-range interaction of power-law type that is connected with difference of non-integer order is suggested. The continuous limit maps the equations of motion of lattice particles into continuum equations with fractional Grünwald–Letnikov–Riesz derivatives. The suggested continuum equations describe fractional generalizations of the gradient and integral elasticity. The proposed type of long-range interaction allows us to have united approach to describe of lattice models for the fractional gradient and fractional integral elasticity. Additional important advantages of this approach are the following: (1) It is possible to use this model of long-range interaction in numerical simulations since this type of interactions and the Grünwald–Letnikov derivatives are defined by generalized finite difference; (2) The suggested model of long-range interaction leads to an equation containing the sum of the Grünwald–Letnikov derivatives, which is equal the Riesz’s derivative. This fact allows us to get particular analytical solutions of fractional elasticity equations.     

The generalized Hamilton’s principle for a non-material volume

Fundamental principles of mechanics were primarily conceived for constant mass systems. Since the pioneering works of Meshcherskii, efforts have been made in order to elaborate an adequate mathematical formalism for variable mass systems. This is a current research field in theoretical mechanics. In this paper, attention is focused on the derivation of the generalized Hamilton’s principle for a non-material volume. First studies on the subject go back at least four decades with the article of McIver (J Eng Math 7(3):249–261, 1973). However, it is curious to note that the extended form of Hamilton’s principle that is derived by McIver does not recover the Lagrange’s equation for a non-material volume which is demonstrated by Irschik and Holl (Acta Mech 153(3–4):231–248, 2002). This does suggest additional theoretical investigations. In the upcoming discussion, Reynolds’ transport theorem is consistently considered regarding the original form of the principle of virtual work, and so the generalized Hamilton’s principle for a non-material volume is properly derived. It is finally shown that the generalization of Hamilton’s principle that is here proposed is in harmony with the Lagrange’s equation which is demonstrated by Irschik and Holl.     

Optical solitons of some fractional differential equations in nonlinear optics

This paper deals with two different forms of the conformable fractional Benjamin–Bona–Mahony (BBM) equations by an analytical method. These physical models have important applications for describing the propagation of optical pulses in non-linear media. The conformable fractional symmetric BBM equation and the conformable time fractional Equal-width (EW) equation are considered. The extended Jacobi’s elliptic function expansion scheme are used to extract explicit solitons.     

Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators endowed with fractional derivatives elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes which rely on a discrete version of the Chapman–Kolmogorov (C–K) equation. This is accomplished by circumventing the solution of the associated Euler–Lagrange equation ordinarily used in the path integral based procedures. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.     

Total energy framework, finite elements, and discretization via Hamilton’s law of varying action

Within the total energy framework which we introduce here for the first time (in contrast to Lagrangian or Hamiltonian mechanics framework), we provide an alternative and have developed in this paper a general numerical discretization for continuum-elastodynamics directly stemming from Hamilton’s law of varying action (HLVA) involving a measurable built-in scalar function, namely, Total Energy ${[{\mathcal{E}}\left({{\boldsymbol{q}},\dot{{\boldsymbol{q}}}}\right): TQ\rightarrow {\mathbb{R}}]}$ . The Total Energy we use herein for enabling the space discretization is defined as the kinetic energy plus the potential energy for N-body systems, and the kinetic energy plus the total potential energy for continuum-body systems. It thereby provides a direct measure and sound physical interpretation naturally, while enabling this framework to permit general numerical discretizations such as with finite elements. In the variational formulation proposed here, we place particular emphasis upon the notion that the scalar function which represents the autonomous total energy of the continuum/N-body dynamical systems can be a crucial mathematical function and physical quantity which is a constant of motion in conservative systems. In addition, we prove that the autonomous total energy possesses the three invariant properties and can be viewed as the so-called total energy version of Noether’s theorem; therefore, the autonomous total energy has time/translational/rotational symmetries for the continuum/N-body dynamical systems. The proposed concepts directly emanating from HLVA inherently involving the scalar function, namely, total energy: (i) can be shown to yield the same governing mathematical model equations of motion that are continuous in space and time together with the natural boundary conditions just as Hamilton’s principle (HP) is routinely used to derive such equations, but without obvious inconsistency via such a principle as explained in the paper; (ii) explain naturally the Bubnov–Galerkin weighted-residual form that is customarily employed for discretization for both space and time, and alternately, (iii) circumvent relying on traditional practices of conducting numerical discretizations starting either from the balance of linear momentum (Newton’s second law) involving Cauchy’s equations of motion (governing equations) arising from continuum mechanics or via (i) and (ii) above if one chooses this option, and instead provides new avenues of discretization for continuum-dynamical systems. The present developments naturally embody the weak form in space and time that can be described by a discrete Total Energy Differential Operator (TEDO). Thereby, a novel yet simple, space-discrete Total Energy formulation proposed here only needs to employ the discrete TEDO which provides new avenues and directly yields the semi-discrete ordinary differential equations in time which can be readily shown to preserve the same physical attributes as the continuous systems for continuum-dynamical applications unlike traditional practices. The modeling of complicated structural dynamical systems such as Euler-Bernoulli beams and Reissner–Mindlin plates is particularly shown here for illustration.     

Evolution equation of moving defects: dislocations and inclusions

Evolution equations, or equations of motion, of moving defects are the balance of the “driving forces”, in the presence of external loading. The “driving forces” are defined as the configurational forces on the basis of Noether’s theorem, which governs the invariance of the variation of the Lagrangean of the mechanical system under infinitesimal transformations. For infinitesimal translations, the ensuing dynamic J integral equals the change in the Lagrangean if and only if the linear momentum is preserved. Dislocations and inclusions are “defects” that possess self-stresses, and the total driving force for these defects consists only of two terms, one expressing the “ self-force” due to the self-stresses, and the other the effect of the external loading on the change of configuration (Peach–Koehler force). For a spherically expanding (including inertia effects) Eshelby (constrained) inclusion with dilatational eigenstrain (or transformation strain) in general subsonic motion, the dynamic J integral, which equals the energy-release rate, was calculated. By a limiting process as the radius tends to infinity, the driving force (energy-release rate) of a moving half-space plane inclusion boundary was obtained which is the rate of the mechanical work required to create an incremental region of eigenstrain (or transformation strain) of a dynamic phase boundary. The total driving force (due to external loading and due to self-forces) must be equal to zero, in the absence of dissipation, and the evolution equation for a plane boundary with eigenstrain is presented. The equation applied to many strips of eigenstrain provides a system to solve for the position/ evolution of strips of eigenstrain.     

Finite element formulations using the theorem of power expended: total energy framework with a differential formulation

Traditional practices involving variational calculus have historically dominated most finite element formulations to-date, and have no doubt served as indispensable tools. Besides these practices, our recent contributions in Acta Mechanica (Har and Tamma, 2009, in press) described new alternatives and developments emanating from Hamilton’s Law of Varying Action (HLVA) as a starting point with a measurable built-in scalar function, namely, the Total Energy. The associated framework (in contrast to Lagrangian or Hamiltonian mechanics framework) demonstrated certain new advances, and also provided some fundamental insight into explaining traditional practices of finite element discretization. Here we additionally provide other advances, new directions, and viable alternatives in contrast to all these past practices which routinely employ variational concepts. In particular, focusing on elastodynamics applications, in this paper we provide for the first time finite element formulations stemming instead from a differential formulation and the theorem of power expended with a measurable built-in scalar function, namely, the Total Energy ${[\mathcal{E}({\varvec q},\dot{{\varvec q}}):TQ\to \mathbb{R}]}$ , as a starting point to capitalize on certain added advantages. The autonomous total energy has time/translational/rotational symmetries for the continuum/N-body dynamical systems. The proposed concepts: (i) can be shown to yield the same governing mathematical model equations of motion that are continuous in space and time together with the natural boundary conditions just as balance laws such as linear momentum or Hamilton’s principle are routinely used to derive such equations, but without resorting to any variational concepts, or approaches such as variational principles, (ii) explain naturally how the classical Bubnov–Galerkin weighted-residual form that is customarily employed for discretization can be readily constructed for both space and time, and alternately, (iii) circumvents relying on traditional practices of conducting numerical discretizations starting either from the balance of linear momentum (Newton’s law) involving Cauchy’s equations of motion (governing equations) arising from continuum mechanics or via (1) and (2) above, and instead provides new avenues of discretization for continuum-dynamical systems. For illustration, numerical discretizations are presented for the modeling of complicated structural dynamical systems.     

The relation between light diffraction and the fractional fourier transform

Abstract

The relation between the diffraction integral and the fractional Fourier transform is analysed in detail from the geometrical and energy points of view. It is shown that the optical matrices associated with the two transformations in the geometrical approximation, although very different, are consistent with the simple relation between the respective integral transforms. Moreover, the meaning of the complex degree of fractionality of the fractional Fourier transform is found to be related to the energy variation of the beam. Its influence on the phase-space representation of the optical beam is shown to be different compared with the diffraction through an energy variant system.