Non-local fractional model of rate independent plasticity

In the paper the generalisation of classical rate independent plasticity using fractional calculus is presented. This new formulation is non-local due to properties of applied fractional differential operator during definition of kinematics. In the description small fractional strains assumption is hold together with additive decomposition of total fractional strains into elastic and plastic parts. Classical local rate independent plasticity is recovered as a special case

Fractional Calculus for Continuum Mechanics - anisotropic non-locality

In this paper the generalisation of previous author's formulation of fractional continuum mechanics to the case of anisotropic non-locality is presented. The considerations include the review of competitive formulations available in literature. The overall concept bases on the fractional deformation gradient which is non-local, as a consequence of fractional derivative definition. The main advantage of the proposed formulation is its analogical structure to the general framework of classical continuum mechanics. In this sense, it allows, to give similar physical and geometrical meaning of introduced objects

One-dimensional dispersion phenomena in terms of fractional media

It is well know that structured solids present dispersive behaviour which cannot be captured by the classical continuum mechanics theories. A canonical problem in which this can be seen is the wave propagation in the Born-Von Karman lattice. In this paper the dispersive effects in a 1D structured solid is analysed using the Fractional Continuum Mechanics (FCM) approach previously proposed by Sumelka (2013). The formulation uses the Riesz-Caputo (RC) fractional derivative and introduces two phenomenological/material parameters: 1) the size of non-local surrounding l(f), which plays the role of the lattice spacing; and 2) the order of fractional continua a, which can be devised as a fitting parameter. The results obtained with this approach have been compared with the reference dispersion curve of Born-Von Karman lattice, and the capability of the fractional model to capture the size effects present in the dynamic behaviour of discrete systems has been proved.The authors wish to acknowledge the Ministerio de Economía y Competitividad de España for the financial support, under grant number DPI2014-57989-P

Numerical Study of Dynamic Properties of Fractional Viscoplasticity Model

The fractional viscoplasticity (FV) concept combines the Perzyna type viscoplastic model and fractional calculus. This formulation includes: (i) rate-dependence; (ii) plastic anisotropy; (iii) non-normality; (iv) directional viscosity; (v) implicit/time non-locality; and (vi) explicit/stress-fractional non-locality. This paper presents a comprehensive analysis of the above mentioned FV properties, together with a detailed discussion on a general 3D numerical implementation for the explicit time integration scheme

Nonlocal vibration analysis of microstretch plates in the framework of space-fractional mechanics-theory and validation

Sumelka, Wojciech/0000-0002-8317-748X; Kiris, Ahmet/0000-0002-3687-6640In this paper, the nonlocal vibration analysis of plates modeled by generalized microstretch theory using Riesz-Caputo fractional derivative concept is presented. The frequency spectrum and the mode shapes of the microstretch plate with two clamped edges and two free edges for different values of the fractional continua order and the material length scale parameter are carried out. The three-dimensional vibration analysis is obtained by Ritz energy method. Moreover, the mode shapes and the absolute differences between classical and fractional eigenvectors for the first six macrofrequencies and additional microfrequencies between them are presented by using contour plots. The main contribution of the paper is that the nonlocal approach utilizing the fractional calculus gives better results compared to the experimental outcomes than the classical local theory. Besides, defining the nonlocality without using the nonlocal kernels is another advantage of the present approach. The overall conclusion is that the fractional mechanics establishes a new model for the nonlocal vibration analysis of microstretch plates.National Science Centre, PolandNational Science Centre, Poland [2017/27/B/ST8/00351]This work was supported by the National Science Centre, Poland, under Grant No. 2017/27/B/ST8/00351

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods

Three-dimensional analysis of nonlocal plate vibration in the framework of space-fractional mechanics & mdash; Theory and validation

Sumelka, Wojciech/0000-0002-8317-748XThis work aims to study the vibration analysis of nonlocal plates utilizing space-fractional mechanics. Riesz? Caputo fractional derivative is used to define nonlocality and the frequency spectrum and mode shapes of the plate with one clamped edge and three free edges (CFFF) are carried out for different values of the fractional continua order (a) and the length scale parameter (1). The 3-D vibration analysis is obtained by well-known Ritz energy method. The frequencies are obtained for different values of fractional material properties (a and 1). Moreover, the modes shapes and absolute differences between classical and fractional eigenvectors for the first nine frequencies are presented by using contour plots. The main contribution of the paper is that the nonlocal approach utilizing the fractional calculus gives better results compared to the experimental outcomes than the classical local theory. The overall conclusion is that fractional mechanics establishes a new model for nonlocal vibration analysis.National Science Centre, PolandNational Science Centre, Poland [2017/27/B/ST8/00351]This work is supported by the National Science Centre, Poland under Grant No. 2017/27/B/ST8/00351

Review on Stress-Fractional Plasticity Models

Fractional calculus plays an increasingly important role in mechanics research. This review investigates the progress of an interdisciplinary approach, fractional plasticity (FP), based on fractional derivative and classic plasticity since FP was proposed as an efficient alternative to modelling state-dependent nonassociativity without an additional plastic potential function. Firstly, the stress length scale (SLS) is defined to conduct fractional differential, which influences the direction and intensity of the nonassociated flow of geomaterials owing to the integral definition of the fractional operator. Based on the role of SLS, two branches of FP, respectively considering the past stress and future reference critical state can be developed. Merits and demerits of these approaches are then discussed, which leads to the definition of the third branch of FP, by considering the influences of both past and future stress states. In addition, some specific cases and potential applications of the third branch can be realised when specific SLS are adopted

Experimental Analysis of Mechanical Anisotropy of Selected Roofing Felts

In this work, four representatives of roofing felts are under consideration. Special attention is paid to the mechanical behaviour under the tensile load of the samples. The results of strength tests for the entire range of material work, from the first load to sample breaking, are shown with respect to a specific direction of sample cutting. Moreover, a unique study of the microstructure obtained with the scanning electron microscope and chemical composition determined by energy dispersive spectroscopy of the tested materials is presented. The significant mechanical material anisotropy is reported and moreover argued by microstructure characteristics. In perspective, the outcomes can give comprehensive knowledge on optimal usage of roofing felt and proper mathematical modelling