Fractional-order uniaxial visco-elasto-plastic models for structural analysis

Abstract

Abstract. Three fractional-order models for uniaxial large strains and rate-dependent plastic behavior of materials in structural analysis are proposed. Our approach is amenable to modeling nonlinear and more sophisticated effects namely visco-elasto-plastic response of materials. This approach seamlessly interpolates between the standard elasto-plastic and visco-plastic models in plasticity, taking into account the history-dependency of the accumulated plastic strain to specify the state of stress. To this end, we propose three models namely i) viscoelasto-plastic with linear hardening plastic model, ii) elastoviscoplastic model, and iii) visco-elasto-plastic model, which combines the first the second models. We employ a fractional-order constitutive law that relates the Kirchhoff stress to its Caputo time-fractional derivative of order α ∈ (0, 1]. When α → 0 the standard elasto-plastic (rate-independent) model and when α = 1, the corresponding visco-plastic model is recovered. Since the material behavior is path-dependent the evolution of the plastic strain is achieved by fractional-order time integration of the plastic strain rate with respect to time. The strain rate is then obtained by means of the corresponding plastic multiplier and deriving proper consistency conditions. Finally, we develop a so called 386

A Petrov–Galerkin spectral element method for fractional elliptic problems

“…Since the grid is non-uniform, equations (33)-(39) do not hold anymore. Thus, using (48), we obtain (1) M (4) M (5) M (3) M (6) M (7) M (8) M (9) M (10) M (11) Ŝ (6) where ∆ε = ε − e > 0, denotes the element difference between the current element ε and the e-th element. Using the same expansion as in (35), we can write (49) as…”

“…Fractional order models open up new possibilities for robust mathematical modeling of complex multi-scale problems and anomalous transport phenomena including: non-Markovian (Lévy flights) processes in turbulent flows [2,3], non-Newtonian fluids and rheology [4], non-Brownian transport phenomena in porous and disordered materials [5,6], non-Gaussian processes in multi-scale complex fluids and multi-phase applications [7], visco-elastic bio-tissues, and visco-elastoplastic materials [6,8,9].…”

A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

“…Over the past decades, anomalous transport has been observed and investigated in a wide range of applications such as turbulence [11,21,43,49], porous media [4,7,16,59,66,67], geoscience [5], bioscience [45][46][47][48], and viscoelastic material [20,40,54,55,60]. The underlying anomalous features, manifesting in memory effects, non-local interactions, powerlaw distributions, sharp peaks, and self-similar structures, can be well described by fractional partial differential equations (FPDEs) [27,41,42,44].…”

On Dynamic Systems in the Frame of Singular Function Dependent Kernel Fractional Derivatives

“…Due to the spatial non-locality of the operators with fractional orders, these operators have become significant tools that enable researchers to bring new aspects to the dynamics of non-local complex systems [1][2][3][4][5][6][7][8]. In addition, there has been an intensive interest in dealing with differential equations embodying derivatives with fractional order from many point of views including the qualitative, theoretical and numerical aspects [1][2][3] and studying the existence and uniqueness of solutions of differential equations in the frame of the traditional fractional derivative has been tackled in many works (see [9] and the references therein).…”