134 research outputs found
Statistical Analysis for Long Term Correlations in the Stress Time Series of Jerky Flow
Stress time series from the PLC effect typically exhibit stick-slips of
upload and download type. These data contain strong short-term correlations of
a nonlinear type. We investigate whether there are also long term correlations,
i.e. the successive up-down patterns are generated by a deterministic
mechanism. A statistical test is conducted for the null hypothesis that the
sequence of the up-down patterns is totally random. The test is constructed by
means of surrogate data, suitably generated to represent the null hypothesis.
Linear and nonlinear estimates are used as test statistics, namely
autocorrelation, mutual information and Lyapunov exponents, which are found to
have proper performance for the test. The test is then applied to three stress
time series under different experimental conditions. Rejections are obtained
for one of them and not with all statistics. From the overall results we cannot
conclude that the underlying mechanism to the PLC effect has long memory.Comment: 42 pages, 6 figures, to appear in the International Journal of
Mechanical Behavio
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
Capturing wave dispersion in heterogeneous and microstructured materials through a three-length-scale gradient elasticity formulation
Abstract
Long-range interactions occurring in heterogeneous materials are responsible for the dispersive character of wave propagation. To capture these experimental phenomena without resorting to molecular and/or atomistic models, generalized continuum theories can be conveniently used. In this framework, this paper presents a three-length-scale gradient elasticity formulation whereby the standard equations of elasticity are enhanced with one additional strain gradient and two additional inertia gradients to describe wave dispersion in microstructured materials. It is well known that continualization of lattice systems with distributed microstructure leads to gradient models. Building on these insights, the proposed gradient formulation is derived by continualization of the response of a non-local lattice model with two-neighbor interactions. A similar model was previously proposed in the literature for a two-length-scale gradient formulation, but it did not include all the terms of the expansions that contributed to the response at the same order. By correcting these inconsistencies, the three-length-scale parameters can be linked to geometrical and mechanical properties of the material microstructure. Finally, the ability of the gradient formulation to simulate wave dispersion in a broad range of materials (aluminum, bismuth, nickel, concrete, mortar) is scrutinized against experimental observations
Applications of Regime-switching in the Nonlinear Double-Diffusivity (D-D) Model
The linear double-diffusivity (D-D) model of Aifantis, comprising two coupled Fick-type partial differential equations and a mass exchange term connecting the diffusivities, is a paradigm in modeling mass transport in inhomogeneous media, e.g. fissures or fractures. Uncoupling of these equations led to a higher order Partial Differential Equation (PDE) that reproduced the non-classical transport terms, analyzed independently through Barenblatt’s pseudoparabolic equation and the Cahn-Hilliard spinodal decomposition equation. In the present article, we study transport in a nonlinearly coupled D-D model and determine the regime-switching of the associated diffusive processes using a revised formulation of the celebrated Lux method that combines forward Fourier transform with a Laplace transform followed by an Inverse Fourier transform of the governing reaction-diffusion (R-D) equations. This new formulation has key application possibilities in a wide range of non-equilibrium biological and financial systems by approximating closed-form analytical solutions of nonlinear models
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