Search results for "equation"
showing 10 items of 4219 documents
Lévy-type diffusion on one-dimensional directed Cantor graphs.
2009
L\'evy-type walks with correlated jumps, induced by the topology of the medium, are studied on a class of one-dimensional deterministic graphs built from generalized Cantor and Smith-Volterra-Cantor sets. The particle performs a standard random walk on the sets but is also allowed to move ballistically throughout the empty regions. Using scaling relations and the mapping onto the electric network problem, we obtain the exact values of the scaling exponents for the asymptotic return probability, the resistivity and the mean square displacement as a function of the topological parameters of the sets. Interestingly, the systems undergoes a transition from superdiffusive to diffusive behavior a…
Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results
2011
In capturing visco-elastic behavior, experimental tests play a fundamental rule, since they allow to build up theoretical constitutive laws very useful for simulating their own behavior. The main challenge is representing the visco-elastic materials through simple models, in order to spread their use. However, the wide used models for capturing both relaxation and creep tests are combinations of simple models as Maxwell and/or Kelvin, that depend on several parameters for fitting both creep and relaxation tests. This paper, following Nutting and Gemant idea of fitting experimental data through a power law function, aims at stressing the validity of fractional model. In fact, as soon as rela…
Filter equation by fractional calculus
2014
Aim of this paper is to represent a causal filter equation for any kind of linear system in the general form L=f(t), where f(t) is the forcing function, x(t) is the output and L is a summation of fractional operators. The exact form of the operator L is obtained by using Mellin transform in complex plane.
Alternative boundary integral equations for fracture mechanics in 2D anisotropic bodies
2017
An alternative dual boundary element formulation for generally anisotropic linear elastic twodimensional bodies is presented in this contribution. The formulation is based on the decomposition of the displacement field into the sum of a vector field satisfying the anisotropic Laplace equation and the gradient of the classic Airy stress function. By suitable manipulation of the integral representation of the anisotropic Laplace equation, a set of alternative integral equations is obtained, which can be used in combination with the displacement boundary integral equation for the solution of crack problems. Such boundary integral equations have the advantage of avoiding hyper-singular integral…
Resonant Tunneling in 2D-Waveguides with Several Resonators
2021
In this chapter, we consider a two-dimensional waveguide that coincides with a strip having \(n+1\) narrows of small diameter \(\varepsilon \). All narrows are of the same shape and are spaced from each other by equal distances. Parts of the waveguide between two neighboring narrows play the role of resonators. The wave function of a free electron satisfies the Dirichlet boundary value problem for the Helmholtz equation in the waveguide. Near a simple eigenvalue of the closed resonator there are n resonant peaks of height close to 1. We let \(\varepsilon \rightarrow 0\) and obtain asymptotic formulas for the resonant energies and for the widths of the resonant peaks at their half-height. Th…
Breakdown of weak-turbulence and nonlinear wave condensation
2009
Abstract The formation of a large-scale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrodinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with small-scale fluctuations superposed (uncondensed particles), which store the information necessary for “time reversal”. We an…
Random vibration of linear and nonlinear structural systems with singular matrices: A frequency domain approach
2017
Abstract A frequency domain methodology is developed for stochastic response determination of multi-degree-of-freedom (MDOF) linear and nonlinear structural systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates/DOFs is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant DOFs can facilitate the formulation of the equations of motion in a less labor intensive manner. Specifically, relying on the generalized matrix inverse theory, a Moore-Penrose (M-P) based f…
Characterization of the dynamic behaviour of flax fibre reinforced composites using vibration measurements
2017
International audience; Experimental and numerical methods to identify the linear viscoelastic properties of flax fibre reinforced epoxy (FFRE) composite are presented in this study. The method relies on the evolution of storage modulus and loss factor as observed through the frequency response. Free-free symmetrically guided beams were excited on the dynamic range of 10 Hz to 4 kHz with a swept sine excitation focused around their first modes. A fractional derivative Zener model has been identified to predict the complex moduli. A modified ply constitutive law has been then implemented in a classical laminates theory calculation (CLT) routine.
Frequency domain identification of the fractional Kelvin-Voigt’s parameters for viscoelastic materials
2019
Abstract In this work, a new innovative method is used to identify the parameters of fractional Kelvin-Voigt constitutive equation. These parameters are: the order of fractional derivation operator, 0 ≤ α ≤ 1, the coefficient of fractional derivation operator, CV, and the stiffness of the model, KV. A particular dynamic test setup is developed to capture the experimental data. Its outputs are time histories of the excitation and excited accelerations. The investigated specimen is a polymeric cubic silicone gel material known as α-gel. Two kinds of experimental excitations are used as random frequencies and constant frequency harmonic excitations. In this study, experimental frequency respon…
Determination of dynamic properties of flax fibres reinforced laminate using vibration measurements
2017
International audience; Experimental and numerical methods to identify the linear viscoelastic properties of flax fibre reinforced polymer (FFRP) composite are presented in this study. The method relies on the evolution of storage modulus and loss factor as observed through the frequency response. Free-free symmetrically guided beams were excited in the dynamic range of 10 Hz to 4 kHz with a swept sine excitation focused around their first modes. A fractional derivative Zener model has been identified to predict the complex moduli. A modified ply constitutive law has been then implemented in a classical laminates theory calculation (CLT) routine. Overall, the Zener model fitted the experime…