No abstract

No abstract

No abstract

Fractional calculus is the mathematical subject dealing with integrals and derivatives of noninteger order. Although its age approaches that of classical calculus, its applications in mechanics are relatively recent and mainly related to fractional damping. Investigations using fractional spatial derivatives are even newer. In the present paper spatial fractional calculus is exploited to investigate a material whose nonlocal stress is defined as the fractional integral of the strain field. The developed fractional nonlocal elastic model is compared with standard integral nonlocal elasticity, which dates back to Eringen's works. Analogies and differences are highlighted. The long tails of the power law kernel of fractional integrals make the mechanical behaviour of fractional nonlocal elastic materials peculiar. Peculiar are also the power law size effects yielded by the anomalous physical dimension of fractional operators. Furthermore we prove that the fractional nonlocal elastic medium can be seen as the continuum limit of a lattice model whose points are connected by three levels of springs with stiffness decaying with the power law of the distance between the connected points. Interestingly, interactions between bulk and surface material points are taken distinctly into account by the fractional model. Finally, the fractional differential equation in terms of the displacement function along with the proper static and kinematic boundary conditions are derived and solved implementing a suitable numerical algorithm. Applications to some example problems conclude the paper.

The brittle crack initiation from a circular hole in an infinite slab under uniaxial remote tensile load is investigated. The analysis consists of two parts. The former is focused on the difference between symmetric and asymmetric crack propagation. Different criteria in the framework of the Theory of Critical Distances are implemented, and the potentiality of coupled Finite Fracture Mechanics (FFM) approaches is highlighted from a theoretical point of view. The latter presents the experimental results obtained by carrying out ad‐hoc tensile tests on polymethyl‐methacrylate (PMMA) and general‐purpose polystyrene (GPPS) notched samples and the related FFM investigation. It is shown that FFM predictions are in very good agreement with the experimental results for both tested materials.

In this paper, the wave propagation in one-dimensional elastic continua, characterized by nonlocal interactions, is investigated by means of a fractional calculus approach. Derivatives of a non-integer order 1 < α < 2 with respect to the spatial variable are involved in the governing equation.

Key words Fractal media, local fractional calculus, long-range interactions, Marchaud fractional derivative. MSC (2000) 03E25Two fractional calculus approaches in the framework of continuum mechanics are revisited and compared. The former is a local approach, which has been proposed to investigate the behaviour of fractal media. The latter is a non-local approach, according to which long-range interactions between material particles are opportunely modelled in the equilibrium equations. Analogies and differences between the two models are outlined.

In this paper, a fractional approach to describe the diffusion process in fractal media is put forward. After introducing anomalous diffusion quantities, the continuity and constitutive equations are derived by means of local fractional calculus, and the problem is formulated both in the steady-state regime and in the transient regime. Eventually, a simple heat conduction problem in the steady-state regime is solved analytically.