Finite Fracture Mechanics at elastic interfaces

In the present paper we provide a method to determine the load causing delamination along an interface in a composite structure. The method is based on the elastic interface model, according to which the interface is equivalent to a bed of linear elastic springs, and on Finite Fracture Mechanics, a crack propagation criterion recently proposed for homogeneous structures. The procedure outlined is general. Details are given for the pull–push shear test. For such geometry, the failure load is obtained and compared with the estimates provided by stress concentration analysis and Linear Elastic Fracture Mechanics. It is seen that Finite Fracture Mechanics provides intermediate values. Furthermore, it is shown that the predictions provided by Finite Fracture Mechanics are almost coincident with the ones provided by the Cohesive Crack Model. As far as we are concerned with the determination of the failure load, the advantage of using Finite Fracture Mechanics with respect to the Cohesive Crack Model is evident, since a troublesome analysis of the softening taking place in the fracture process zone is not necessary. A final comparison with classical fracture criteria based on critical distances, such as the average stress criterion, concludes the paper.     

Size effect upon grained materials tensile strength: The increase of the statistical dispersion at the smaller scales

The present paper provides a statistical model to the size effect on grained materials tensile strength; it is based on an Extreme Value Theory approach. Since the weakest link in grained materials is usually represented by the interface between the matrix and the aggregates, it is assumed that the flaw distribution can be represented by the aggregate distribution, expressed as a probability density function (pdf) of the grain diameters. Under the hypothesis that the strength of the material depends on the largest flaw, the tensile strength is computed as a function of the specimen size. In this way, two remarkable results are obtained: (i) a size effect for the average tensile strength that substantially agrees with the multifractal scaling law (MFSL) proposed by the first author and (ii) an increase of scatter of the tensile strength values when testing small specimens. Both these trends are confirmed by experimental data available in the literature.     

Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach

In the present paper, the wave propagation in one-dimensional elastic continua, characterized by nonlocal interactions modeled by fractional calculus, is investigated. Spatial derivatives of non-integer order 1 < α < 2 are involved in the governing equation, which is solved by fractional finite differences. The influence of long-range interactions is then analyzed as α varies: the resonant frequencies and the standing waves of a nonlocal bar are evaluated and the deviations from the classical (local) ones, recovered by imposing α = 2, are discussed.     

On the most dangerous V-notch

The determination of the failure load for brittle or quasi-brittle specimens containing a re-entrant corner has been faced by several authors, whose approaches are available in the Scientific Literature. However, up to now, little attention has been paid to the presence of a minimum, i.e. an angle at which the critical load attains its minimum value. Even if the minimum was detected in several experiments, it was not highlighted or it was considered as a mere consequence of the scattering of experimental data. Restricting the analysis to a sharp V-notched infinite slab under a remote tensile load, the problem is fully investigated in this paper. It is shown that a minimum, more or less pronounced according to the brittleness number, is always present. It means that the edge crack is not the most dangerous configuration, although the notch opening angle providing the minimum failure load tends to vanish for large notch depths as well as for very brittle materials.     

Diffusion problems on fractional nonlocal media

In this paper, the nonlocal diffusion in one-dimensional continua is investigated by means of a fractional calculus approach. The problem is set on finite spatial domains and it is faced numerically by means of fractional finite differences, both for what concerns the transient and the steady-state regimes. Nonlinear deviations from classical solutions are observed. Furthermore, it is shown that fractional operators possess a clear physical-mechanical meaning, representing conductors, whose conductance decays as a power-law of the distance, connecting non-adjacent points of the body.     

The problem of the critical angle for edge and center V-notched structures

In the present paper, the problem of detecting the critical notch angle, i.e. the angle providing the minimum failure load, for brittle or quasi-brittle structures containing either edge or center V-notches is investigated. The expression of the generalized fracture toughness is obtained according to Finite Fracture Mechanics. It is shown that a critical angle is always present: its value depends, through the brittleness number, on both material and geometric characteristics. Thus, the crack is not the most dangerous configuration. The result is supported by experimental results presented in the Literature.     

Nonlocal elasticity: an approach based on fractional calculus

Fractional calculus is the mathematical subject dealing with integrals and derivatives of non-integer 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.     

A fractional calculus approach to nonlocal elasticity

If the attenuation function of strain is expressed as a power law, the formalism of fractional calculus may be used to handle Eringen nonlocal elastic model. Aim of the present paper is to provide a mechanical interpretation to this nonlocal fractional elastic model by showing that it is equivalent to a discrete, point-spring model. A one-dimensional geometry is considered; the static, kinematic and constitutive equations are presented and the governing fractional differential equation highlighted. Two numerical procedures to solve the fractional equation are finally implemented and applied to study the strain field in a finite bar under given edge displacements.     

Non-local criteria for the borehole problem: Gradient Elasticity versus Finite Fracture Mechanics

Two nonlocal approaches are applied to the borehole geometry, herein simply modelled as a circular hole in an infinite elastic medium, subjected to remote biaxial loading and/or internal pressure. The former approach lies within the framework of Gradient Elasticity (GE). Its characteristic is nonlocal in the elastic material behaviour and local in the failure criterion, hence simply related to the stress concentration factor. The latter approach is the Finite Fracture Mechanics (FFM), a well-consolidated model within the framework of brittle fracture. Its characteristic is local in the elastic material behaviour and non-local in the fracture criterion, since crack onset occurs when two (stress and energy) conditions in front of the stress concentration point are simultaneously met. Although the two approaches have a completely different origin, they present some similarities, both involving a characteristic length. Notably, they lead to almost identical critical load predictions as far as the two internal lengths are properly related. A comparison with experimental data available in the literature is also provided.