Science JournalFeeds

Plastic grains are found to form fractal patterns in elastic-hardening plastic materials in two dimensions, made of locally isotropic grains with random fluctuations in plastic limits or elastic/plastic moduli. The spatial assignment of randomness follows a strict-white-noise random field on a square lattice aggregate of square-shaped grains, whereby the flow rule of each grain follows associated plasticity. Square-shaped domains (comprising 256x256 grains) are loaded through either one of three macroscopically uniform boundary conditions admitted by the Hill–Mandel condition. Following an evolution of a set of grains that have become plastic, we find that it is monotonically plane filling with an increasing macroscopic load. The set’s fractal dimension increases from 0 to 2, with the response under kinematic loading being stiffer than that under mixed-orthogonal loading, which, in turn, is stiffer than the traction controlled one. All these responses display smooth transitions but, as the randomness decreases to zero, they turn into the sharp response of an idealized homogeneous material. The randomness in yield limits has a stronger effect than that in elastic/plastic moduli. On the practical side, the curves of fractal dimension versus applied stress—which indeed display a universal character for a range of different materials—offer a simple method of assessing the inelastic state of the material. A qualitative explanation of the morphogenesis of fractal patterns is given from the standpoint of a correlated percolation on a Markov field on a graph network of [...]

We consider the linear stability of two-dimensional nonlinear magnetohydrodynamic basic states to long-wavelength three-dimensional perturbations. Following Hughes & Proctor (Hughes & Proctor 2009 Proc. R. Soc. A 465, 1599–1616 (doi:10.1098/rspa.2008.0493)), the two-dimensional basic states are obtained from a specific forcing function in the presence of an initially uniform mean field of strength
. By extending to the nonlinear regime the kinematic analysis of Roberts (Roberts 1970 Phil. Trans. R. Soc. Lond. A 266, 535–558 (doi:10.1098/rsta.1970.0011)), we show that it is possible to predict the growth rate of these perturbations by applying mean-field theory to both the momentum and the induction equations. If
, these equations decouple and large-scale magnetic and velocity perturbations may grow via the kinematic -effect and the anisotropic kinetic alpha instability, respectively. However, if
, the momentum and induction equations are coupled by the Lorentz force; in this case, we show that four transport tensors are now necessary to determine the growth rate of the perturbations. We illustrate these situations by numerical examples; in particular, we show that a mean-field description of the nonlinear regime based solely on a quenched coefficient is [...]

Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time t needs
computations owing to the repeated evaluation of integrals over intervals that grow like t. Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled infinite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives (Singh & Chatterjee 2006 Nonlinear Dyn. 45, 183–206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with
computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e.g. in stability [...]

The steady axisymmetric behaviour of a relatively small bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations, using a Galerkin finite-element method with boundary-fitted mesh, for wide ranges of capillary number Ca and Reynolds number Re. Here a bubble is considered relatively small when its volume-equivalent radius is less than that of the tube. At small values of Re, the velocity of a bubble increases with bubble size for large values of Ca but decreases with bubble size for small values of Ca. At large values of Re, however, a bubble of large size appears to move at a slower velocity for any given value of Ca. When Re is large (e.g. Re = 100) and Ca > 0.1, a bubble of radius greater than half of the tube radius moves at a velocity that seems to be independent of bubble size. The strong inertial effect at large Re makes a small bubble of radius greater than a quarter of the tube radius to deform into a noticeable oblate shape as Ca increases from very small value, and then to be elongated into a bullet shape with further increasing Ca after Ca reaches an intermediate value. Even very small bubbles (e.g. of radius equal to one-tenth of the tube radius) can still be significantly deformed provided that the value of Ca is adequately large. Despite significant shape deformations that may still occur, bubbles of radius less than a quarter of that of the tube almost always move at the same velocity as that of the local liquid flow at the tube centreline (i.e. twice that of the average liquid velocity), regardless the values of Re and Ca. This fact suggests that very small bubbles are basically carried by the local liquid [...]

This paper proposes a super resolution near-field radio frequency focusing device consisting of a thin planar layer of a particular ferrite characterized by negative permeability. Radiation focusing is investigated and it is established that the resulting non-structured lens is characterized by a resolving power 2–3 times the lens thickness, regardless of the wavelength. The resulting near field lens can be used as a magnetic field device for imaging inside non-magnetic [...]

Explicit expressions of Green’s function and its derivative for three-dimensional infinite solids are presented in this paper. The medium is allowed to exhibit a fully magneto-electro-elastic (MEE) coupling and general anisotropic behaviour. In particular, new explicit expressions for the first-order derivative of Green’s function are proposed. The derivation combines extended Stroh formalism, Radon transform and Cauchy’s residue theory. In order to cover mathematical degenerate and non-degenerate materials in the Stroh formalism context, a multiple residue scheme is performed. Expressions are explicit in terms of Stroh’s eigenvalues, this being a feature of special interest in numerical applications such as boundary element methods. As a particular case, simplifications for MEE materials with transversely isotropic symmetry are derived. Details on the implementation and numerical stability of the proposed solutions for degenerate cases are [...]

In the framework of classical plasticity, even when limit multipliers and collapse mechanisms associated with different loads independently acting on a solid or structure are known, not much can be inferred on the limit multiplier of the combined loading. Frame structures under the action of dead loads and seismic forces, soil–foundation interaction problems, tunnels under a variety of loads, deepwater pipelines subject to bending and pressure constitute only a few selected examples for which some sort of superposition rule, as well as bounding techniques, would be extremely useful. The present paper introduces a set of theorems for bounding limit multipliers for combined loads. In particular, ranging from a minimum knowledge about the critical state under a particular loading to a reasonable guess of the kinematics of the problem under combined loads, more and more refined bounds for the overall limit multiplier are derived. The results, to the best of the authors’ knowledge, are novel and a few examples showing their practical value are presented and [...]

This paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental. The method involves replacing each transcendental term in the dispersion relation by a finite-product polynomial. The finite products chosen must be consistent with the low-frequency, low-wavenumber limit; but the method is nevertheless accurate up to high frequencies and high wavenumbers. Full details of the method are presented for a non-trivial example, that of anti-symmetric elastic waves in a layer; the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers. It is proved that the method is accurate because certain gamma-function expressions, which occur as ratios of transcendental terms to finite products, largely cancel out, nullifying Runge’s phenomenon. The polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are ideal for numerical computation. The method is potentially useful for a very wide range of problems in wave theory and stability [...]

Development and employment of dynamic elements can result in substantial gain in solution convergence for vibration analysis, when compared with the usual finite element discretization. Efficient solution of the associated quadratic matrix eigenproblem is crucial in achieving a superior and relatively economical solution. This paper first describes a novel eigensolution of the quadratic matrix equation by a progressive simultaneous iteration method. Free vibration analysis results of a rectangular prestressed membrane are next presented in detail that provides an assessment of relative solution convergence and computing expenses of the two idealization [...]

In the present paper, we investigate a model for propagating progressive waves associated with the voids within the framework of a linear theory of porous media. Owing to the use of lighter materials in modern buildings and noise concerns in the environment, such models for progressive waves are of much interest to the building industry. The analysis of such waves is also of interest in acoustic microscopy where the identification of material defects is of paramount importance to the industry and medicine. Our analysis is based on the strong ellipticity of the poroelastic materials. We illustrate the model of progressive wave propagation for isotropic and transversely isotropic porous materials. We also study the propagation of harmonic plane waves in porous materials including the thermal [...]