Stress Intensity Factor of Mode III Cracks in Thin Sheets

The stress field at the tip of a crack of a thin plate of elastic material that is broken due to a mode III shear tearing has a universal form with a non-universal amplitude, known as the stress intensity factor, which depends on the crack length and the boundary conditions. We present in this paper exact analytic results for this stress intensity factor, thus enriching the small number of exact results that can be obtained within Linear Elastic Fracture Mechanics (LEFM).Comment: 5 pages, 2 figure

Elastic Moduli in Nano-Size Samples of Amorphous Solids: System Size Dependence

This Letter is motivated by some recent experiments on pan-cake shaped nano-samples of metallic glass that indicate a decline in the measured shear modulus upon decreasing the sample radius. Similar measurements on crystalline samples of the same dimensions showed a much more modest change. In this Letter we offer a theory of this phenomenon; we argue that such results are generically expected for any amorphous solid, with the main effect being related to the increased contribution of surfaces with respect to bulk when the samples get smaller. We employ exact relations between the shear modulus and the eigenvalues of the system's Hessian matrix to explore the role of surface modes in affecting the elastic moduli

Conformal Theory of the Dimensions of Diffusion Limited Aggregates

We employ the recently introduced conformal iterative construction of Diffusion Limited Aggregates (DLA) to study the multifractal properties of the harmonic measure. The support of the harmonic measure is obtained from a dynamical process which is complementary to the iterative cluster growth. We use this method to establish the existence of a series of random scaling functions that yield, via the thermodynamic formalism of multifractals, the generalized dimensions D(q) of DLA for q >= 1. The scaling function is determined just by the last stages of the iterative growth process which are relevant to the complementary dynamics. Using the scaling relation D(3) = D(0)/2 we estimate the fractal dimension of DLA to be D(0) = 1.69 +- 0.03.Comment: 5 pages, 3 figures, submitted to Phys. Rev. Let

Extended Self-Similarity in Turbulent Systems: an Analytically Soluble Example

In turbulent flows the $n$'th order structure functions $S_n(R)$ scale like $R^{\zeta_n}$ when $R$ is in the "inertial range". Extended Self-Similarity refers to the substantial increase in the range of power law behaviour of $S_n(R)$ when they are plotted as a function of $S_2(R)$ or $S_3(R)$. In this Letter we demonstrate this phenomenon analytically in the context of the ``multiscaling" turbulent advection of a passive scalar. This model gives rise to a series of differential equations for the structure functions $S_n(R)$ which can be solved and shown to exhibit extended self similarity. The phenomenon is understood by comparing the equations for $S_n(R)$ to those for $S_n(S_2)$.Comment: Phys. Rev. Lett., submitted, RevTeX, 4 pages, two figures by request from Daniel Segel: [email protected]

Elasticity with Arbitrarily Shaped Inhomogeneity

A classical problem in elasticity theory involves an inhomogeneity embedded in a material of given stress and shear moduli. The inhomogeneity is a region of arbitrary shape whose stress and shear moduli differ from those of the surrounding medium. In this paper we present a new, semi-analytic method for finding the stress tensor for an infinite plate with such an inhomogeneity. The solution involves two conformal maps, one from the inside and the second from the outside of the unit circle to the inside, and respectively outside, of the inhomogeneity. The method provides a solution by matching the conformal maps on the boundary between the inhomogeneity and the surrounding material. This matching converges well only for relatively mild distortions of the unit circle due to reasons which will be discussed in the article. We provide a comparison of the present result to known previous results.Comment: (10 pages, 10 figures

Fokker-Planck equation with memory: the crossover from ballistic to diffusive processes in many-particle systems and incompressible media

The unified description of diffusion processes that cross over from a ballistic behavior at short times to normal or anomalous diffusion (sub- or superdiffusion) at longer times is constructed on the basis of a non-Markovian generalization of the Fokker-Planck equation. The necessary non-Markovian kinetic coefficients are determined by the observable quantities (mean- and mean square displacements). Solutions of the non-Markovian equation describing diffusive processes in the physical space are obtained. For long times, these solutions agree with the predictions of the continuous random walk theory; they are, however, much superior at shorter times when the effect of the ballistic behavior is crucial.Comment: 18 pages, 11 figure