Biomedical subjects

Joachim Mathiesen

Publications and source records attributed to Joachim Mathiesen.

10 recordsLinked to original sources

PubMed2006

Analytical approach to continuous and intermittent bottleneck flows.

We propose a many-particle-inspired theory for granular outflows from a hopper and for the escape dynamics through a bottleneck based on a continuity equation in polar coordinates. If the inflow is below the maximum outflow, we find an asymptotic stationary solution. If the inflow is above this value, we observe queue formation, which can be described by a shock wave equation. We also address the experimental observation of intermittent outflows, taking into account the lack of space in the merging zone by a minimum function and coordination problems by a stochastic variable. This results in avalanches of different sizes even if friction, force networks, inelastic collapse, or delay-induced stop-and-go waves are not assumed. Our intermittent flows result from a random alternation between particle propagation and gap propagation. Erratic flows in congested merging zones of vehicle traffic may be explained in a similar way.

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PubMed2005

Diffusion, fragmentation, and coagulation processes: analytical and numerical results.

We formulate dynamical rate equations for physical processes driven by a combination of diffusive growth, size fragmentation, and fragment coagulation. Initially, we consider processes where coagulation is absent. In this case we solve the rate equation exactly leading to size distributions of Bessel type which fall off as exp(-x(3/2)) for large x values. Moreover, we provide explicit formulas for the expansion coefficients in terms of Airy functions. Introducing the coagulation term, the full nonlinear model is mapped exactly onto a Riccati equation that enables us to derive various asymptotic solutions for the distribution function. In particular, we find a standard exponential decay exp(-x) for large x and observe a crossover from the Bessel function for intermediate values of x . These findings are checked by numerical simulations, and we find perfect agreement between the theoretical predictions and numerical results.

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PubMed2005

Void formation and roughening in slow fracture.

Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the microm-mm and the nm scales, respectively, reveals self-affine cracks with anomalous scaling exponent zeta approximately = 0.8 in 3 dimensions and zeta approximately = 0.65 in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.

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PubMed2005

Branching instabilities in rapid fracture: dynamics and geometry.

We propose a theoretical model for branching instabilities in 2-dimensional fracture, offering predictions for when crack branching occurs, how multiple cracks develop, and what is the geometry of multiple branches. The model is based on equations of motion for crack tips which depend only on the time dependent stress intensity factors. The latter are obtained by invoking an approximate relation between static and dynamic stress intensity factors, together with an essentially exact calculation of the static ones. The results of this model are in qualitative agreement with a number of experiments in the literature.

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PubMed2004

Roughening of fracture surfaces: the role of plastic deformation.

Post mortem analysis of fracture surfaces of ductile and brittle materials on the microm-mm and the nm scales, respectively, reveal self-affine cracks with anomalous scaling exponent zeta approximately 0.8 in three dimensions and zeta approximately 0.65 in two dimensions. Attempts to use elasticity theory to explain this result failed, yielding exponent zeta approximately 0.5 up to logarithms. We show that when the cracks propagate via plastic void formations in front of the tip, followed by void coalescence, the void positions are positively correlated to yield exponents higher than 0.5.

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PubMed2004

Stress field around arbitrarily shaped cracks in two-dimensional elastic materials.

The calculation of the stress field around an arbitrarily shaped crack in an infinite two-dimensional elastic medium is a mathematically daunting problem. With the exception of few exactly soluble crack shapes the available results are based on either perturbative approaches or on combinations of analytic and numerical techniques. We present here a general solution of this problem for any arbitrary crack. Along the way we develop a method to compute the conformal map from the exterior of a circle to the exterior of a line of arbitrary shape, offering it as a superior alternative to the classical Schwartz-Cristoffel transformation. Our calculation results in an accurate estimate of the full stress field and in particular of the stress intensity factors K(I) and K(II) and the T-stress which are essential in the theory of fracture.

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PubMed2003

Competition between diffusion and fragmentation: an important evolutionary process of nature.

We investigate systems of nature where the common physical processes diffusion and fragmentation compete. We derive a rate equation for the size distribution of fragments. The equation leads to a third order differential equation which we solve exactly in terms of Bessel functions. The stationary state is a universal Bessel distribution described by one parameter, which fits perfectly experimental data from two very different systems of nature, namely, the distribution of ice-crystal sizes from the Greenland ice sheet and the length distribution of alpha helices in proteins.

Crystallization↗

PubMed2003

Scaling exponent of the maximum growth probability in diffusion-limited aggregation.

An early (and influential) scaling relation in the multifractal theory of diffusion limited aggregation (DLA) is the Turkevich-Scher conjecture that relates the exponent alpha(min) that characterizes the "hottest" region of the harmonic measure and the fractal dimension D of the cluster, i.e., D=1+alpha(min). Due to lack of accurate direct measurements of both D and alpha(min), this conjecture could never be put to a serious test. Using the method of iterated conformal maps, D was recently determined as D=1.713+/-0.003. In this paper, we determine alpha(min) accurately with the result alpha(min)=0.665+/-0.004. We thus conclude that the Turkevich-Scher conjecture is incorrect for DLA.

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PubMed2002

Tip splittings and phase transitions in the dielectric breakdown model: mapping to the diffusion-limited aggregation model.

We show that the fractal growth described by the dielectric breakdown model exhibits a phase transition in the multifractal spectrum of the growth measure. The transition takes place because the tip splitting of branches forms a fixed angle. This angle is eta dependent but it can be rescaled onto an "effectively" universal angle of the diffusion-limited aggregation branching process. We derive an analytic rescaling relation which is in agreement with numerical simulations. The dimension of the clusters decreases linearly with the angle and the growth becomes non-ractal at an angle close to 74 degrees (which corresponds to eta = 4.0+/-0.3).

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PubMed2002

Multifractal structure of the harmonic measure of diffusion-limited aggregates.

The method of iterated conformal maps allows one to study the harmonic measure of diffusion-limited aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as 10(-35), and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions D(q) are infinite for q<q*, where q* is of the order of -0.2. In the language of f(alpha) this means that alpha(max) is finite. The f(alpha) curve loses analyticity (the phenomenon of "phase transition") at alpha(max) and a finite value of f(alpha(max)). We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of q* and f(alpha(max)). We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.

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