Fracture scaling relations for scratch tests of axisymmetric shape

Scratch testing and scratch test analysis continues to gain momentum in Applied Mechanics, due to the possibility offered by this method to assess fracture properties at very fine scales. In this paper, we derive general scratch force scaling relations for axisymmetric scratch probes defined by single variable monomial functions. These relations are used to define fracture criteria with and without consideration of the development of shear stresses at the probe–material interface. The approach is illustrated for common scratch probe geometries: conical probe, flat punch, and hemi-spherical probe. Application of the proposed method to micro-scratch tests on two materials (an aluminum alloy and a thermoplastic polymer) using a Rockwell probe (a conical probe ending in a hemi-spherical shape) illustrates the versatility of the approach: First, the scratch force-depth scaling relations provide a means to determine the degree of the homogeneous function characterizing the scratch probe. Second, the fracture criteria enable an experimental assessment of the fracture toughness. The good agreement between the fracture toughness determined by scratching and values reported in the open literature show the potential of the proposed method for determining fracture properties of materials at even smaller scales.     

Reactivity of 3-silyloxy-1,4-enynes: Gold(III)-catalyzed regioselective nucleophilic substitution

Gold-catalyzed reactions of 3-silyloxy-1,4-enynes with alcohols afford primary, secondary, and tertiary pent-2-en-4-ynyl ethers in moderate to excellent yields. The substitution proceeds with high regioselectivity. An initial cyclization providing five-membered carbocycles instead was not observed under the reaction conditions. Control experiments show that these reactions are also catalyzed by Brønsted- and Lewis-acids, although scope and yields are markedly reduced.     

Cross section of electron-nucleus bremsstrahlung near the short-wavelength limit

Using the partial-wave formalism the cross section of bremsstrahlung for a pure Coulomb field is calculated for various atomic numbers of the target nucleus and various incident electron energies. The results are compared to those of previous theories. For intermediate and large atomic numbers the latter cross sections undervalue significantly the present ones. Except for low atomic numbers the cross sections in Born approximation with the Coulomb correction by Elwert yield more accurate results than the cross sections obtained by using the approximate Sommerfeld-Maue wave function. The effect of screening is small for low atomic numbers. At nonrelativistic energies the number of required partial waves is rather large for target nuclei with high atomic numbers, contrary to the expectations. The pertinent results exceed the numerical values including screening by factors of up to five. They verify the assumption that the nonrelativistic cross section of Sommerfeld is fairly accurate even for high atomic numbers.     

An index 0 differential-algebraic equation formulation for multibody dynamics: Nonholonomic constraints

A method is presented for formulating and numerically integrating index 0 differential-algebraic equations of motion for multibody systems with holonomic and nonholonomic constraints. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation. Orthogonal dependent coordinates and velocities are used to enforce position, velocity, and acceleration constraints to within specified error tolerances. Explicit and implicit numerical integration algorithms are presented and used in solution of three examples: one planar and two spatial. Numerical results verify that accurate results are obtained, satisfying all three forms of kinematic constraint to within error tolerances embedded in the formulation.     

Simulation of friction and stiction in multibody dynamics model problems

A continuous model of Coulomb friction is used with a tangent space formulation of differential algebraic equations of motion for simulation of multibody dynamic model problems. Characteristics of the model problems studied are similar to those encountered in broad classes of multibody systems, without the associated geometric and analytical complexities. An implicit trapezoidal numerical solution algorithm is used to simulate dynamic response that includes the onset of stiction, its progression, and its termination, avoiding stiff behavior that is reported in the literature when index 3 formulations are used. Analytical criteria for stiction are derived for a three mass Coulomb friction model problem that defines the onset of and departure from stiction events with redundant equations of constraint. The tangent space formulation with implicit trapezoidal integration is applied to this analytical model to compute dynamic response, determine ranges of constraint forces that may occur during periods of stiction, and demonstrate that dynamic response is a discontinuous function of model parameters when stiction occurs. Accuracy of the continuous model of Coulomb friction is established, through comparison of results with those of the analytical model. Cartesian coordinate models of higher dimension are presented for three and four mass model problems that encounter a higher degree of redundancy in constraints during periods of stiction. Simulation of the Cartesian coordinate models, which have characteristics similar to more general multibody systems, yields accurate solutions, without any indication of stiffness in the tangent space equations of motion. Methods successfully demonstrated in model problems provide a foundation for simulation of spatial multibody dynamic systems with friction.     

On stable torsion-free nilpotent groups

Summary We show that an infinite field is interpretable in a stable torsion-free nilpotent groupG of classk, k>1. Furthermore we prove thatG/Z _k-1 (G) must be divisible. By generalising methods of Belegradek we classify some stable torsion-free nilpotent groups modulo isomorphism and elementary equivalence.Supported by the Deutsche Forschungsgemeinschaft     

Numerical Methods for Constrained Equations of Motion in Mechanical System Dynamics

ABSTRACT

ABSTRACT A new class of numerical methods for solving equations of motion of constrained mechanical systems is presented, the framework of which is based on manifold theoretic methods. Rewriting the system of differential-algebraic equations (DAEs) that describe constrained motion is ordinary differentia] equations (ODEs) on a constraint manifold, the theoretical framework for solving equations of motion is constructed, using a local     

Probing the coherent transport through coupled quantum dots

We investigate the photoconductance through a double quantum dot or ‘artificial molecule’ induced by a broadband millimeter wave source. This source functions as a heterodyne interferometer, consisting of two nonlinear transmission lines generating harmonic output in the range of 2–400 GHz, and, being coherent, allows tracking of the induced current of the sample in both magnitude and phase. This enables us to monitor effects related to coherent electronic transport through these ‘artificial molecules’.     

Higher-Order Airy Scaling in Deformed Dyck Paths

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, ‘jumps’ orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half-length, area and number of jumps. This represents the first example of an exactly solvable two-dimensional lattice vesicle model showing a higher-order multicritical point. Applying the generalized method of steepest descents, we see that the associated two-variable scaling function is given by the logarithmic derivative of a generalized (higher-order) Airy integral.     

The kinetics of hole-burning in semiconductor lasers

It is shown that the nonequilibrium electron distributions of a lasing semiconductor can be calculated from a linearized Boltzmann equation for the intraband Coulombcollision rate. A successive expansion for small momentum transfer yields an integro-differential equation, whose differential part is the usual Fokker-Planck equation. The additional integral part assures the conservation of the total momentum and energy of the electron gas. A formal solution of the resulting kinetic equation is given from which one can calculate the hole burned into the electron distribution by the laser field.