Thermal stability of self-supported nanolayered Cu/Nb films

We report the development of thermally stable nanoscale layered structures in sputter deposited Cu/Nb multilayered films with 75?nm individual layer thickness, vacuum annealed at temperatures of 800°C or lower. The continuity of the layered structure was maintained and layer thickness unchanged in the annealed films. The nanolayers were observed to be offset by shear at the triple-point junctions that had equilibrium groove angles and were aligned in a zigzag pattern. A mechanism is proposed for the evolution of this ‘anchored’ structure that may be resistant to further morphological instability.     

Hypoelliptic vector fields and almost periodic motions on the torus Tn

Abstract: In this paper we prove that a hypoelliptic vector fields on the torus Tⁿ may be transformed by a smooth diffeomorphism of torus Tⁿ into a vector field with constant coefficients, which moreover satisfy a Diophantine condition. We also discuss the relation between the hypoelliptic vecttor fields and the almost periodic motions on the torus Tⁿ      

Quantized fracture mechanics

A new energy-based theory, quantized fracture mechanics (QFM), is presented that modifies continuum-based fracture mechanics; stress- and strain-based QFM analogs are also proposed. The differentials in Griffith's criterion are substituted with finite differences; the implications are remarkable. Fracture of tiny systems with a given geometry and type of loading occurs at ‘quantized’ stresses that are well predicted by QFM: strengths predicted by QFM are compared with experimental results on carbon nanotubes, β-SiC nanorods, α-Si₃N₄ whiskers, and polysilicon thin films; and also with molecular mechanics/dynamics simulation of fracture of carbon nanotubes and graphene with cracks and holes, and statistical mechanics-based simulations on fracture of two-dimensional spring networks. QFM is self-consistent, agreeing to first-order with linear elastic fracture mechanics (LEFM), and to second-order with non-linear fracture mechanics (NLFM). For vanishing crack length QFM predicts a finite ideal strength in agreement with Orowan's prediction. In contrast to LEFM, QFM has no restrictions on treating defect size and shape. The different fracture Modes (opening I, sliding II and tearing III), and the stability of the fracture propagations, are treated in a simple way.     

Perturbation Theory for Discrete Volterra Equations

Fixed point theory is used to investigate nonlinear discrete Volterra equations that are perturbed versions of linear equations. Sufficient conditions are established (i) to ensure that stability (in a sense that is defined) of the solutions of the linear equation implies a corresponding stability of the zero solution of the nonlinear equation and (ii) to ensure the existence of asymptotically periodic solutions.     

Experimental investigations of the normal loading of elastic spherical and conical indenters on to elastic flats

Results obtained from a series of experimental investigations are described in which an elastic polyisoprene hemisphere and elastic rubber cones of included angles 60°, 90°, 120° and 150° were loaded normally on to smooth blocks of soda–lime glass and polydimethylsiloxane (PDMS) containing 10% and 20% by volume of the curing agent. The load versus displacement data were continuously recorded with an instrumented indentation machine. It is shown that, whereas the loading behaviour of the hemisphere on to the blocks of the soda–lime glass and PDMS closely follows the theory of Hertz (see equation (1)), the load versus displacement behaviour of the rubber cones of included angles 60°, 90° and 120° could not be fitted by the Sneddon equation (see equation (5)) for rigid conical indenters loading on to an elastic half-space or by the modified Sneddon equation (see equation (6)) employing the combined moduli of the indenter and the half-space. The discrepancy between the predictions of the modified Sneddon equation and the experimental measurements is very significant, thus confirming our recent concern about the validity of using the modified Sneddon equation for analysing the experimental data obtained from nanoindentation experiments. Estimates of the errors caused by the use of the modified Sneddon equation have been made to further illustrate our contention. On the other hand, the behaviour of the 150° included angle cone loading on to the blocks of rubber, PDMS (1?:?10) and PDMS (1?:?20), has been shown to be particularly striking, as the rubber cone behaved as if it were rigid; moreover, the experimental data are well fitted by the Sneddon equation corresponding to a 150° rigid cone loading on to an elastic half-space. Finally, it has been proposed that, in order to determine the elastic modulus of a very stiff solid (i.e. Young's modulus close to that of the indenter) correctly using the technique of instrumented indentation, including nanoindentation, the included angle of the indenter, made of diamond, should be 150° and the measured load versus displacement data should be analysed using the Sneddon equation corresponding to a rigid cone of an included angle of 150°.     

Coupled Intervals in the Discrete Optimal Control

In this paper, we investigate the nonnegativity and positivity of a quadratic functional ? with variable (i.e. separable and jointly varying) endpoints in the discrete optimal control setting. We introduce a coupled interval notion, which generalizes (i) the conjugate interval notion known for the fixed right endpoint case and (ii) the coupled interval notion known in the discrete calculus of variations. We prove necessary and sufficient conditions for the nonnegativity and positivity of ? in terms of the nonexistence of such coupled intervals. Furthermore, we characterize the nonnegativity of ? in terms of the (previously known notions of) conjugate intervals, a conjoined basis of the associated linear Hamiltonian system, and the solvability of an implicit Riccati equation. This completes the results for the nonnegativity that are parallel to the known ones on the positivity of ?. Finally, we define partial quadratic functionals associated with ? and a (strong) regularity of ?, which we relate to the positivity and nonnegativity of ?.     

Optimal control for a parabolic problem in a domain with highly oscillating boundary

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.     

Vibrational entropy of dislocations in Al

The region nearest to a lattice defect must be described by an atomistic model, while a continuum model suffices further away from the defect. We study such a separation into two regions for an edge dislocation. In particular we focus on the excess defect energy and vibrational entropy, when the dislocation core is described by a cluster of about 500–100?atoms, embedded in a large discrete and relaxed, but static, lattice. The interaction between the atoms is given by a potential of the embedded-atom model type referring to Al. The dynamic matrix of the vibrations in the cluster is fully diagonalized. The excess entropy ΔS near the core has positive and negative contributions, depending on the sign of the local strain. Typically, ΔS/k _B ≈ 2 per atomic repeat length along the dislocation core in fcc Al. In the elastic continuum region far from the dislocation core the excess entropy shows the same logarithmic divergence as the elastic energy. Although the work refers to a specific material and defect type, the results are of a generic nature.     

Phase diagrams of decomposing nanoalloys

The thermodynamics of nucleation and decomposition in small isolated particles are considered. There exist three possibilities: phase separation, prohibition of decomposition and a metastable state. We investigate the peculiarities of phase diagrams related to depletion of the nanosize parent phase even at the nucleation stage. For small particles the equilibrium diagram becomes split (and shifted and size dependent). Concentration, size and temperature hystereses take place. Size-dependent ‘critical supersaturation’, increasing with decreasing size, has been analysed.