Institute of Applied Mathematics and Mechanics

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Camassa–Holm equation for decaying initial data, completing previous results by Boutet de Monvel and Shepelsky.

The crucial role of mechanical stress in voltage hysteresis of lithium ion batteries in charge-discharge cycles is investigated theoretically and experimentally. A modified Butler-Volmer equation of electrochemical kinetics is proposed to account for the influence of mechanical stresses on electrochemical reactions in lithium ion battery electrodes. It is found that the compressive stress in the surface layer of active materials impedes lithium intercalation, and therefore, an extra electrical overpotential is needed to overcome the reaction barrier induced by the stress. The theoretical formulation has produced a linear dependence of the height of voltage hysteresis on the hydrostatic stress difference between lithiation and delithiation, under both open-circuit conditions and galvanostatic operation. Predictions of the electrical overpotential from theoretical equations agree well with the experimental data for thin film silicon electrodes.

As a new two-dimensional (2D) material, phosphorene has drawn growing attention owing to its novel electronic properties, such as layer-dependent direct bandgaps and high carrier mobility. Herein we investigate the in-plane and cross-plane thermal conductivities of single- and multi-layer phosphorene, focusing on geometrical (sample size, orientation and layer number) and strain (compression and tension) effects. A strong anisotropy is found in the in-plane thermal conductivity with its value along the zigzag direction being much higher than that along the armchair direction. Interestingly, the in-plane thermal conductivity of multi-layer phosphorene is insensitive to the layer number, which is in strong contrast to that of graphene where the interlayer interactions strongly influence the thermal transport. Surprisingly, tensile strain leads to an anomalous increase in the in-plane thermal conductivity of phosphorene, in particular in the armchair direction. Both the in-plane and cross-plane thermal conductivities can be modulated by external strain; however, the strain modulation along the cross-plane direction is more effective and thus more tunable than that along the in-plane direction. Our findings here are of great importance for the thermal management in phosphorene-based nanoelectronic devices and for thermoelectric applications of phosphorene.

Flow field designs for the bipolar plates of the proton exchange membrane fuel cell are reviewed; including the serpentine, parallel, interdigitated, mesh type or their mixtures, furthermore 2D circular and 3D tubular geometries, porous, fractal and biomimetic flow fields. The advantages / disadvantages and tendencies from field optimizations are discussed. The performance of each flow field design is compared to the conventional serpentine flow field. Good flow field plates give uniform gas distributions, low pressure drop for transport and sufficient rib area to provide high electronic conductivity. A good field should also prevent water condensation, remove water efficiently, and provide sufficiently high moisture content in the membrane. The demands on design are sometimes contradictory. Future work should aim for a flow field geometry and topology that produces uniform gas delivery at a low pressure-drop, and at the same time has an optimal channel shape for better water removal. It is concluded that for an area-filling gas distributor, the developments should aim to find a flow field in accordance with minimum entropy production, making an emphasis on multi-criteria optimization methods.

The objective of this paper is twofold: first to illustrate that nonlinear modal interactions, namely, a two-to-one internal resonance energy pump, can be exploited to improve the steady-state bandwidth of vibratory energy harvesters; and, second, to investigate the influence of key system’s parameters on the steady-state bandwidth in the presence of the internal resonance. To achieve this objective, an L-shaped piezoelectric cantilevered harvester augmented with frequency tuning magnets is considered. The distance between the magnets is adjusted such that the second modal frequency of the structure is nearly twice its first modal frequency. This facilitates a nonlinear energy exchange between these two commensurate modes resulting in large-amplitude responses over a wider range of frequencies. The harvester is then subjected to a harmonic excitation with a frequency close to the first modal frequency, and the voltage–frequency response curves are generated. Results clearly illustrate an improved bandwidth and output voltage over a case which does not involve an internal resonance. A nonlinear model of the harvester is developed and validated against experimental findings. An approximate analytical solution of the model is obtained using perturbation methods and utilized to draw several conclusions regarding the influence of key design parameters on the harvester’s bandwidth.

Negative Poisson's ratio (NPR) materials have drawn significant interest because the enhanced toughness, shear resistance, and vibration absorption that typically are seen in auxetic materials may enable a range of novel applications. In this work, we report that single-layer graphene exhibits an intrinsic NPR, which is robust and independent of its size and temperature. The NPR arises due to the interplay between two intrinsic deformation pathways (one with positive Poisson's ratio, the other with NPR), which correspond to the bond stretching and angle bending interactions in graphene. We propose an energy-based deformation pathway criteria, which predicts that the pathway with NPR has lower energy and thus becomes the dominant deformation mode when graphene is stretched by a strain above 6%, resulting in the NPR phenomenon.

A nonlinear energy sink (NES) approach is proposed for whole-spacecraft vibration reduction. Frequency sweeping tests are conducted on a scaled whole-spacecraft structure without or with a NES attached. The experimental transmissibility results demonstrate the significant reduction of the whole-spacecraft structure vibration over a broad spectrum of excitation frequency. The NES attachment hardly changes the natural frequencies of the structure. A finite element model is developed, and the model is verified by the experimental results. A two degrees-of-freedom (DOF) equivalent model of the scaled whole-spacecraft is proposed with the two same natural frequencies as those obtained via the finite element model. The experiment, the finite element model, and the equivalent model predict the same trends that the NES vibration reduction performance becomes better for the increasing NES mass, the increasing NES viscous damping, and the decreasing nonlinear stiffness. The energy absorption measure and the energy transition measure calculated based on the equivalent model reveals that an appropriately designed NES can efficiently absorb and dissipate broadband-frequency energy via nonlinear beats, irreversible targeted energy transfer (TET), or both for different parameters.

In this study, we considered the unsteady peristaltic motion of a non-Newtonian nanofluid under the influence of a magnetic field and Hall currents. The simultaneous effects of ion slip and chemical reaction were also taken into consideration. The flow problem was suggested on the basis of the continuity, thermal energy, linear momentum, and nanoparticle concentration, which were further reduced with the help of Ohm's law. Mathematical modelling was executed using the lubrication approach. The resulting highly nonlinear partial differential equations were solved semi-analytically using the homotopy perturbation technique. The impacts of all the pertinent parameters were investigated mathematically and graphically. Numerical calculations have been used to calculate the expressions for the pressure increase and friction forces along the whole length of the channel. The results depict that for a relatively large value of the Brownian parameter, the chemical reaction has a dual behaviour on the concentration profile. Moreover, there is a critical point of the magnetic parameter at which the behaviours of the pressure increase and friction forces are reversed for progressive values of the power law index. The present investigation provides a theoretical model that estimates the impact of a wide range of parameters on the characteristics of blood-like fluid flows.

<italic>In situ</italic> observed electrodeposition and dissolution of lithium dendrite of commercial graphite electrode.

The Kreĭn-Naĭmark formula provides a parametrization of all selfadjoint exit space extensions of a (not necessarily densely defined) symmetric operator in terms of maximal dissipative (in ℂ+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct generalized resolvents from given parameter families. A general version of the coupling method is introduced and the role of the boundary relations and their Weyl families in the Kreĭn-Naĭmark formula is investigated and explained. These notions lead to several new results and new types of solutions to problems involving generalized resolvents and their applications, e.g., in boundary-value problems for (ordinary and partial) differential operators. For instance, an old problem going back to M. A. Naĭmark and concerning the analytic characterization of the so-called Naĭmark extensions is solved.

We study the conformality problems associated with quasiregular mappings in space. Our approach is based on the concept of the infinitesimal space and some new Grötzsch‐Teichmüller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients.

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Measurements of mechanical properties of copper current collector foils were performed using three techniques: a standard microtensile testing machine equipped with a laser sensor, dynamic mechanical analysis (DMA), and nanoindentation. For electrolytic copper (E-Cu) foils, we find elastic moduli of approximately 70 GPa, and for rolled copper (R-Cu) foils, we find elastic moduli of approximately 50 GPa.

We prove a priori supremum bounds for solutions to \begin{equation*} u_{t} - {\text{\rm div}} \big(u^{m-1} | {Du}| ^{\lambda -1} Du \big) = f(x) u^{p}\,, \end{equation*} as $t$ approaches the time when $u$ becomes unbounded. Such bounds are universal in the sense that they do not depend on $u$. Here $f$ may become unbounded, or vanish, as $x\to 0$. When $f\equiv1$, we also prove a bound below, as well as uniform localization of the support, for subsolutions to the corresponding Cauchy problem.

Diverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ0⩽|α|,|β|⩽m (−1) α D α a α,β (x)D β , a α,β (·) ∈ C ∞( $$ \bar \Omega $$ ) on smooth (bounded or unbounded) domains Ω in ℝ n with compact boundary ∂Ω are considered. Trace-ideal properties of powers of resolvent differences for these closed realizations of A are proved by using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, and estimates for negative eigenvalues of certain selfadjoint extensions of the nonnegative minimal operator are derived. Our results extend classical theorems due to Vishik, Povzner, Birman, and Grubb.

Abstract In this article, we study a free boundary problem for a system of two partial differential equations, one parabolic and other elliptic. The system models the growth of a tumor with arbitrary initial shape. We establish the existence and uniqueness of a solution for some time interval. In the special case where we only have the elliptic equation, the problem coincides with the Hele–Shaw problem.

It is shown that, under a Caldern type condition on the function , the continuous open mappings that belong to the Orlicz-Sobolev classes W 1, loc have total differential almost everywhere; this generalizes the well-known theorems of Gehring-Lehto-Menchoff in the case of R 2 and of Visl in R n , n 3. Appropriate examples show that the Caldern type condition is not only sufficient but also necessary. Moreover, under the same condition on , it is also proved that the continuous mappings of class W 1, loc and, in particular, of class W 1,p loc for p > n-1 have Lusin's (N )-property on a.e. hyperplane. On that basis, it is shown that, under the same condition on , the homeomorphisms f with finite distortion of class W 1, loc and, in particular, those belonging to W 1,p loc for p > n-1, are what is called lower Q-homeomorphisms, where Q is equal to their outer dilatation K f ; also, they are so-called ring Q * -homeomorphisms with Q * = K n-1 f . The latter fact makes it possible to fully apply the theory of the boundary and local behavior of the ring and lower Q-homeomorphisms, as developed earlier by the authors, to the study of mappings in the Orlicz-Sobolev classes.

We study the Hamiltonians HX,α,q with δ-type point interactions at the centers xk on the positive half line in terms of energy forms. We establish analogs of some classical results on operators Hq=−d2/dx2+q with locally integrable potentials q∊Lloc1(R+). In particular, we prove that the Hamiltonian HX,α,q is self-adjoint if it is lower semibounded. This result completes the previous results of Brasche [“Perturbation of Schrödinger Hamiltonians by measures—selfadjointness and semiboundedness,” J. Math. Phys. 26, 621 (1985)] on lower semiboundedness. Also we prove the analog of Molchanov’s discreteness criteria, Birman’s result on stability of a continuous spectrum, and investigate discreteness of a negative spectrum. In the recent paper [Kostenko, A. and Malamud, M., “1–D Schrödinger operators with local point interactions on a discrete set,” J. Differ. Equations 249, 253 (2010)], it was shown that the spectral properties of HX,α≔HX,α,0 correlate with the corresponding spectral properties of a certain class of Jacobi matrices. We apply the above mentioned results to the study of spectral properties of these Jacobi matrices.

[No abstract available]

A simple model of avascular solid tumor dynamics is studied in the paper. The model is derived on the basis of reaction-diffusion dynamics and mass conservation law. We introduce time delay in a cell proliferation process. In the case studied in this paper, the model reduces to one ordinary functional-differential equation of the form that depends on the existence of necrotic core. We focus on the process of this necrotic core formation and the possible influence of delay on it. Basic mathematical properties of the model are studied. The existence, uniqueness and stability of steady state are discussed. Results of numerical simulations are presented.