Abstract:
The paper is devoted to the homogenization of the special type of natural elastic curved shells called lipid membranes. The homogenization of the model is achieved via methods of statistical mechanics. The homogenized model is then solved for particular settings of the model constants and obtained solutians are demonstrated.

Abstract:
We investigate the thermal conductivity of four types of deformed carbon nanotubes by using the nonequilibrium molecular dynamics method. It is reported that various deformations have different influence on the thermal properties of carbon nanotubes. For the bending carbon nanotubes, the thermal conductivity is independent on the bending angle. However, the thermal conductivity increases lightly with XY-distortion and decreases rapidly with Z-distortion. The thermal conductivity does not change with the screw ratio before the breaking of carbon nanotubes but decreases sharply after the critical screw ratio.

Abstract:
In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by a epsilon-periodic lattice of highly conducting thin rods. The estimates are derived only at a distance epsilon^{1+tau} (for some tau>0) away from the fibres. This distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in L^p as soon as p>2. One key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a Fourier series expansion with respect to the fibres direction, and weighted by the high-contrast conductivity. The dependence on powers of epsilon of these two-dimensional estimates is shown to be sharp. The initial motivation for this work comes from imaging, and enhanced resolution phenomena observed experimentally in the presence of micro-structures. We use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium assuming that the heterogeneities stay at a distance epsilon^{1+tau} away from the fibres.

Abstract:
We have measured the thermal properties of suspended membranes from 10 K to 300 K for two amplitudes of internal stress (about 0.1 GPa and 1 GPa) and for two different thicknesses (50 nm and 100 nm). The use of the original 3 \omega -Volklein method has allowed the extraction of both the specific heat and the thermal conductivity of each SiN membrane over a wide temperature range. The mechanical properties of the same substrates have been measured at helium temperatures using nanomechanical techniques. Our measurements show that the thermal transport in freestanding SiN membranes is not affected by the presence of internal stress. Consistently, mechanical dissipation is also unaffected even though Qs increase with increasing tensile stress. We thus demonstrate that the theory developed by Wu and Yu [Phys. Rev. B 84, 174109 (2011)] does not apply to this amorphous material in this stress range. On the other hand, our results can be viewed as a natural consequence of the "dissipation dilution" argument [Y. L. Huang and P. R. Saulson, Rev. Sci. Instrum. 69, 544 (1998)] which has been introduced in the context of mechanical damping.

Abstract:
Graphene is convenient material for nanomechanichal applications since high-frequency oscillations are easily accessible. In this Article, we consider graphene on a rough substrate attached to imperfections at random locations. We explore the statistics of low-lying phonon modes, which exert most influence on the conductivity of graphene. We find that the nearest neighbor spacings of low lying eigenfrequencies have the Wigner-Dyson probability distribution after averaging over the random configurations of disorder. Due to interaction of electrons with the oscillations of the membrane, an electron can be transfered to higher or lower energies, which is a manifestation of the phonon-assisted Tien-Gordon effect. The Tien-Gordon effect suppresses the conductivity of graphene. In the regime of low Fermi energies and small sizes of the sample an increase of conductivity is observed which we refer to Klein tunneling and electron-hole pair creation. Eventually, when the increase of the transmission becomes too prominent, the pair creation changes the ground state of the system, signalizing the limit of applicability of the single-particle Dirac equation used in this paper.

Abstract:
The computation of the optical conductivity of strained and deformed graphene is discussed within the framework of quantum field theory in curved spaces. The analytical solutions of the Dirac equation in an arbitrary static background geometry for one dimensional periodic deformations are computed, together with the corresponding Dirac propagator. Analytical expressions are given for the optical conductivity of strained and deformed graphene associated with both intra and interbrand transitions. The special case of small deformations is discussed and the result compared to the prediction of the tight-binding model.

Abstract:
The description of experimentally observed phenomenon of abnormally high electrical conductivity—\"superconductivity\" (SC) at the room and higher temperatures is represented. The effect was observed in metallic monospirals of small radius curvature with high density and regular distribution of dislocations. Transition into state of SC has been observed experimentally in the range from –50 up to 3000°C at the density of transmitting current up to 2·10^{9} A/cm^{2}. The experimental data confirming the watched phenomenon are represented. The explanations of this phenomenon are being proposed in the framework of the dislocation model.

Abstract:
We evaluate effective dielectric permittivity and electric conductivity for water-saturated rocks based on a realistic model of a representative cell of the pore space which has periodical structure. We have applied the method of two-scale homogenization of the Maxwell equations, which results in up-scaling coupled equations at the microscale to equations valid at the macroscale. We have analyzed the interfacial Maxwell-Wagner dispersion effect and the Archie law as well.

Abstract:
Previous experimental studies of the thermal conductivity of plastically deformed lead crystals in the superconducting state have shown strong anomalies in the thermal conductivity. Similar effects were also found for the thermal conductivity of bent ${}^4\text{He}$ samples. Until now, a theoretical explanation for these results was missing. In this paper we will introduce the process of phonon-kink scattering and show that it qualitatively explains the anomalies that experiments had found.

Abstract:
Knowledge of the mean free path distribution of heat-carrying phonons is key to understanding phonon-mediated thermal transport. We demonstrate that thermal conductivity measurements of thin membranes spanning a wide thickness range can be used to characterize how bulk thermal conductivity is distributed over phonon mean free paths. A non-contact transient thermal grating technique was used to measure the thermal conductivity of suspended Si membranes ranging from 15 to 1500 nm in thickness. A decrease in the thermal conductivity from 74% to 13% of the bulk value is observed over this thickness range, which is attributed to diffuse phonon boundary scattering. Due to the well-defined relation between the membrane thickness and phonon mean free path suppression, combined with the range and accuracy of the measurements, we can reconstruct the bulk thermal conductivity accumulation vs. phonon mean free path, and compare with theoretical models.