In order to examine the excitation of intrinsic localized modes (ILMs) in a real material, we analyze two-dimensional vibration of a graphene sheet using molecular-dynamics simulations based on the Brenner potential. Energy localization is observed in a few regions for 1.0 ps. Fairly constant vibration modes, in which the two neighboring atoms vibrate in the opposite direction, are generated locally. The frequency of the localized vibration exceeds the upper bound of the phonon band and the mode is similar to one of the eigenmodes. The localized vibration remains at the same location and maintains the mode for a fairly long time (26 cycles). Since all of these results correspond to the characteristic features of ILM, it is concluded that the stationary ILM is excited in the graphene sheet.
In order to examine the excitation of intrinsic localized modes ͑ILMs͒ in a three-dimensional material, we conducted molecular dynamics numerical simulations on ͑12,0͒, ͑10,0͒, and ͑8,0͒ zigzag carbon nanotubes ͑CNTs͒ and ͑7,7͒, ͑6,6͒, and ͑5,5͒ armchair CNTs based on the Brenner potential. While energy localization is observed in several regions in the zigzag CNTs, it is not seen in the armchair CNTs. In the former, fairly constant modes, where two neighboring atoms vibrate in the opposite direction along the axial direction, are found in the energy-localized region, and their frequencies exceed the upper bound of the phonon band. In the armchair CNTs, atomic vibrations in the circumferential direction within high-energy regions cannot last a long time. These results indicate that the ILM is excited in the three zigzag CNTs but not in the three armchair ones. This is because the bond along the tube-axial direction has stronger nonlinearity under vibrations than that in the circumferential one, and the bond direction depends on the structure of the CNT.
The Cu through-hole is a structure of electroplated Cu thin film, which penetrates the substrate. Because of the mismatch of the thermal expansion coefficient between the Cu thin film and the substrate along the thickness direction, thermal strain occurs repeatedly at the Cu through-hole part with the variation of temperature. As a result, the thermal fatigue failure of Cu through-hole part is one of the failure modes of the substrate. In this study, the effects of thermal cycle conditions on the thermal fatigue life of the substrate with Cu through-hole were investigated by thermal cycle tests and Finite Element Method (FEM)-based analyses. Thermal cycle tests of the substrate with Cu through-hole were conducted under different thermal conditions. The effects of dwell time, temperature range and maximum temperature were investigated. Among these factors, the maximum temperature shows the greatest influence on the thermal fatigue life of Cu through-hole part. FEM-based thermal cycle analyses were also carried out to understand the effects of thermal cycle conditions. The glass cloth structures of the substrate should be considered in the analyses, because their rigid properties probably affect the generation of the failure at the through-hole part. In this study, glass cloth structures were modeled by taking advantage of a homogenization method. On the other hand, the inelastic constitutive model of the electroplated Cu thin film was introduced in the analyses in order to describe the creep deformation during the dwell process of thermal cycles. The inelastic strain range of the Cu through-hole during thermal cycles was calculated from the analysis results and the effectiveness of the Coffin-Manson law was evaluated. The results showed that the fatigue life prediction using the Coffin-Manson model was effective in the range of the same substrate thickness and the same maximum temperature. Additionally the influences of material model and material constants of epoxy resin were investigated to expand the range of application of the fatigue life prediction.
In this study, the thermal fatigue life of substrate with Cu through-hole is evaluated by considering the mechanical properties of Cu thin film and glass fiber cloths structure. We first conducted tensile tests of Cu thin film and found that the rate-dependence of inelastic property varies abruptly with temperature. An inelastic constitutive model for Cu thin film is then proposed by combining both rate-independent and rate-dependent models. The proposed inelastic constitutive model is introduced in a Finite Element Method based analysis of glass epoxy substrate with Cu through-hole. Moreover, low cycle tests of Cu thin film are carried out by using repeated 4-point bending to evaluate its isothermal fatigue properties. Through our analysis we verified the capability of the proposed model to predict thermal fatigue life of Cu through-hole using the isothermal fatigue properties. The results show that the glass fiber cloths structure of the substrate needs to be considered in order to successfully predict the thermal fatigue life of the Cu through-hole.
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