预应力弹性格构的增量本构张量和应变局部化：II部分--增量动力学
Incremental constitutive tensors and strain localization for prestressed elastic lattices: Part II -- incremental dynamics
摘要：
Floquet-Bloch波渐近性用于均匀化弹性Rayleigh杆的周期性格栅的平面内机械响应，具有分布的质量密度，连同转动惯量。网格受到增量时间谐波动态运动，叠加到任意大小的轴向载荷的给定状态。与准静态能量匹配（在本研究的第一部分中讨论）相反，晶格的振动性质直接由等效固体的声学张量表示（不通过本构张量）。的声学张量被证明是独立的杆的转动惯量，并允许分析强椭圆的等效连续，证明符合宏观分岔晶格。另一方面，微分叉对应于晶格的最低色散分支的消失频率的振动，发生在有限波长处。动态均匀化揭示了接近椭圆度损失的声学分支的结构，并且受迫振动的分析（在物理空间和傅立叶空间两者中）示出了低频波局部化。的比较之间发生的应变局部化附近的椭圆度损失和受迫振动的晶格，都对应于一个集中的脉动力的应用。比较表明，均匀化技术允许一个几乎完美的表示的晶格，与微分叉发生的情况下，这是不受影响的等效固体的例外。因此，所提出的结果铺平了道路的建筑蜂窝材料的设计中使用的极端变形的应用。
英文摘要：
Floquet-Bloch wave asymptotics is used to homogenize the in-plane mechanical response of a periodic grillage of elastic Rayleigh rods, possessing a distributed mass density, together with rotational inertia. The grid is subject to incremental time-harmonic dynamic motion, superimposed to a given state of axial loading of arbitrary magnitude. In contrast to the quasi-static energy match (addressed in Part I of the present study), the vibrational properties of the lattice are directly represented by the acoustic tensor of the equivalent solid (without passing through the constitutive tensor). The acoustic tensor is shown to be independent of the rods' rotational inertia and allows the analysis of strong ellipticity of the equivalent continuum, evidencing coincidence with macro-bifurcation in the lattice. On the other hand, micro-bifurcation corresponds to a vibration of vanishing frequency of the lowest dispersion branch of the lattice, occurring at finite wavelength. Dynamic homogenization reveals the structure of the acoustic branches close to ellipticity loss and the analysis of forced vibrations (both in physical space and Fourier space) shows low-frequency wave localizations. A comparison is presented between strain localization occurring near ellipticity loss and forced vibration of the lattice, both corresponding to the application of a concentrated pulsating force. The comparison shows that the homogenization technique allows an almost perfect representation of the lattice, with the exception of cases where micro-bifurcation occurs, which is shown to leave the equivalent solid unaffected. Therefore, the presented results pave the way for the design of architected cellular materials to be used in applications where extreme deformations are involved.
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