Nonperiodic arrangements of inclusions with incremental linear negative stiffness embedded within a host material offer the ability to achieve unique and useful material properties on the macroscale. In an effort to study such types of inclusions, the present paper develops a time-domain model to capture the nonlinear dynamic response of a heterogeneous medium containing a dilute concentration of subwavelength nonlinear inclusions embedded in a lossy, nearly incompressible medium. Each length scale is modeled via a modified Rayleigh-Plesset equation, which differs from the standard form used in bubble dynamics by accounting for inertial and viscoelastic effects of the oscillating spherical element and includes constitutive equations formulated with incremental deformations. The two length scales are coupled through the constitutive relations and viscoelastic loss for the effective medium, both dependent on the inclusion and matrix properties. The model is then applied to an example nonlinear inclusion with incremental negative linear stiffness stemming from microscale elastic instabilities embedded in a lossy, nearly incompressible host medium. The macroscopic damping performance is shown to be tunable via an externally applied hydrostatic pressure with the example system displaying over two orders of magnitude change in energy dissipation due to changes in prestrain. The numerical results for radial oscillations versus time, frequency spectra, and energy dissipation obtained from the coupled dynamic model captures the expected response for quasistatic and dynamic regimes for an example buckling inclusion for both constrained and unconstrained negative stiffness inclusions.

中文翻译：

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包含亚波长设计的微结构不稳定性的非线性耦合多尺度模型的声响应

在主体材料中嵌入具有增量线性负刚度的夹杂物的非周期性排列提供了在宏观上实现独特且有用的材料特性的能力。为了研究此类夹杂物，本论文开发了一种时域模型，以捕获包含稀疏浓度的亚波长非线性夹杂物的非均质介质的非线性动态响应，该亚波长非线性夹杂物嵌入有损，几乎不可压缩的介质中。每个长度尺度都是通过修改后的Rayleigh-Plesset方程建模的，该方程与气泡动力学中使用的标准形式有所不同，因为它考虑了振荡球面元件的惯性和粘弹性效应，并且包括具有增量变形的本构方程。对于有效介质，这两个长度尺度通过本构关系和粘弹性损失耦合，这两者都取决于内含物和基质性质。然后将该模型应用于示例非线性包含物，该非线性包含物由于嵌入有损，几乎不可压缩的宿主介质中的微尺度弹性不稳定性而具有增量负线性刚度。宏观阻尼性能显示为可通过外部施加的静水压力来调节，示例系统显示由于预应变的变化，能量耗散的变化超过两个数量级。径向振动与时间，频谱，