We study the interaction of in-plane elastic waves with imperfect interfaces composed of a periodic array of voids or cracks. An effective model is derived from high-order asymptotic analysis based on two-scale homogenization and matched asymptotic technique. In two-dimensional elasticity, we obtain jump conditions set on the in-plane displacements and normal stresses; the jumps involve in addition effective parameters provided by static, elementary problems being the equivalents of the cell problems in classical two-scale homogenization. The derivation of the model is conducted in the transient regime and its stability is guarantied by the positiveness of the effective interfacial energy. Spring models are envisioned as particular cases. It is shown that massless-spring models are recovered in the limit of small void thicknesses and collinear cracks. By contrast, the use of mass-spring model is justified at normal incidence, otherwise unjustified. We provide quantitative validations of our model and comparison with spring models by means of comparison with direct numerical calculations in the harmonic regime.

中文翻译：

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重新审视二维弹性动力学的不完美界面定律


我们研究面内弹性波与由周期性空隙或裂纹阵列组成的不完美界面的相互作用。基于二尺度均质化和匹配渐近技术的高阶渐近分析推导出有效模型。在二维弹性中，我们获得了在面内位移和法向应力上设置的跳跃条件；跳跃还涉及由静态基本问题提供的额外有效参数，这些问题相当于经典两尺度均质化中的细胞问题。模型的推导是在瞬态状态下进行的，有效界面能的正值保证了模型的稳定性。 Spring 模型被设想为特殊情况。结果表明，无质量弹簧模型在小空隙厚度和共线裂纹的限制下得到了恢复。相比之下，质量-弹簧模型的使用在法向入射时是合理的，否则是不合理的。我们通过与谐波状态下的直接数值计算进行比较，对我们的模型进行了定量验证，并与弹簧模型进行了比较。