In this work, the concept of high-frequency homogenization is extended to the case of one-dimensional periodic media with imperfect interfaces of the spring-mass type. In other words, when considering the propagation of elastic waves in such media, displacement and stress discontinuities are allowed across the borders of the periodic cell. As is customary in high-frequency homogenization, the homogenization is carried out about the periodic and antiperiodic solutions corresponding to the edges of the Brillouin zone. Asymptotic approximations are provided for both the higher branches of the dispersion diagram (second-order) and the resulting wave field (leading-order). The special case of two branches of the dispersion diagram intersecting with a non-zero slope at an edge of the Brillouin zone (occurrence of a so-called Dirac point) is also considered in detail, resulting in an approximation of the dispersion diagram (first-order) and the wave field (zeroth-order) near these points. Finally, a uniform approximation valid for both Dirac and non-Dirac points is provided. Numerical comparisons are made with the exact solutions obtained by the Bloch–Floquet approach for the particular examples of monolayered and bilayered materials. In these two cases, convergence measurements are carried out to validate the approach, and we show that the uniform approximation remains a very good approximation even far from the edges of the Brillouin zone.

中文翻译：

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具有不完美界面的周期性介质中的高频均匀化

在这项工作中，高频均匀化的概念被扩展到一维周期性介质的情况，该介质具有不完美的弹簧质量型界面。换句话说，当考虑弹性波在这种介质中的传播时，允许跨越周期单元边界的位移和应力不连续。按照高频均化的惯例，均化是针对对应于布里渊区边缘的周期解和反周期解进行的。为色散图的较高分支（二阶）和产生的波场（前导阶）提供了渐近近似。还详细考虑了色散图的两个分支与布里渊区边缘的非零斜率相交的特殊情况（出现所谓的狄拉克点），从而得到色散图的近似值（第一-阶）和这些点附近的波场（零阶）。最后，提供了对狄拉克点和非狄拉克点均有效的统一近似。对于单层和双层材料的特定示例，与通过 Bloch-Floquet 方法获得的精确解进行数值比较。在这两种情况下，进行收敛测量以验证该方法，并且我们表明即使远离布里渊区的边缘，均匀近似仍然是非常好的近似。导致这些点附近的色散图（一阶）和波场（零阶）的近似值。最后，提供了对狄拉克点和非狄拉克点均有效的统一近似。对于单层和双层材料的特定示例，与通过 Bloch-Floquet 方法获得的精确解进行数值比较。在这两种情况下，进行收敛测量以验证该方法，并且我们表明即使远离布里渊区的边缘，均匀近似仍然是非常好的近似。导致这些点附近的色散图（一阶）和波场（零阶）的近似值。最后，提供了对狄拉克点和非狄拉克点均有效的统一近似。对于单层和双层材料的特定示例，与通过 Bloch-Floquet 方法获得的精确解进行数值比较。在这两种情况下，进行收敛测量以验证该方法，并且我们表明即使远离布里渊区的边缘，均匀近似仍然是非常好的近似。对于单层和双层材料的特定示例，与通过 Bloch-Floquet 方法获得的精确解进行数值比较。在这两种情况下，进行收敛测量以验证该方法，并且我们表明即使远离布里渊区的边缘，均匀近似仍然是非常好的近似。对于单层和双层材料的特定示例，与通过 Bloch-Floquet 方法获得的精确解进行数值比较。在这两种情况下，进行收敛测量以验证该方法，我们表明即使远离布里渊区的边缘，均匀近似仍然是非常好的近似。