This article focuses on the time-domain propagation of elastic waves through a 1D periodic medium that contains non-linear imperfect interfaces, i.e. interfaces exhibiting a discontinuity in displacement and stress governed by a non-linear constitutive relation. The array considered is generated by a, possibly heterogeneous, cell repeated periodically and bonded by interfaces that are associated with transmission conditions of non-linear “spring–mass” type. More precisely, the imperfect interfaces are characterized by a linear dynamics but a non-linear elasticity law. The latter is not specified at first and only key theoretical assumptions are required. In this context, we investigate transient waves with both low-amplitude and long-wavelength, and aim at deriving homogenized models that describe their effective motion. To do so, the two-scale asymptotic homogenization method is deployed, up to the first-order. To begin, an effective model is obtained for the leading zeroth-order contribution to the microstructured wavefield. It amounts to a wave equation with a non-linear constitutive stress–strain relation that is inherited from the behavior of the imperfect interfaces at the microscale. The next first-order corrector term is then shown to be expressed in terms of a cell function and the solution of a linear elastic wave equation. Without further hypothesis, the constitutive relation and the source term of the latter depend non-linearly on the zeroth-order field, as does the cell function. Combining these zeroth- and first-order models leads to an approximation of both the macroscopic behavior of the microstructured wavefield and its small-scale fluctuations within the periodic array. Finally, particularizing for a prototypical non-linear interface law and in the cases of a homogeneous periodic cell and a bilaminated one, the behavior of the obtained models are then illustrated on a set of numerical examples and compared with full-field simulations. Both the influence of the dominant wavelength and of the wavefield amplitude are investigated numerically, as well as the characteristic features related to non-linear phenomena.

本文重点介绍弹性波在包含非线性不完美界面的一维周期介质中的时域传播，即，界面表现出位移和应力的不连续性（受非线性本构关系控制）。所考虑的阵列是由可能周期性地重复并由与非线性“弹簧-质量”类型的传输条件相关联的接口结合的可能异类的单元生成的。更精确地，不完美的界面的特征在于线性动力学，但非线性弹性定律。最初没有指定后者，仅需要关键的理论假设。在这种情况下，我们研究具有低振幅和长波长的瞬态波，并旨在推导描述其有效运动的均质模型。为此，采用了二阶渐近均匀化方法，直到一阶。首先，获得对微结构波场的领先零阶贡献的有效模型。它相当于一个具有非线性本构应力-应变关系的波动方程，该波动方程是从微观尺度上不完善界面的行为继承而来的。然后，显示下一个一阶校正项以单元函数和线性弹性波方程的解表示。在没有进一步假设的情况下，本构关系及其源项与单元函数一样，非线性地依赖于零阶场。将这些零阶和一阶模型结合起来，可以得到微结构化波场的宏观行为及其在周期阵列内的小规模波动的近似值。最后，专门针对典型的非线性界面定律，在均质周期单元和双层单元的情况下，然后在一组数值示例中说明获得的模型的行为，并与全场模拟进行比较。数值研究了主波长和波场振幅的影响，以及与非线性现象有关的特征。然后，在一组数值示例上说明获得的模型的行为，并与全场模拟进行比较。数值研究了主波长和波场振幅的影响，以及与非线性现象有关的特征。然后，在一组数值示例上说明获得的模型的行为，并与全场模拟进行比较。数值研究了主波长和波场振幅的影响，以及与非线性现象有关的特征。