Nonlinear effects can enrich the propagation of elastic waves in mechanical metamaterials, which makes it possible to extend classical phenomena and functions in linear systems to nonlinear ones. In this work, rather than monochromatic waves in similar linear structures, the negative refraction is realized by mixing waves which are generated in nonlinear elastic wave metamaterials. Based on the stiffness matrix and plane wave expansion methods, dispersion curves of in–plane modes resulting from the collinear and non–linear mixings of two longitudinal waves are calculated. In the frequency spectrum, two propagating modes coalesce at exceptional points due to the coupling of in–plane modes, and those points at which the refraction type changes are also exceptional ones. Two kinds of negative refraction can be found in the mixing modes near exceptional points, but each of them needs to be induced in a specific configuration. Moreover, experiments are performed to support the pure negative refraction and beam splitting of the nonlinear elastic waves. Particularly, the parallel configuration is able to separate and extract the nonlinear mode when the single–mode negative refraction occurs, which shows the possibility to design elastic wave device by the negative refraction of nonlinear mixing waves.

非线性效应可以丰富弹性波在机械超材料中的传播，这使得将线性系统中的经典现象和函数扩展到非线性系统成为可能。在这项工作中，负折射不是通过类似线性结构中的单色波，而是通过非线性弹性波超材料中产生的混合波来实现。基于刚度矩阵和平面波展开方法，计算了两个纵波共线和非线性混合产生的面内模式的色散曲线。在频谱中，由于面内模式的耦合，两种传播模式在异常点处合并，并且折射类型发生变化的点也是异常点。在异常点附近的混合模式中可以发现两种负折射，但每种负折射都需要在特定的配置中引起。此外，还进行了实验来支持非线性弹性波的纯负折射和分束。特别是，当单模负折射发生时，并联结构能够分离和提取非线性模式，这表明利用非线性混合波的负折射设计弹性波器件的可能性。